Vincent Cossart

Olivier Piltant

Shankar Abhyankar

Resolution of Singularities of Arithmetical Threefolds

Keywords: AMS Classification: 11G25, 11G35, 14B05, 14E15

We prove Grothendieck's conjecture on Resolution of Singularities for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = SpecA, A a reduced complete Noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three. Similarly, if F is a number field, a complete discretely valued field or more generally the quotient field of any excellent Dedekind domain O, any regular projective surface X/F has a proper and flat model X over O which is everywhere regular.

The Resolution of Singularities conjecture has been, and still is a long standing open problem since it was formulated by A. Grothendieck in the 1960's [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF](7.9.6). Grothendieck emphasized its importance for studying homological and homotopical properties of schemes. Even since H. Hironaka's celebrated Theorem [START_REF] Hironaka | Resolution of singularities of an algebraic variety over a field of characteristic zero[END_REF] proved fifty years ago, some new results have bettered our understanding of the problem in equal characteristic zero [START_REF] Bierstone | Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant[END_REF][81] [START_REF] Wlodarczyk | Simple Hironaka resolution in characteristic zero[END_REF]. These results focus on the constructivity and functoriality of their algorithms for Resolution in contrast with Hironaka's.

In arbitrary characteristic, a major advance towards Grothendieck's conjecture is due to A.J. de Jong [START_REF] De Jong | Smoothness, semistability and Alterations[END_REF] Theorem 4.1 and Theorem 6.5. He proved a weaker form of the above conjecture for varieties X over a field or a complete discrete valuation ring. A significant difference with Grothendieck's formulation is that de Jong's alterations allow a finite extension of the function field. Furthermore, de Jong's result does not in general provide a regular compactification X of some étale covering U of the regular locus RegX.

Resolution of Singularities in its full birational form was to this date restricted to surfaces [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF][5] [START_REF] Hironaka | Desingularization of excellent surfaces[END_REF][62] [START_REF] Cutkosky | Resolution of singularities[END_REF][36] [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF], only to mention some contributions. In dimension three, some partial results do exist for algebraic varieties over an algebraically closed field k of positive characteristic p ≥ 7 [START_REF] Abhyankar | Resolution of singularities of embedded algebraic surfaces[END_REF] [START_REF] Cutkosky | Resolution of singularities for 3-folds in positive characteristic[END_REF]. These results extend to all characteristics p > 0 and fields k with [k : k p ] < +∞ [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Theorem on p. 1839. For arithmetical schemes (unequal residue characteristic), birational Resolution of Singularities was so far restricted to surfaces. The first and main purpose of this article is to prove: Theorem 1.1. Let X be a reduced and separated Noetherian scheme which is quasi-excellent and of dimension at most three. There exists a proper birational morphism π : X ′ → X with the following properties: (i) X ′ is everywhere regular;

(ii) π induces an isomorphism π -1 (RegX ) ≃ RegX ;

(iii) π -1 (SingX ) is a strict normal crossings divisor on X ′ .

Since the class of quasi-excellent schemes is stable by morphisms of finite type, Theorem 1.1 applies in particular to algebraic varieties and to arithmetical varieties over excellent Dedekind rings. Another application of Theorem 1.1 concerns formal geometry. Indeed, Theorem 1.1 applies to reduced completions of affine Noetherian schemes along quasi-excellent subschemes (O. Gabber [START_REF] Illusie | Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasiexcellents[END_REF] p. 17, Theorem 9.2, see also C. Rotthaus [START_REF] Rotthaus | Komplettierung semilokaler quasiausgezeichneter Ringe[END_REF] Theorem 3 in the semilocal case).

Corollary 1.3. Let O be an excellent Dedekind domain with quotient field F and Σ/F be a regular projective surface. There exists a proper and flat O-scheme X with generic fiber X F = Σ which is everywhere regular.

Remark 1.4. The morphism π provided by Theorem 1.1 is not constructed as a composition of Hironaka-permissible blowing ups, i.e. with regular centers along which the successive strict transforms of X are normally flat ( called Hironaka Resolution for short). Taking global sections of an appropriate exceptional divisor with exceptional support, a Hironaka Resolution provides an ideal sheaf I ⊆ O X whose blowing up is regular, with zero locus V (I) = SingX . When X is affine, our theorem states that there exists an ideal sheaf I ′ ⊆ O X whose blowing up is regular. In contrast, I ′ O RegX is locally principal but not necessarily trivial.

On the other hand, a certain local version of Theorem 1.1 is proved using only local Hironaka-permissible blowing ups in Theorem 1.5 below. This fact appears to be a piece of evidence that Hironaka Resolution could be true for threefolds of nonzero residue characteristic, vid. also [START_REF] Cossart | Polyèdre caractéristique d'une singularité[END_REF] [START_REF] Moh | On a Newton polygon approach to the uniformization of singularities of characteristic p[END_REF] in positive characteristic. It is however restricted to certain hypersurface threefolds of multiplicity not bigger than the residue characteristic and the problem remains widely open even in dimension three.

In higher dimensions n ≥ 4, the Resolution of Singularities conjecture for algebraic varieties over a field is considered in several recent papers [START_REF] Villamayor | Monoidal transforms and invariants of singularities in positive characteristic[END_REF] [START_REF] Villamayor | On elimination of variables in the study of singularities in positive characteristic[END_REF] [START_REF] Bravo | Singularities in positive characteristic, stratification and simplification of the singular locus[END_REF][51] [START_REF] Hironaka | Three key theorems on infinitely near singularities, Singularités Franco-Japonaises Paris[END_REF] [START_REF] Kawanoue | Toward resolution of singularities over a field of positive characteristic I. Foundation; the language of the idealistic filtration[END_REF][57] [START_REF] Moh | On the bound of d 2[END_REF] but remains open to this date. Its local variant for valuations is also considered in [START_REF] Illusie | Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasiexcellents[END_REF] [START_REF] Knaf | Every place admits local uniformization in a finite extension of the function field[END_REF][69] [START_REF] Teissier | Valuations, Deformations and Toric Geometry[END_REF][79] [START_REF] Temkin | Inseparable local uniformization[END_REF] but remains equally unsolved. The case of arithmetical schemes has apparently attracted less attention.

The second purpose of this article is to explore the Resolution of Singularities Conjecture as formulated by A. Grothendieck [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] (7.9.6). The text includes numerous examples and prospective remarks aimed at preparing the ground for further research in higher dimension. For this purpose, we consider finite morphisms η : X → SpecS, where S is an arbitrary excellent regular local ring. A test case for Resolution if S has positive characteristic p > 0 is when η is purely inseparable; this was already recognized by O.Zariski [START_REF] Zariski | The fundamental ideas of abstract algebraic geometry[END_REF] p.88 and S. Abhyankar [START_REF] Abhyankar | Resolution of singularities of embedded algebraic surfaces[END_REF] and recently confirmed by M. Temkin's purely inseparable Local Uniformization Theorem [START_REF] Temkin | Inseparable local uniformization[END_REF] Theorem 1.3.2, vid. Remark 1.3.5 (iii). In residue characteristic p > 0, we also include Galois coverings of degree p to this test case. The main step in proving Theorem 1.1 consists in proving the following result. Assumption (i) below is the purely inseparable case for charS = p. Assumption (ii) below is the cyclic Galois case. For charS = p, Artin-Schreier polynomials h = X pg p-1 X + f , f, g ∈ S, g = 0 satisfy assumption (ii). For charS = 0, S containing the group µ p of p th -roots of unity, cyclic polynomials h = X pf , f ∈ S, f = 0 satisfy assumption (ii). The total quotient ring L = Tot(S[X]/(h)) is a direct product of fields. By a valuation of L, we mean a valuation of one of these fields.

Theorem 1.5. Let (S, m S , k) be an excellent regular local ring of dimension n = 3, quotient field K := QF (S) and residue characteristic chark = p > 0.

Let h := X p + f 1 X p-1 + • • • + f p ∈ S[X], f 1 , . . . , f p ∈ S (1.1)
be a reduced polynomial, X := Spec(S[X]/(h)) and L := Tot(S[X]/(h)) be its total quotient ring. Assume that h satisfies one of the following assumptions:

(i) charK = p and f 1 = • • • = f p-1 = 0, or (ii) X is G-invariant, where G := Aut K (L) = Z/p.

Let µ be a valuation of L which is centered in m S . There exists a composition of local Hironaka-permissible blowing ups:

(X =: X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ), (1.2)
where x i ∈ X i is the center of µ, such that (X r , x r ) is regular.

We develop an approach to the Resolution of Singularities Conjecture for hypersurface singularities defined by an equation (1.1) such that (i) or (ii) holds (condition (G) in the text) in any dimension n := dimS ≥ 1. No other assumption on S is required here than excellence of S; we do not even assume that [k : k p ] < +∞ as suggested by A. Grothendieck loc.cit. An extra condition (E) on η (Definition 2.32) is also assumed: (i) purely inseparable case: the image in SpecS of the locus Sing p X of multiplicity p, is contained in a normal crossings divisor E; (ii) cyclic Galois case: the discriminant locus of X → SpecS is contained in a normal crossings divisor E. If charS = 0 ( so 1.5(ii) holds), E has characteristic p.

This extra condition (E) can be achieved by preparatory blowing ups in dimension three (Corollary 4. [START_REF] Cossart | Desingularization of embedded excellent surfaces[END_REF]), applying known Resolution theorems for two-dimensional schemes.

The basic structure we work with is the triple (S, h, E) thus defined. The main combinatorial data attached with the singularity X is a characteristic polyhedron [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] [START_REF] Cossart | Characteristic polyhedra of singularities without completion[END_REF]: ∆ S (h; u 1 , . . . , u n ; Z) ⊆ R n ≥0 ,

where Z := Xφ, φ ∈ S, is a linear coordinate change minimizing this polyhedron (beginning of chapter 2). Resolution for hypersurface singularities in residue characteristic zero uses two primary invariants: the multiplicity function x → m(x) and the (normalized) slope function x → ǫ(x). The latter is not well-behaved in residue characteristic p > 0: it is in general not a constructible function on X ; the pair (m(x), ǫ(x)) in general increases after performing Hironaka-permissible blowing ups. This pair is denoted (ν, ǫ) for surfaces in [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] p.253.

In contrast, we construct a numerical function (Definition 2.68) ι : X → {1, . . . , p} × N × {1, ≥ 2} : x → (m(x), ω(x), κ(x)) (1.4) which refines the multiplicity function at those points x ∈ X such that m(x) = p. This function is differential in nature and has "expected" properties: ι is invariant by regular base change S ⊂ S, S excellent (Theorem 2.74) and is constructible on X (Corollary 3.23).

Remark 1.6. The differential multiplicity ω(x) sprouts from Hironaka's ǫ(x) if one requires invariance by smooth base change, vid. Theorem 2.74. A difference takes place between (i) the purely inseparable case, and (ii) the Galois case considered in Theorem 1.5: eventually ι is uppersemicontinuous in case (i) but only constructible in general in case (ii), vid. Corollary 3.23 and following Example 3.25.

The proof of Theorem 1.4 relies mainly on the properties of our function ω: constructiblity and behavior under a family of permissible blowing ups. It is defined using equation (1.1) and both (i) and (ii). We know no analogue of ω with these properties in the apparently similar case

h = X p e + f p e ∈ S[X], e ≥ 2.
This equation is studied by Moh in [START_REF] Kawanoue | Toward resolution of singularities over a field of positive characteristic I. Foundation; the language of the idealistic filtration[END_REF] with a related open problem "On the bound of d 2 " and by H. Hauser and S. Perlega in [START_REF] Hauser | Cycles of singularities appearing in the resolution problem in positive characteristic[END_REF].

We develop a notion of permissible blowing up for ι refining that of H. Hironaka. Permissible centers Y ⊂ X are of two different kinds (Definitions 3.1 and 3.5), first kind being "ǫ-constant". They also extend to permissible centers under regular base change (Theorem 3.8). The function ι is nonincreasing with respect to permissible blowing ups (Theorem 3.13). Differential multiplicities and permissible centers have a similar behavior to adapted multiplicities and permissible blowing ups considered in Resolution of Singularities for differential forms and vector fields [START_REF] Seidenberg | Reduction of singularities of the differential equation Ady = Bdx[END_REF][15] [START_REF] Cano | Reduction of the singularities of codimension one singular foliations in dimension three[END_REF] [START_REF] Cano | Reduction of singularities of three-dimensional line foliations[END_REF][64] [START_REF] Panazzolo | Resolution of singularities of real-analytic vector fields in dimension three[END_REF] and for toroidalization of morphisms [START_REF] Cutkosky | A simpler proof of toroidalization of morphisms from 3-folds to surfaces[END_REF] [START_REF] Cutkosky | Toroidalization of dominant morphisms of 3-folds[END_REF].

Remark 1.7. Our notion of permissible blowing up also sprouts from Hironaka's ǫ-constant blowing ups if one requires invariance by smooth base change, vid. Theorem 3.8. Permissibility at a point y ∈ X implies permissibility on a nonempty Zariski open subset U ⊆ Y := {y} (Theorem 3.22). Example 3.6 shows the relevance of permissible blowing ups of the second kind whenever X has dimension n ≥ 3. Section 3.3 includes further results intended to serve as a guideline for n ≥ 4.

Beginning from chapter 4, dimension n = 3 is assumed and we focus on the proof of Theorem 1.1. Chapter 4 reduces the proof of Theorem 1.1 to that of Theorem 1.5 and is adapted from [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] to our arbitrary characteristic context. The main issue for proving Theorem 1.5 is to achieve m(x r 1 ) < p for some r 1 ≥ 0; achieving (X r , x r ) regular, i.e. m(x r ) = 1, is then relatively easy and has been proved in [START_REF] Cossart | Resolution of Singularities of Threefolds in Mixed Characteristics. Case of small multiplicity[END_REF].

The last four chapters contain the technical bulk of this article. In chapter 5, the function κ in (1.4) is refined with values in {1, 2, 3, 4}. For fixed ι(x), we attach a generic projection from SpecS to dimension two. In contrast with residue characteristic zero, there is no obvious way to attach a projected two-dimensional structure similar to (S, h, E). This difficulty (no reasonable notion of "maximal contact") seems to be inherent to residue characteristic p > 0 and has proved to be quite a match. Our method consists in projecting only the combinatorial structure provided by the characteristic polyhedron given in (1.3), say:

p 2 : [∆ S (h; u 1 , u 2 , v; Z) ⊆ R 3 ≥0 ] → [∆ 2 (h; u 1 , u 2 ; v; Z) ⊆ R 2 ≥0 ]. (1.5)
Here, p 2 is a linear projection and v := u 3φ 2 , φ 2 ∈ S, is a linear coordinate change minimizing the image polygon. New combinatorial invariants are associated to the right-hand side polygon; their control under permissible blowing ups eventually leads to a smaller value ι(x ′ ) < ι(x). This is the content of the Projection Theorem 5.5 from which Theorem 1.5 follows easily by induction on ι(x) (Corollary 5.6). The strategy follows that of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] but also contains very substantial improvements:

• the sequence (1.2) which is constructed involves Hironaka-permissible blowing ups only, in contrast with [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF]. It does not depend on the given valuation µ and can be considered as a version of Hironaka's Local Control (Hironaka's A/B Game, in residue characteristic zero) for equations (1.1). Precise statements use the notion of independent sequence (Definition 2.77) and Theorem 5.5 is stated in these terms.

The authors hope that Theorem 1.5 could be extended to a Resolution of Singularities π : X ′ → X , π a composition of Hironaka-permissible (global) blowing ups (and with G-invariant centers under assumption (ii)).

• all resolution invariants used in this text are defined in terms of initial form polynomials in σ h w.r.t. certain faces σ of the characteristic polyhedron attached to h. Furthermore, these initial form polynomials provide control for the invariants under blowing up. These facts are the main reason why our proof is characteristic free: in σ h is a polynomial with coefficients in the residue field k(x). They are also the reason why the extra assumption [k(x) : k(x) p ] < +∞ is not required in the proof.

• the role played by small residue characteristics is very minor (essentially the extra twist in Lemma 7. [START_REF] Cutkosky | Resolution of singularities for 3-folds in positive characteristic[END_REF] for p = 2). Difficulties caused by nonperfect residue fields k(x) appear mostly technical in nature, because one is led to carry along (absolute) p-bases (λ l ) l∈Λ in the construction (section 2.4). Nontrivial issues are related to regular base change (Proposition 2.15, Theorem 2.74 and Theorem 3.8), the Hilbert-Samuel stratum (Proposition 2.55) and Zariski closure of formal centers (Proposition 3.17) in arbitrary dimension n ≥ 1. For n = 3, vid. Remark 2.56, Proposition 5.8 and section 7.5; real difficulties come from Lemma 7.22(3)(3') for inseparable extensions of degree d = p = 2.

The proof of Theorem 5.5 is spread along chapters 6 (κ(x) = 1), 7 (κ(x) = 2), 8 and 9 (κ(x) = 3, 4). Chapter 9 uses blowing ups along Hironakapermissible curves which are not necessarily of the first or second kind. The authors do not know if such blowing ups are required in general in order to achieve Resolution (in contrast with permissible blowing ups of the second kind, vid. Example 3.6). They do not appear in [START_REF] Cossart | Polyèdre caractéristique d'une singularité[END_REF].

Quoting H. Hironaka's euphemism from [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] p.254: "in the case of dimension 3 or more, the behavior of [the characteristic polyhedron] appears to be far more complicated and has not yet been fully investigated [...] a little experiments lead us to an aphorism: Reduction of singularities is sharpening of polyhedra."

When the hypersurface singularity X has dimension 3 and satisfies the assumptions of Theorem 1.5, our results give a precise content to this aphorism:

(1) the numerical character ι(x) = (m(x), ω(x), κ(x)) is attached to the initial form polynomial in m S h w.r.t. the initial face of the characteristic polyhedron;

(2) permissible blowing ups produce a smaller value ι(x ′ ), or a monic form for the new initial in m S ′ h ′ , with (m(x ′ ), ω(x ′ )) = (m(x), ω(x)). This monic form corresponds to a certain vertex v ′ of the characteristic polyhedron;

(3) projecting from v ′ produces a characteristic polygon with numerical character γ(x ′ ) ∈ N;

(4) further Hironaka-permissible blowing ups either produce a smaller value ι(x ′′ ) < ι(x), or achieve ι(x ′′ ) = ι(x ′ ), in m S ′′ h ′′ in monic form with γ(x ′′ ) < γ(x ′ ).

Acknowledgement: the authors acknowledge many stimulating discussions held during the "Fall School on Resolution of Threefolds in Positive Characteristic", University of Regensburg, October 1-11/2013. They hereby thank H. Kawanoue, S. Perlega, S. Saito, M. Spivakovsky, A. Voitovitch, A. Weber and J. W lodarczyk for numerous questions and suggestions, with very special thanks to the organizers U. Jannsen and B. Schober. Very special thanks also to H. Mourtada, D. Rydh and B. Schober who organized the workshop "Two approaches towards local uniformization and resolution of singularities in characteristic p > 0", at the Mittag-Leffler Institute, Djursholm, May 23-27/2016.

Extending the proof of Corollary 1.3 to that of Theorem 1.1 was suggested to the authors by D. Cutkosky who is thanked warmly about it.

Both authors thank the referees for their accurate readings, for the lot of time and energy they have devoted to this paper: 36 pages of reports. Their comments have been very helpful for amending the text, clarifying its content and making it more accessible to readers.

Overview of the content and proof of Theorem 1.1.

This article is organized as follows: in chapter 2, we introduce our main tool which is the Hironaka Characteristic Polyhedron [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] (Definition 2.8). This is performed for any polynomial equation

h := X m + f 1,X X m-1 + • • • + f m,X ∈ S[X], f 1,X , . . . , f m,X ∈ S
where S is an excellent regular local ring of dimension n ≥ 1.

Our notation ∆ S (h; {u j } j∈J ; X) for polyhedra (Definition 2.2) slightly differs from Hironaka's because we focus our attention on the variation of the characteristic polyhedron along regular subschemes W := ({u j } j∈J ) ⊆ SpecS, J ⊆ {1, . . . , n}.

A basic algebraic object attached to W is the graded algebra:

G(W ) := i≥0 I(W ) i /I(W ) i+1 ≃ S/({u j } j∈J )[{U j } j∈J ].
(1.6)

To a given face σ = σ α defined by a weight vector α ∈ R n ≥0 , an initial form polynomial in α h is attached (Definition 2.3). Proposition 2.12 is imported from [START_REF] Cossart | Characteristic polyhedra of singularities without completion[END_REF] and is an essential tool for studying these variations along W . It states that ∆ S (h; u 1 , . . . , u n ; X) ⊆ R n ≥0 can be made minimal by a suitable linear coordinate change Z := Xφ, φ ∈ S. Denote X := Spec(S[Z]/(h)), η : X -→ SpecS.

If x ∈ η -1 (m S ) is a point of multiplicity m(x) = m, then η -1 (m S ) = {x}, k(x) = S/m S .
Hironaka's slope for ∆ S (h; u 1 , . . . , u n ; Z) is denoted by δ(x) ≥ 1 when this polyhedron is minimal (Proposition 2.10 and Definition 2.11).

Assume that a reduced normal crossings divisor

E = div(u 1 • • • u e ) ⊆ SpecS (1.7)
is specified. Well adapted coordinates (u 1 , . . . , u n ; Z) are coordinates such that (1.7) holds and ∆ S (h; u 1 , . . . , u n ; Z) is minimal (Definition 2.24). Relevant numerical data are defined for well adapted coordinates only. For such coordinates, h has weights d j := min{x j : (x 1 , . . . , x n ) ∈ ∆ S (h; u 1 , . . . , u n ; Z)}, 1 ≤ j ≤ e.

When m = p, assumptions (i) or (ii) of Theorem 1.5 (condition (G) in the text) and (E) (Definition 2.32) imply that pδ(x), H j := pd j ∈ N (Corollary 2.30) (1.8) and provide the structure Theorem 2.36 for the initial form polynomials in α h with respect to its compact faces (Definition 2.3). This fact allows us to reproduce part of the equicharacteristic p > 0 constructions used in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF].

Note that E is always assumed to be equicharacteristic p > 0 (Definition 2.32). For example when α = 1 := (1, . . . , 1), σ 1 is the initial face of the polyhedron ∆ S (h; u 1 , . . . , u n ; Z); the corresponding homogeneous polynomial

in 1 h ∈ G(m S )[Z], G(m S ) := gr m S S ≃ k(x)[U 1 , . . . , U n ]
(denoted by in m S h in the text) has degree pδ(x), setting degZ := δ(x). Theorem 2.36 can be stated as follows: assume that ∆ S (h; u 1 , . . . , u n ; Z) is not an orthant with vertex in R e (ǫ(x) = 0 in the text); then

in m S h = Z p -G p-1 Z + F p,Z ∈ G(m S )[Z].
(1.9)

Let H := e j=1 U H j j ∈ G(m S ) with notations as in (1.8). We denote (Definition 2.25):

ǫ(x) := deg(in m S h) -degH = pδ(x) - e j=1 H j ∈ N.
This leads us to define the function ι in (1.4) (Definition 2.68). The function ω is a differential version of Hironaka's ǫ-function [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] and requires introducing a differential structure (S, h, E) adapted to the normal crossings divisor E ⊂ SpecS (section 2.5). This is done by considering the G(m S )module Ω G(m S ) (log U 1 • • • U e ) of absolute logarithmic differentials and its dual space of derivatives D(m S ). The derivatives

H -1 ∂ ∂Z , {H -1 D} D∈D(m S ) (1.10)
act on in m S h. If G = 0, we simply let κ(x) ≥ 2, vid. (1.4), and .11) If G = 0, the definition is more delicate but only relies on elementary linear algebra. We then have (ω(x) = ǫ(x), κ(x) = 1) or (ω(x) = ǫ(x) -1, κ(x) ≥ 2).

ω(x) :=    ǫ(x) if ∂F p,Z ∂U j = 0, e + 1 ≤ j ≤ n ǫ(x) -1 otherwise . ( 1
(1.12)

In order to deal with blowing ups along Hironaka-permissible subschemes Y ⊂ X , the above construction is performed in a more general setup; we introduce logarithmic Nagata derivatives D(W ) on the graded algebras G(W ) = G( Ŵ ) for W ⊂ E having normal crossings with E (note that charW = p since charE = p). The main definitions are given in (2.49): homogeneous submodules

V (F, E, W ) ⊂ G(W ) d-d W -1 , J(F, E, W ) ⊂ G(W ) d-d W
are attached to a homogeneous element F ∈ G(W ) d and a monomial ideal H W ⊂ G(W ) d W . This construction plays a fundamental role in this article and is used passim.

Another important notion is that of the affine cone Max(x) and the affine space Dir(x) (Definition 2.72). These are respectively the stratum and the directrix of the space of forms of degree ω(x) obtained by applying those derivatives in (1.10). Once again, the definition is more delicate when G = 0 but elementary in nature. For applications to dimension three, we always have Max(x) = Dir(x), vid. Remark 2.56.

When ω(x) = 0 in (1.4), a simple combinatorial blowing up algorithm (similar to residue characteristic zero) makes the value of the multiplicity function smaller than p at all points of the blown up space mapping to x (Theorem 2.81). It remains to deal with points x ∈ X such that m(x) = p, ω(x) > 0.

Chapter 3 develops a notion of permissible blowing up π : X ′ → X which refines that of H. Hironaka. Roughly speaking, a Hironaka permissible center Y ⊂ X is permissible in our sense if X is "differentially equimultiple" along Y (Definition 3.1 and Definition 3.5). The notion is somewhat subtle but has good properties, the main result being Theorem 3.13: ι is nonincreasing along permissible blowing ups. Furthermore, ι decreases except possibly at exceptional points x ′ ∈ π -1 (x) belonging to some embedded projective cone

P C(x, Y) ⊂ π -1 (x)
given in Definition 3.12. The cone P C(x, Y) is the projectivization of a certain cone containing Max(x) and coincides with it when ω(x) = ǫ(x). We also mention:

• persistence of permissibility under regular base change (Theorem 3.8);

• the strict transform Z ′ ⊂ X ′ of a permissible center Z ⊂ X under a permissible blowing up π with center Y ⊂ Z is permissible (Theorem 3.15);

• the support of a formal arc can be made permissible at its special point by performing permissible blowing ups (Proposition 3.17);

• Hironaka permissible centers are permissible in a dense open subset of their support (Theorem 3.22).

Remark 1.8. Example 3.18 points out a substantial difference between permissibility for ι and Hironaka-permissibility when n ≥ 4. It states that the support Z ⊆ X of a formal arc cannot in general be made permissible for ι at its special point x by iterated quadratic transforms. This phenomenon also occurs for n = 3 but only for ω(x) = 1; it is then easily dealt with.

The section concludes with the constructibility on X of the function ι (Corollary 3.23). Dimension n = 3 is assumed in the next chapters.

Chapter 4 contains what can be deduced from known Embedded Resolution results in excellent regular threefolds. We also adapt some of the equal characteristic p > 0 material from [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] to our arbitrary characteristic context and prove: Chapter 5 collects together all previous results. A projection number κ(x) ∈ {1, 2, 3, 4} (Definition 5.1) is associated to a singular point x ∈ X such that m(x) = p, ω(x) > 0. This function basically expresses the transverseness or tangency of the initial form (1.9) of the characteristic polyhedron with respect to the initial face. For convenience of the reader, we give a sample of the main types of initial form polynomials occurring when E = div(u 1 ); we take ω(x) > 0, λ ∈ k(x) and all exponents are integers in these formulae. Furthermore, we have λ = 0, λ ∈ k(x) p if (d 1 , ω(x)/p) ∈ N 2 (resp. if d 1 + ω(x)/p ∈ N) in the second (resp. fifth) formula:

in m S h =                                            Z p -λU d 1 + ω(x) p 1 p-1 Z κ(x) = 1 Z p + λU pd 1 1 U ω(x) 3 ω(x) ≡ 0modp κ(x) = 2 Z p + λU pd 1 1 U 2 U ω(x) 3 ω(x) ≡ 0modp κ(x) = 2 Z p + λU pd 1 1 U 1+ω(x) 3
1 + ω(x) ≡ 0modp κ(x) = 3

Z p + λU pd 1 +ω(x) 1 κ(x) = 4 Z p + λU pd 1 +ω(x) 1 U 2 κ(x) = 4
The complete definition of κ(x) takes into account all possible in m S h and E which may occur. The simpler forms listed above are "monic forms" in the sense that a certain monomial computing ω(x) occurs in in m S h. We now explain these definitions and the hierarchy between them: for fixed ω(x), the singularity is considered as milder as κ(x) decreases. To begin with, ω(x) is computed from in m S h by applying certain derivatives (1.10)-(1.12).

• when this derivative is transverse to the base SpecS, i.e. applying H -1 ∂ ∂Z in (1.10), we set κ(x) = 1; otherwise κ(x) ≥ 2.

• when κ(x) ≥ 2, we set κ(x) = 4 if the directrix affine space Dir(x) has equations in U 1 , . . . , U e , i.e. in those coordinates corresponding to E. Otherwise, Dir(x) has an equation which is transverse to E, say U 3 = 0 with e = 1 or e = 2. The very transverse case κ(x) = 2 means that a derivative transverse to U 3 is involved in (1.10), i.e. a derivative w.r.t. another variable U 1 , U 2 or to a constant in k(x):

D = H -1 U 1 ∂ ∂U 1 , D = H -1 ∂ ∂U 2 (e = 1), or D = H -1 ∂ ∂λ .
We set κ(x) = 3 if none of the cases before holds. Theorem 5.5 states that ι(x) can be made smaller by performing local Hironaka permissible blowing ups. Theorem 1.5 then follows easily by descending induction on ι(x).

The proof of Theorem 5.5 is very long and intricate. For κ(x) = 1 (resp. κ(x) = 2, 3, 4), the proof is given in Corollary 6.4 (resp. Theorem 7.26, Theorem 9.6, ibid.). Three main phenomena are responsible for these intricacies:

(i) no obvious way shows up for reducing Theorem 5.5 for (S, h, E) to some statement on the coefficients of the polynomial h. When this is possible (for κ(x) = 1 and in part for κ(x) = 3, 4), the proofs are notably simplified. This is done in section 6 where some weak form of maximal contact with a component of E is assumed for ι.

(ii) reducing Theorem 5.5 to the "monic forms" corresponding to κ(x) is achieved by a casuistic analysis which seems for the moment out of reach in higher dimensions. Sections 7.2, 8.3 and part of 8.1, 8.2 are concerned with this problem.

(iii) blowing up a monic form along a permissible center (e.g. a closed point) may lead to a bigger value ι(x ′ ) = (p, ω(x), 4) > ι(x) when κ(x) = 2, 3. These situations are also dealt with by a casuistic analysis whose extension to higher dimensions seems out of reach. Section 7.1 and part of 8.1, 8.2 are concerned with this problem.

Chapter 6 proves Theorem 5.5 for sequences of permissible blowing ups with centers lying inside the successive strict transforms of a fixed irreducible component of E. This proves Theorem 5.5 in the case κ(x) = 1 and prepares the ground in the cases κ(x) = 3, 4. The proof is similar to that of Resolution for excellent surfaces [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] [START_REF] Cossart | Desingularization of embedded excellent surfaces[END_REF] [START_REF] Cossart | Resolution of surface singularities[END_REF], but does not follow from it.

Chapter 7 proves Theorem 5.5 when κ(x) = 2. The above phenomenon (iii) is studied in section 7.1. The proofs are essentially the same as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 2.II except that all statements and proofs are phrased only in terms of initial form polynomials in α h w.r.t. certain faces σ α of ∆ S (h; u 1 , u 2 , u 3 ; Z). Section 7.2 defines the "monic forms" (Definition 7.10) and deals with the above phenomenon (ii) in Proposition 7.11. No obvious reduction to Resolution for surfaces is available (phenomenon (i)). The proof then follows our strategy as indicated at the end of the previous section (3) and (4). Section 7.3 builds up the projected polygon ∆ 2 (h; u 1 , u 2 ; v; Z) of (1.5) (Theorem 7.18) and defines secondary numerical invariants (Definition 7. [START_REF] Cossart | Desingularization of embedded excellent surfaces[END_REF]). The main invariant is denoted by γ(x) ∈ N. Two main difficulties arise here: rationality over S (i.e. v can be chosen in S and not only in Ŝ), and independence of choices of coordinates. Section 7.4 studies the behavior of the invariants under blowing up a closed point. Finally, section 7.5 proves that permissible blowing ups produce some point x ′ with ι(x ′ ) ≤ (p, ω(x), 1) (Theorem 7.26). The algorithm blows up permissible curves only when γ(x) = 0, 1.

Chapters 8 and 9 prove Theorem 5.5 for κ(x) = 3, 4. Since only Hironakapermissible centers are used, this chapter contains many new features in comparison with the corresponding [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 3.II. Definition 8.1 states what is required of the "monic forms", called respectively (**) (κ(x) = 3, 4) and (T**) (κ(x) = 4). Phenomenon (iii) seems to be untractable here and is the reason for these stronger conditions imposed on h. Reduction to these monic forms is harder than in chapter 7 and is spread along sections 8.1, 8.2 and 8.3 (Propositions 8.9 and 8.11).

Section 9.2 reduces a monic form (T**) to (**) or to κ(x) ≤ 2 (Proposition 9.1). The proof is an application of Theorem 6.3 since a weak form of maximal contact with a component of E holds for this reduction. Section 9.3 finally proves that monic forms (**) can be reduced to κ(x) ≤ 2 (Proposition 9.5). When ω(x) ≥ p, this reduction is achieved by blowing up along Hironakapermissible curves, not necessarily permissible of the first or second kind, but contained in the locus Ω + (X ) := {y ∈ X : ω(y) > 0}.

In order to ensure Hironaka-permissibility, the condition E = η(Sing p X ) is required (section 9.2.1, condition (E') in the text). Section 9.2.2 builds up the projected polygon ∆ 2 (h; u 1 , u 2 ; v; Z) (Definition 9.12 and Proposition 9.13) and defines secondary numerical invariants (Definition 9.15). Said blowing ups along Hironaka-permissible curves are performed mostly in Propositions 9.18 and 9.20.

2 Adapted structure and primary invariants.

All along this article, we will denote by S a regular local ring of arbitrary dimension n ≥ 1, and by (u 1 , . . . , u n ) a regular system of parameters (r.s.p. for short) of S. Its maximal ideal is denoted by m S := (u 1 , . . . , u n ) and its formal completion w.r.t. m S by Ŝ. The order function ord m S on S is defined by: ord m S f := sup{n ∈ N : f ∈ m n S } ∈ N ∪ {+∞}, f ∈ S.

This order function extends to a discrete valuation on the quotient field K := QF (S) of S.

We will assume that char(S/m S ) > 0 except for the next three sections. We also assume that S is excellent beginning from Proposition 2.12 on. The basic reference for excellent rings is [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] 7.8 and 7.9. A useful compendium is [START_REF] Matsumura | Commutative ring theory[END_REF] pp. 255-260; some extensions and examples of non excellent regular local rings can be found in [START_REF] Illusie | Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasiexcellents[END_REF] pp. 7-22. Let

h := X m + f 1,X X m-1 + • • • + f m,X ∈ S[X], f 1,X , . . . , f m,X ∈ S (2.1)
be a monic polynomial of degree m ≥ 2. We denote by

X := Spec(S[X]/(h)) and η : X -→ SpecS (2.2)
respectively the corresponding hypersurface and induced projection. The total ring of fractions X is denoted by L := Tot(S[X]/(h)). Given a point y ∈ X , its ideal, residue field and multiplicity are respectively denoted by m y , k(y) and m(y).

For convenience of the reader, we make the definition of m(y) explicit. Let s := η(y) ∈ SpecS, k(s) be the residue field of S s and h ∈ k(s)[X] be the reduction of h. The point y corresponds to a certain irreducible factor T of h with k(y) = k(s)[X]/(T ). One defines m(y) by:

m(y) := ord m S[X]y h ≥ 1.
By definition of regular local rings, we thus have:

O X ,y is a regular local ring ⇔ m(y) = 1.
The singular (i.e. not regular) locus (resp. locus of multiplicity m) of X is denoted by : SingX = {y ∈ X : m(y) ≥ 2} (resp. Sing m X := {y ∈ X : m(y) = m}).

Both are viewed as reduced embedded subschemes of X . Nontrivial material concerning regularity and the multiplicity function is normally accompanied with a reference to [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] or [START_REF] Matsumura | Commutative ring theory[END_REF]. A basic, but especially important property is Proposition 2.10.

Given a "linear change of" (one also says "translation on") the X-coordinate, say X ′ := Xφ, φ ∈ Ŝ, we still denote by

h = X ′ m + f 1,X ′ X ′ m-1 + • • • + f m,X ′ ∈ Ŝ[X ′ ]
the corresponding expansion of h(X ′ + φ), f 1,X ′ , . . . , f m,X ′ ∈ Ŝ. The explicit formula for this change of coordinate is :

f i,X ′ = m i φ i + i j=1 m -j i -j f j,X φ i-j , 1 ≤ i ≤ m. ( 2
cl d I := Vect({cl d φ} φ∈I ) ⊆ S/m S [U 1 , . . . , U n ] d . Suppose that a weight vector α = (α 1 , . . . , α n ) ∈ R n ≥0 is given. Let Γ α := Zα 1 + • • • + Zα n ⊂ R. For x = (x 1 , . . . , x n ) ∈ R n ≥0 , denote | x | α := α 1 x 1 + • • • + α n x n ∈ (Γ α ) ≥0 .
An associated valuation µ α of K is defined by setting for f ∈ S, f = 0:

µ α (f ) := max{a ∈ Γ α : f ∈ I α (a) := ({u x 1 1 • • • u xn n :| x | α ≥ a})}.
It easily follows from the Noetherianity of S that µ α (f ) is well defined. One sets

µ α (f /g) := µ α (f ) -µ α (g) for f, g ∈ S, f g = 0.
Note that ord m S = µ 1 , where 1 = (1, 1, . . . , 1) ∈ R n >0 . We will systematically use the graded ring gr α S of S w.r.t. µ α :

gr α S ≃ S/({u i : α i > 0})[{U i : α i > 0}].
(2.4)

If a ∈ Γ α and φ ∈ S is given with a ≤ µ α (φ), its initial form cl α,a φ ∈ gr α S is defined as before. Similarly, if I ⊂ S and a ≤ µ α (I), we associate a (gr α S) 0 -module denoted by cl α,a I := Span({cl α,a φ} φ∈I ) ⊆ (gr α S) a .

Characteristic polyhedron and first invariants.

Polygons have been used since Newton to compute the leading terms in Puiseux parametrizations of plane branches. Nowadays, Newton polyhedra are classical tools for the study of singularities. They encode certain numerical data attached with a given singular germ and provide a rough approximation of the geometry of the singularity. For toric singularities [START_REF] Kempf | Toroidal embeddings. I. Lecture Notes in Mathematics[END_REF], resolution of singularities is recovered from their polyhedron. In general, it only provides invariants for bettering singularities by using a stepwise blowing up process [START_REF] Schober | A polyhedral approach to the invariant of Bierstone and Milman[END_REF], [START_REF] Lipman | Quasi-ordinary singularities of surfaces in C3. Singularities[END_REF] for quasi-ordinary singularities.

Hironaka showed how Newton polyhedra can be used to construct resolution of singularities for surfaces [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF]. For singularities of any dimension, one first projects the Newton polyhedron from a special face related to a transversal projection, then minimizes the image by suitable changes of coordinates. Hironaka's characteristic polyhedron is defined to be the closure of the image.

Each face of the characteristic polyhedron is defined by some monomial valuation which leads to associated graded rings and initial forms. From these data we will define our main invariants. We also study the behavior by base change and blowing ups.

Let S and (u 1 , . . . , u n ) be fixed as above. Given a subset J ⊆ {1, . . . , n}, we denote by I J := ({u j } j∈J ) ⊂ S and S J := S/I J .

We also use the notation s J ∈ SpecS to denote the point s J = I J , reserving the idealistic notation I J to commutative algebraic formulae. The next proposition will be applied to each coefficient of h in (2.1).

Proposition 2.1. Let f ∈ S. There exists a unique finite set S J (f ) ⊂ N J such that the following holds:

(i) the set of monomials { j∈J u a j j : a = ({a j } j∈J ) ∈ S J (f )} forms a minimal system of generators of the ideal

I(f ) := j∈J u a j j : a = ({a j } j∈J ) ∈ S J (f ) ;
(ii) there is an expansion

f = a∈S J (f ) γ(f, a) j∈J u a j j ∈ S, γ(f, a) ∈ S (2.5)
such that γ(f, a) ∈ I J (i.e. γ(f, a) is a unit in S I J ) for every a ∈ S J (f ).

Proof. Let S J be the formal completion of S along I J . Since I J ⊆ m S , S J is faithfully flat over S [START_REF] Matsumura | Commutative ring theory[END_REF] Theorem 8.14 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF]. Thus I S J ∩ S = I for any ideal I ⊆ S, in particular for any monomial ideal in {u j } j∈J . One deduces that property (i) and existence of an expansion (2.5) descend from S J to S.

Suppose that an expansion (2.5) exists for a given S J (f ) satisfying (i). Each S/I n+1 J , n ≥ 0 has a structure of free S J -module with basis j∈J u a j j : a = ({a j } j∈J ) and j∈J a j ≤ n .

Therefore the class γ(f, a) + I J is independent of the chosen expansion (2.5) by the minimality property in (i). This proves that the property γ(f, a) ∈ I J in (ii) also descends from S J to S. In other terms, we may assume that S is I J -adically complete.

Independent monomial generators in S/I n J lift to independent monomial generators in S/I n+1 J for every n ≥ 1. One easily deduces the existence of an expansion (ii) satisfying (i) for some finite subset S J (f ) ⊂ N J , since S is I J -adically complete and Noetherian.

Uniqueness of S J (f ) is also checked by taking images in S/I n+1 J for some n >> 0.

Given an equation h ∈ S[X] (2.1) and a r.s.p. (u 1 , . . . , u n ) of S, let us write a finite expansion:

h := i,A c i,A X i u a 1 1 • • • u an n , A = (a 1 , • • • , a n ) ⊂ N n , c i,A ∈ S (2.6) c i,A invertible in S.
The Newton polyhedron NP (h; u 1 , . . . , u n ; X) w.r.t. the variables (u 1 , . . . , u n , X) associated to h is defined as:

NP (h; u 1 , . . . , u n ; X) := convex hull of c i,A (A, i) + R n+1 ≥0 ⊆ R n+1 ≥0 .
Let P := (0, . . . , 0, 1) ∈ R n+1 ≥0 , so P ∈ 1 m NP (h; u 1 , . . . , u n ; X), and p : R n+1 \{P } -→ R n be the projection on the (u 1 , . . . , u n )-space from the point P . We define a polyhedron by:

∆ S (h; u 1 , . . . , u n ; X) := p 1 m NP (h; u 1 , . . . , u n ; X) ∩ {x n+1 < 1} ⊆ R n ≥0 .
The characteristic polyhedron is introduced in a more general context in [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF].

In our setting, it consists in minimizing ∆ S (h; u 1 , . . . , u n ; X ′ ) over all linear changes of coordinates

X ′ = X -φ, φ ∈ Ŝ (2.
3): see Definition 2.8 below.

In this section, we review and adapt notations to fit our purposes. A fundamental algebraicity result is borrowed from [START_REF] Cossart | Characteristic polyhedra of singularities without completion[END_REF] in Proposition 2.12 below.

Definition 2.2. (Associated Polyhedron). Given an equation h ∈ S[X]

(2.1) and J ⊆ {1, . . . , n}, we define a rational polyhedron:

∆ S (h; {u j } j∈J ; X) := Conv   m i=1 a∈S J (f i,X ) a i + R J ≥0   ⊆ R J ≥0 .
Definition 2.3. (Initial forms). Let α = ({α j } j∈J ) ∈ R J >0 be a weight vector. We define δ α (h; {u j } j∈J ; X) := min{| y | α : y ∈ ∆ S (h; {u j } j∈J ; X)}.

The weight vector defines a compact face σ α of ∆ S (h; {u j } j∈J ; X) compact face by:

σ α := {y ∈ ∆ S (h; {u j } j∈J ; X) : | y | α = δ α (h; {u j } j∈J ; X)}.
Given h and α, the grading of gr α S (2.4) can be extended to gr α (S[X]) = (gr α S)[X] by setting:

degX := δ α (h; {u j } j∈J ; X).

Then the initial form in

α h of h w.r.t. α is the polynomial in α h := X m + m i=1 F i,X,α X m-i ∈ (gr α S)[X], (2.7)
where

F i,X,α := y∈σα γ(f i,X , iy)U iy ,
and bars denote images in (gr α S) 0 = S J , i.e.

γ(f i,X , iy) := cl α,0 γ(f i,X , iy) ∈ (gr α S) 0 = S J .
By convention, we take γ(f i,X , iy) = 0 in these formulae whenever iy ∈ S J (f i,X ). Note that the polynomial in α h is homogeneous for this grading of degree mδ α (h; {u j } j∈J ; X).

Remark 2.4. Any vertex of ∆ S (h; {u j } j∈J ; X) has coordinates in 1 m! N. We have:

∆ S (h; {u j } j∈J ; X) = ∅ ⇔ h = X m .
Remark 2.5. It is worth emphasizing that the polynomial in α h only depends on the face σ α and not on the specific weight vector α defining it. Given a vertex y ∈ ∆ S (h; {u j } j∈J ; X), in α h is the y-initial in y h defined in [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] Definition (3.7) for any α ∈ R J >0 such that σ α = y. This motivates the need to consider weights α = 1.

We now briefly review the behaviour of polyhedra and initial forms under basic operations such as formal completion, localization and projection onto a regular subscheme. The case of regular local morphisms S ⊂ S, S excellent will be considered further on. With notations as above, let α ∈ R J >0 be a weight vector and

σ α ⊂ ∆ S (h; {u j } j∈J ; X), in α h ∈ (gr α S)[X].
Formal Completion: Ŝ is excellent [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Theorem 7.8.3(iii). Proposition 2.1 and Definition 2.2 give an identification

∆ S (h; {u j } j∈J ; X) = ∆ Ŝ (h; {u j } j∈J ; X). (2.8)
This identification preserves the initial form in α h for each weight vector α via the inclusion gr α S ⊆ gr α Ŝ ≃ gr α S ⊗ S Ŝ.

Localization: the regular local ring S s J is excellent if S is excellent [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Theorem 7.4.4. Similarly, the identifications ∆ S (h; {u j } j∈J ; X) = ∆ S s J (h; {u j } j∈J ; X) (2.9) also preserve the initial form in α h (2.7) via the inclusion gr α S ⊆ gr α S s J ≃ (gr α S) ⊗ S QF (S J ).

Projection: let J ⊆ {1, . . . , n} and denote by J ′ := {1, . . . , n}\J its complement. The regular local ring

S J is excellent if S is excellent. A r.s.p. of S J is ({u j ′ } j ′ ∈J ′ )
, where bars denote images in S J . With notations as above, we have:

∆ S (h; {u j } j∈J ; X) = pr J ∆ S (h; u 1 , . . . , u n ; X), (2.10)
where pr J : R n → R J , x → y = ({x j } j∈J ) denotes the projection. Let

f i,X = a∈S(f i,X ) γ(f i,X , a)u a 1 1 • • • u an n ∈ S,
be an expansion (2.5) (for the subset {1, . . . , n}, where S(f i,X ) here stands for S {1,...,n} (f i,X )), 1 ≤ i ≤ m. Then (2.7) is given by

F i,X,α := y∈σα   pr J (x)=y γ(f i,X , ix) j ′ ∈J ′ u ix j ′ j ′   j∈J U iy j j , (2.11)
where bars denote images in (gr α S) 0 = S J as before (recall that by convention, we take γ(f i,X , ix) := 0 in this formula if ix ∈ S(f i,X )). In other terms:

γ(f i,X , iy) = pr J (x)=y γ(f i,X , ix) j ′ ∈J ′ u ix j ′ j ′ .
Definition 2.6. (Solvable vertices). Let y ∈ R J be a vertex of the polyhedron ∆ S (h; {u j } j∈J ; X), that is, a 0-dimensional face σ = {y}. Following Hironaka (cf. Remark 2.5), we denote by

in y h = X m + m i=1 F i,X,y X m-i ∈ (gr α S)[X]
the initial form polynomial (2.7) w.r.t. any defining weight vector α. We will say that y is solvable if y ∈ N J and there exists λ ∈ S J such that

in y h = (X -λU y ) m .
Explicitly, with notations as in (2.7) sqq., the latter equality means that

γ(f i,X , iy) = (-1) i m i λ i ∈ S J , 1 ≤ i ≤ m.
Note that m i ∈ S J is not a unit in general when char(S/m S ) > 0.

The following result is a rewriting of [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] in this hypersurface situation.

Proposition 2.7. (Hironaka). There exists a linear change of the Xcoordinate Z := Xθ, with θ ∈ Ŝ, such that

∆ Ŝ (h; {u j } j∈J ; Z) = min X ′ ∆ Ŝ (h; {u j } j∈J ; X ′ ), (2.12)
where the minimum is taken w.r.t. inclusions and over all possible linear changes of coordinates

X ′ := X -φ, φ ∈ Ŝ. Given X ′ := X -φ, φ ∈ Ŝ, ∆ Ŝ (h; {u j } j∈J ; X ′ ) achieves equality in (2.
12) if and only if it has no solvable vertex.

If S is excellent, there is an equivalence

∆ Ŝ (h; {u j } j∈J ; Z) = ∅ ⇔ ∃g ∈ S : h = (X -g) m .
Proof. This is respectively [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] Hironaka's Vertex Preparation Lemma (3.10) and Theorem (4.8), and [START_REF] Cossart | Characteristic polyhedra of singularities without completion[END_REF] Lemma II.1.

Definition 2.8. (Characteristic Polyhedron). For X ′ := Xφ, φ ∈ Ŝ, we will say that the polyhedron ∆ Ŝ (h; {u j } j∈J ; X ′ ) is minimal if it has no solvable vertex.

Example 2.9. Let p be a prime number and n ∈ Z not divisible by p. We take:

S := Z (p) and h := X p -np a ∈ S[X], a ≥ 0.
The following holds:

(1) if a ∈ pZ, then ∆ Zp (h; p; X) = [a/p, +∞[ is minimal;

(2) if a ∈ pZ, then ∆ Zp (h; p; Z) is minimal, where Z := Xnp a/p and we have:

∆ Zp (h; p; Z) =    [ a+1 p , +∞[ if n p -n ∈ p 2 Z [ a p + 1 p-1 , +∞[ if n p -n ∈ p 2 Z .
Proposition 2.10. With notations and conventions as in (2.1) and (2.2), assume that J = {1, . . . , n} and α = 1 (so µ 1 = ord m S ) [48] [23]. Then the rational number δ 1 (h; u 1 , . . . , u n ; Z) is independent of the r.s.p. (u 1 , . . . , u n ) and Z = Xθ, θ ∈ Ŝ such that ∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal. If ∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal, the following characterizations hold:

(i) δ 1 (h; u 1 , . . . , u n ; Z) > 0 ⇔ (η -1 (m S ) = {x} and k(x) = S/m S ); (ii) δ 1 (h; u 1 , . . . , u n ; Z) ≥ 1 ⇔ η -1 (m S ) ∩ Sing m X = ∅. Proof. Let (Z ′ , u ′ 1 , . . . , u ′ n ) and (Z, u 1 , . . . , u n ) be two systems of coordinates such that both polyhedra ∆ Ŝ (h; u ′ 1 , . . . , u ′ n ; Z ′ ) and ∆ Ŝ (h; u 1 , . . . , u n ; Z) are minimal. Suppose that δ 1 (h; u ′ 1 , . . . , u ′ n ; Z ′ ) > δ 1 (h; u 1 , . . . , u n ; Z). Then f m! i i,Z ′ ∈ m m!δ 1 (h;u ′ 1 ,...,u ′ n ;Z ′ ) S for each i, 1 ≤ i ≤ m, hence δ 1 (h; u 1 , . . . , u n ; Z ′ ) ≥ δ 1 (h; u ′ 1 , . . . , u ′ n ; Z ′ ) > δ 1 (h; u 1 , .
. . , u n ; Z). This contradicts the assumption ∆ Ŝ (h; u 1 , . . . , u n ; Z) minimal. The first assertion follows by symmetry.

Let h ∈ S/m S [Z] be the reduction of h modulo m S . Since

η -1 (m S ) = Spec(S/m S [Z]/(h)),
(i) and the "only if" part in (ii) are immediate from the definitions. We have

ord x h(Z) ≤ ord x h(Z) ≤ m, hence x ∈ Sing m X implies h(Z) = (Z -λ) m for some λ ∈ S/m S . Since ∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal, 0 ∈ R n
is not a solvable vertex and therefore we have λ = 0. This proves that (i) holds, the "if" part in (ii) being then obvious.

Definition 2.11. Let s ∈ SpecS, (v 1 , . . . , v n(s) ) be a r.s.p. of S s and y ∈ η -1 (s). Let Z := Xθ, θ ∈ S s be such that ∆ Ss (h; v 1 , . . . , v n(s) ; Z) is minimal, where S s denotes the formal completion of S s w.r.t. its maximal ideal. We let:

δ(y) := δ 1 (h; v 1 , . . . , v n(s) ; Z) = min 1≤i≤m ord m Ss f i,Z i ∈ 1 m! N.
This invariant is classical and appears in e.g. [START_REF] Cossart | Sur le polyèdre caractéristique d'une singularité[END_REF], [START_REF] Cossart | Desingularization of embedded excellent surfaces[END_REF] and [START_REF] Villamayor | Monoidal transforms and invariants of singularities in positive characteristic[END_REF] Definition 4.2 and Proposition 4.8 in an equal characteristic context. Our main resolution invariants will be defined in terms of coordinates (u 1 , . . . , u n ) and Z = Xθ, θ ∈ Ŝ such that ∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal. Since minimizing polyhedra involves in principle choosing formal coordinates, an algebraic version will be useful for proving the constructibility of our invariants. The following proposition is fundamental for this purpose. When charS/m S = 0, the first statement in the proposition easily follows from Proposition 2.7 by applying the Tschirnhausen transformation (take θ = -1 m f 1,X below).

We assume from this point on that S is excellent.

Proposition 2.12. [START_REF] Cossart | Characteristic polyhedra of singularities without completion[END_REF] Given h ∈ S[X] (2.1) and a r.s.p. (u 1 , . . . , u n ) of S, there exists Z := Xθ, θ ∈ S such that ∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal.

For any such Z, the following holds: for every subset J ⊆ {1, . . . , n}, the polyhedron ∆ S s J (h; {u j } j∈J ; Z) is also minimal and is computed by:

∆ S s J (h; {u j } j∈J ; Z) = pr J ∆ Ŝ (h; u 1 , . . . , u n ; Z), (2.13)
where pr J : R n → R J , x → y = ({x j } j∈J ) denotes the projection. In particular, we have

δ(y) = min 1 i j∈J a j , a ∈ S {1,...,n} (f i,Z ), 1 ≤ i ≤ m , y ∈ η -1 (s J ).
Proof. The proposition is trivial if 0 ∈ R n is a nonsolvable vertex of the polyhedron ∆ Ŝ (h; u 1 , . . . , u n ; Z), taking Z := X. Otherwise it can be assumed that f i,X ∈ m S , 1 ≤ i ≤ m. The first statement is [START_REF] Cossart | Characteristic polyhedra of singularities without completion[END_REF] Corollary II.4. Formula (2.13) follows from (2.8) (2.9) (2.10). To prove minimality, suppose that y ∈ N J is a solvable vertex of ∆ S s J (h; {u j } j∈J ; Z) defined by some α ∈ R J >0 . By definition,

∃λ ∈ QF (S J ) : in y h = (Z -λU y ) m . (2.14)
By (2.11), we have λ m = (-1) m U -my F m,Z,α ∈ S J . Hence λ ∈ S J , since the regular ring S J is integrally closed. By (2.13), there exists a vertex

x ∈ ∆ Ŝ (h; u 1 , . . . , u n ; Z) such that y = pr J (x). Lifting up, there exists β ∈ R n >0 , α = pr J (β) defining x, and we let α ′ := pr J ′ (β). There is an induced valuation µ α ′ on S J . The initial form of λ in gr α ′ S J has the form

λ j ′ ∈J ′ U x j ′ j ′ , λ ∈ S/m S , λ = 0, {x j ′ } j ′ ∈J ′ ∈ N J ′ .
Collecting together (2.11) Remark 2.13. This proposition allows us to skip the reference to formal completion when stating that a certain polyhedron is minimal, i.e. given Z := Xφ, φ ∈ S, the statement "∆ S (h; u 1 , . . . , u n ; Z) is minimal" stands for "∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal". On the other hand, we will keep the reference to the regular local ring S since we are also interested in base change.

Let S ⊆ S be a local base change which is regular, i.e. flat with geometrically regular fibers [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Definition 6.8.1(iv). In particular S is regular [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Proposition 6.5.1(ii) and faithfully flat over S. The ring S is not excellent in general, but this certainly holds in the following cases:

(i) S = Ŝ [42] 7.8.3(iii);
(ii) S is ind-étale over S [START_REF] Illusie | Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasiexcellents[END_REF] Theorem I.8.1(iv), or (iii) S is essentially of finite type over S, i.e. smooth over S [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Proposition 7.8.6(i).

An important special case of (ii) is when S is the Henselization or strict Henselization of S. When regular base changes are concerned, we always assume that S is excellent. These conditions are preserved by localizing, i.e. replacing S ⊆ S by S s ⊆ Ss , s ∈ Spec S and s ∈ SpecS its image. Notation 2.14. Let S ⊆ S be a local base change which is regular, S excellent, s ∈ Spec S with image m S ∈ SpecS. Any r.s.p. (u 1 , . . . , u n ) of S can be extended to a r.s.p. (u 1 , . . . , u ñ) of S. We let h ∈ S[X] be the image of h and η : X = X × S Spec S → Spec S.

It follows from Definition 2.6 that, if x ∈ R n ≥0 is a nonsolvable vertex of ∆ S (h; u 1 , . . . , u n ; Z), the vertex (x, 0, . . . , 0

ñ-n ) ∈ ∆ S (h; u 1 , . . . , u ñ; Z) ⊆ R ñ ≥0
is nonsolvable provided that (S/m S ) ∩ ( S/m S ) p = (S/m S ) p . This is of course always satisfied when S/m S is perfect (e.g. charS/m S = 0). An obvious consequence of the second statement in Proposition 2.7 is: Proposition 2.15. (Behavior under regular base change). Let S ⊆ S be a local base change which is regular, S excellent. Assume that

(S/m S ) ∩ ( S/m S ) p = (S/m S ) p . Let Z = X -θ, θ ∈ S, be such that ∆ S (h; u 1 , . . . , u n ; Z) is minimal. Then ∆ S (h; u 1 , . . . , u ñ; Z) = ∆ S (h; u 1 , . . . , u n ; Z) × R ñ-n ≥0 ⊆ R ñ ≥0
and this polyhedron is minimal.

Note that the assumptions of the proposition are satisfied in the above situation (ii): S is ind-étale over S. In situation (iii), i.e. S smooth over S, the following example will make the situation clear:

Example 2.16. Let (S, m S , k) be an excellent DVR, chark = p > 0, and γ ∈ S be a unit. Let λ ∈ k be the residue of γ and assume furthermore that

h := X p -γu pa 1 ∈ S[X], a ≥ 1, λ ∈ k\k p . Then ∆ S (h; u 1 ; X) = [a, +∞[ and is minimal. Take S = S[t] (u 1 ,P (t))
, where P is a monic polynomial with irreducible residue

P (t) ∈ k[t] (resp. P = 0). Let u 2 := P (t), so (u 1 , u 2 ) (resp. (u 1 )) is a r.s.p. of S. Let k := S/m S = k[t]/(P (t)) (resp. k = k(t))
be the residue field of S. Setting {x} = η-1 (m S ), we have

δ(x) = a if λ ∈ kp δ(x) = a + 1 p if λ ∈ kp .
This is obvious if λ ∈ kp (in particular when P = 0); if λ ∈ kp , take

Z := X -γu a 1 , with ṽ := γp -γ ∈ m S .
We claim that (u 1 , ṽ) is a r.s.p. of S. Indeed, S is smooth over S and S[t]/(t pγ) is regular at its closed point. Hence

S ⊗ S S[t]/(t p -γ) = S[t]/(t p -γ) ≃ S[t ′ ]/(t ′ p + ṽ)
is also regular. So (u 1 , ṽ, t ′ ) is a r.s.p. at (x, 0). We have:

∆ S ( h; u 1 , ṽ; Z) = (a, 1/p) + R 2 ≥0 .
In particular, the function

A 1 k = {x} × A 1 k ⊂ X × k A 1 k → 1 p N, x → δ(x)
is not a constructible function.

Proposition 2.12 and Proposition 2.15 suggest the following question. An affirmative answer would be very useful in order to build geometrical invariants from characteristic polyhedra. Proposition 2.15 answers in the affirmative only when S/m S is perfect, taking S := S in the answer to the following question.

Question 2.17. Let S be an excellent regular local ring with r.s.p. (u 1 , . . . , u n ) and h ∈ S[X] (2.1). Does there exist a smooth local base change S ⊆ S, a r.s.p. (u 1 , . . . , u ñ) of S extending (u 1 , . . . , u n ) and Z = X -φ, φ ∈ S, such that the following holds:

"for every smooth local base change S ⊆ S ′ and r.s.p. (u 1 , . . . , u n ′ ) of S ′ extending (u 1 , . . . , u ñ), the polyhedron ∆ S ′ (h; u 1 , . . . , u n ′ ; Z) is minimal"?

Uncovering transformation rules for the characteristic polyhedron under blowing up is a major problem, vid. [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] p.254. A good behavior is known in the special case of a blowing up along a Hironaka permissible subscheme (cf. Definition 2.20) and an exceptional point at the origin of some standard chart.

Proposition 2.18. (Behavior under blowing up). With notations as before, let J ⊆ {1, . . . , n}, y ∈ η -1 (s J ) and assume that δ(y) ≥ 1. Fix j 0 ∈ J and let

S ′ := S[{u ′ j } j∈J ] (u ′ 1 ,...,u ′ n ) , where u ′ j := u j /u j 0 if j ∈ J\{j 0 }; u ′ j := u j if j ∈ J ′ ∪ {j 0 }.
Let Z = Xθ, θ ∈ S, with ∆ S (h; u 1 , . . . , u n ; Z) minimal and define:

h ′ (Z ′ ) := u -m j 0 h(Z) = Z ′ m + u -1 j 0 f 1,Z Z ′ m-1 + • • • + u -m j 0 f m,Z ∈ S ′ [Z ′ ], (2.15)
where Z ′ := Z/u j 0 . Define a map l : R n -→ R n by

x = (x 1 , . . . , x n ) → x ′ = (x 1 , . . . , x j 0 -1 , j∈J x j -1, x j 0 +1 , . . . , x n ). (2.16) Then l(∆ S (h; u 1 , . . . , u n ; Z)) = ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′
) and this polyhedron is minimal.

Proof. The assumption δ(y) ≥ 1 forces f i,Z ∈ I i J by the last statement in Proposition 2.12. Therefore (2.15) makes sense, i.e. h ′ (Z ′ ) ∈ S ′ [Z ′ ]. Since l is one-to-one, we have

1 i S {1,...,n} (f i,Z ′ ) ⊆ l 1 i S {1,...,n} (f i,Z ) , 1 ≤ i ≤ m,
with notations as in Proposition 2.1. By Definition 2.2, we get:

l(∆ S (h; u 1 , . . . , u n ; Z)) = ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ). Let x ′ = l(x) be a vertex of ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ). Denote in x h = Z m + λ 1 U x Z m-1 + • • • + λ m U mx , λ 1 , . . . , λ m ∈ S/m S ,
with the convention as before that

λ i = 0 if ix ∈ N n , 1 ≤ i ≤ m. Applying l (2.16), we get in x ′ h = Z ′ m + λ 1 U ′ x ′ Z ′ m-1 + • • • + λ m U ′ mx ′ . Since S ′ /m S ′ = S/m S ,

Normal crossings divisors.

We now introduce a normal crossings divisor E ⊆ SpecS. This section fixes the terminology and notations for blowing ups and base changes with respect to E, then introduces the Hironaka ǫ function on X . Hironaka-permissible centers are classically defined as regular subschemes along which a given Noetherian scheme is normally flat.

E if E = div(u 1 • • • u e )
for some e, 0 ≤ e ≤ n.

We emphasize that we allow e = 0, i.e. E = ∅ in this definition.

In this context, we use the following notion of Hironaka permissible center: We remind the reader that an integral closed subscheme W ⊆ SpecS has normal crossings with E = div(u 1 • • • u e ) if the family (u 1 , . . . , u e ) can be extended to a r.s.p. (u 1 , . . . , u n ) of S such that the ideal I(W ) of W is of the form I J = ({u j } j∈J ) ⊆ S, for some J ⊆ {1, . . . , n}.

Note that a Hironaka-permissible center w.r.t. any E (e.g. E = ∅) is Hironaka-permissible: we have m(y) = m(x) = m and y ∈ η -1 (w) ∩ Sing m X , where w is the generic point of W ; by Proposition 2.10 applied to S w , the map Y → W is birational, hence an isomorphism since W is regular.

Since the notion is local on X , a Hironaka-permissible blowing up (w.r.t. E) is simply the blowing up along a center Y ⊂ X which is Hironakapermissible (w.r.t. E) at each point of its support. By a local Hironakapermissible blowing up, we simply mean the localization at some point of the exceptional divisor π -1 (Y) of the blowing up π along a Hironaka-permissible center. The important fact is that Hironaka-permissible blowing ups w.r.t. E preserve our structure: Proposition 2.22. Let S, h ∈ S[X] (2.1), X and E = div(u 1 • • • u e ) be as above. Let π : X ′ → X be a Hironaka-permissible blowing up w.r.t. E at x ∈ X . There exists a commutative diagram

X π ←-X ′ ↓ ↓ SpecS σ ←-S ′ (2.17)
where σ : S ′ → SpecS is the blowing up along W .

For every

s ′ ∈ σ -1 (s), S ′ := O S ′ ,s ′ , there exists h ′ ∈ S ′ [X ′ ] monic of degree m such that X ′ s ′ = Spec(S ′ [X ′ ]/(h ′ )). Furthermore, there exists a r.s.p. (u ′ 1 , . . . , u ′ n ) of S ′ adapted to the stalk E ′ s ′ , E ′ := σ -1 (E ∪ W ) red .
Proof. By the above remarks, there exists J ⊆ {1, . . . , n} such that I(W ) = I J = ({u j } j∈J ). By Proposition 2.12, there exists Z := Xθ, θ ∈ S, such that ∆ S (h; u 1 , . . . , u n ; Z) is minimal. Since x, y ∈ Sing m X , we have

η -1 (s) = {x}, η -1 (W ) = Y and δ(x) ≥ 1, δ(y) ≥ 1
by Proposition 2.10. In particular, the ideal of Y at x is

I(Y) = (Z, {u j } j∈J ).
Since δ(y) ≥ 1, the point at infinity (1 : 0 :

• • • : 0) does not belong to X ′ so ({u j } j∈J )O X ′ is invertible.
By the universal property of blowing up, there is a commutative diagram (2.17). Let s ′ ∈ σ -1 (s) and j 0 ∈ J be such that u j 0 is a local equation of π -1 0 (W ). We take X ′ := Z/u j 0 and

h ′ := u -m j 0 h(Z) = X ′ m + u -1 j 0 f 1,Z X ′ m-1 + • • • + u -m j 0 f m,Z . (2.

18)

Note that h ′ ∈ S ′ [X ′ ] follows from the last statement in Proposition 2.12. The last statement is obvious because

E ′ = σ -1 (E ∪ W ) red is a normal crossings divisor on S ′ .
We will stick to these notations when local Hironaka-permissible blowing ups are concerned, or compositions of such local blowing ups. We always refer to the reduced total transform of E on the blown up base SpecS.

Suppose a base change is given as considered in the previous section, i.e. formal completion S ⊆ Ŝ, localization at a prime S ⊆ S s or regular local base change S ⊆ S, S excellent. Notation 2.23. Given S ⊆ S ′ such a base change, we denote

E ′ := E × S SpecS ′ , η ′ : X ′ = X × S SpecS ′ → SpecS ′ . The image of h in S ′ [X] is denoted h ′ ∈ S ′ [X]
. This notation is used consistently with Notation 2.14.

For instance if s ∈ SpecS, there exists a r.s.p. (v 1 , . . . , v n(s) ) of S s which is adapted to E s , where E s is the stalk of E at s. We then have

E s = div(v 1 • • • v e(s)
) and may choose v j = u ϕ(j) for some injective map ϕ : {1, . . . , e(s)} → {1, . . . , e}. It is of course not possible in general to extend a given (v 1 , . . . , v n(s) ) to a r.s.p. (u 1 , . . . , u n ) of S. We let h s ∈ S s [X] be the image of h. Definition 2.24. Let s ∈ SpecS and (v 1 , . . . , v n(s) ) be a r.s.p. of S s which is adapted to

E s , E s = div(v 1 • • • v e(s)
). We say that coordinates

(v 1 , . . . , v n(s) ; Z s ), Z s := X -φ s , φ s ∈ S s , are well adapted at y ∈ η -1 (s) if ∆ Ss (h; v 1 , . . . , v n(s) ; Z s ) is minimal.
As remarked after Definition 2.11, the invariant ǫ(y) introduced below is classically used in Resolution of Singularities. Although the situation is subtle in positive residue characteristic, a general purpose is performing Hironaka-permissible blowing ups to get smaller values of the function ǫ at singular points. Definition 2.25. Let (u 1 , . . . , u n ) be a r.s.p. of S which is adapted to E. Let j, 1 ≤ j ≤ e, and let Y j ⊂ X be an irreducible component of η -1 (div(u j )) with generic point y j ∈ X . We let

d j := δ(y j ) ∈ 1 m! N.
For any s ∈ SpecS and y ∈ η -1 (s), we let

ǫ(y) := m   δ(y) - div(u j )⊆Es d j   ∈ 1 (m -1)! Z.
Summing up results from the previous section, we have:

Proposition 2.26. Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x ∈ η -1 (m S ). With notations as above, we have

d j = min a j i , a ∈ S {1,...,n} (f i,Z ), 1 ≤ i ≤ m , 1 ≤ j ≤ e.
For s ∈ SpecS and y ∈ η -1 (s), we have ǫ(y) ≥ 0.

Proof. The first (resp. second) statement follows from the last one in Proposition 2.12 applied to S and J := {j} (resp. to S s and each J := {j} with div(u j ) ⊆ E s ).

2.3

The Galois or purely inseparable assumption.

In this section, we introduce the assumptions of Theorem 1.5: the polynomial h is either purely inseparable (char(S) = p > 0) or Galois (char(S) = p > 0 or char(S) = 0). This is phrased as condition (G) below. This condition (G) plays an important role in this article for two main reasons: Firstly, (G) is stable under Hironaka permissible blowing ups (Definitions 2.20 and 2.21, Proposition 2.28 below).

Secondly, the initial form polynomials in α h from Definition 2.3 satisfy again (G) (Proposition 2.29(a)). This implies that in α h is either an Artin-Schreier polynomial or a purely inseparable polynomial (Theorem 2.36).

We recall the notations:

h := X p + f 1 X p-1 + • • • + f p ∈ S[X], X := Spec(S[X]/(h)), K := QF (S) and L := Tot(S[X]/(h)).
From now on, we assume furthermore that the following property holds:

(G) m = p is a prime number, h is reduced, the ring extension L|K is normal and X is G-invariant, where G := Aut K (L).
Assumption (G) is maintained up to the end of this chapter.

Since [L : K] = p is a prime number, we have either G = Z/p (L|K separable, cases (a) and (b) below) or G = (1) (L|K inseparable, case (c) below). Case (a) is included here for the sake of completeness and because residue actions in case (b) may lead to case (a). The three cases to be considered are:

(a) h is totally split (product of p pairwise distinct linear factors) over K;

(b) h is irreducible and Galois over K with group G = Z/p;

(c) h is irreducible, charS = p, f i,X = 0, 1 ≤ i ≤ p -1.
Assumption (G) is also preserved by those base changes considered in the previous sections, i.e. formal completion S ⊆ Ŝ, localization at a prime S ⊆ S s or regular local base change S ⊆ S, S excellent. Note that in any case, h reduced implies respectively h s , ĥ (since S is excellent) and h reduced (Notation 2.23). Recall notations and definitions of initial forms from Definition 2.3.

Proposition-Definition 2.27. Assume that charS/m S = p. Let (u 1 , . . . , u n ) be a given r.s.p. of S and α ∈ R n >0 be a weight vector. The integer i 0 (α) := min{i ∈ {1, . . . p} : F i,Z,α = 0}

does not depend on Z = Xθ, θ ∈ Ŝ such that ∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal. If i 0 (α) < p, the form F i 0 (α),Z,α is also independent of the choice of Z = Xθ as above.

In case α = 1, the integer i 0 (1) (also denoted by i 0 (x) for x ∈ η -1 (m S )) and the forms F i 0 (1),Z = F i 0 (1),Z,1 (if i 0 (1) < p) are also independent of the choice of the r.s.p. (u 1 , . . . , u n ) of S and Z = Xθ, θ ∈ Ŝ such that ∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal.

Proof. Take Z ′ = Zφ such that both polyhedra ∆ Ŝ (h; u 1 , . . . , u n ; Z) and ∆ Ŝ (h; u 1 , . . . , u n ; Z ′ ) are minimal. By minimality, we have µ α (φ) ≥ a := δ α (h; u 1 , . . . , u n ; Z).

The initial forms in

α h(Z) ∈ (gr α S)[Z] and in α h(Z ′ ) ∈ (gr α S)[Z ′ ] are related by in α h(Z ′ ) = in α h(Z -cl α,a φ).
The first statement follows from the elementary fact that µ α p i > 0 for 1 ≤ i ≤ p -1, since p ∈ m S . The second statement then follows from Proposition 2.10.

Proposition 2.28. Let x ∈ SingX , s := η(x). Then we have:

η -1 (s) = {x}, k(x) = k(s) and δ(x) > 0. (2.19)
Assume furthermore that a normal crossings divisor E = div(u 1 • • • u e ) ⊂ SpecS is specified and let π : X ′ → X be a Hironaka-permissible blowing up w.r.t. E at x. Then, with notations as in Proposition 2.22, for every s ′ ∈ σ -1 (s), X ′ s ′ satisfies again (G). Proof. It can be assumed that s = m S . Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x and h(Z) ∈ S/m S [Z] be the reduction of h modulo m S . By (G), G acts transitively on the fiber η -1 (s). Then h(Z) is either a p th -power or satisfies again (G) w.r.t. the zero-dimensional regular local ring S/m S .

If h(Z) satisfies (G), then (h(Z), u 1 , . . . , u n ) is a r.s.p. of the local ring S[Z] mx , so x is a regular point of X .

Assume now that h(Z) = (Zλ) p for some λ ∈ S/m S . Now (0, . . . , 0) is a solvable vertex of ∆ S (h; u 1 , . . . , u n ; Z) unless λ = 0. Since (u 1 , . . . , u n ; Z) are well adapted coordinates at x, we have λ = 0.

To prove the last statement, let us first note that x is G-invariant by (2.19). Let Y ⊂ X be Hironaka-permissible w.r.t. E and y be its generic point. Applying again (2.19), y is also G-invariant (i.e. g(I(Y)) = I(Y) ⊂ O X for every g ∈ G): with notations as in Proposition 2.22, the blow up

X ′ of X along Y is then G-invariant.
The following proposition prepares the proof of Theorem 2.36. Statement (i) is the main ingredient to get this structure theorem about the polynomial in α h. Proposition 2.29. Let x ∈ η -1 (m S ) and (u 1 , . . . , u n ; Z) be well adapted coordinates at x. For α ∈ R n >0 a weight vector, the following statements hold:

(i) the polynomial in α h ∈ (gr α S)[Z] satisfies again (G) w.r.t. the local ring (gr α S) (U 1 ,...,Un) ;

(ii) if (charS/m S = p and i 0 (α) < p), then

δ α (h; u 1 , . . . , u n ; Z) ∈ Γ α = Zα 1 + • • • + Zα n ; (iii) if charS/m S = 0 or if (charS/m S = p and i 0 (α) = p), then δ α (h; u 1 , . . . , u n ; Z) ∈ 1 p Γ α .
Proof. If δ(x) = 0, we have δ α (h; u 1 , . . . , u n ; Z) = 0 and in α h = h(Z) with notations as in the previous proof, so the proposition is trivial. Assume that δ(x) > 0. By Proposition 2.7, we have ∆ S (h; u 1 , . . . , u n ; Z) = ∅ and this polyhedron has no solvable vertex. Therefore in α h is not a p th -power. Let z ∈ L be the image of Z and ν α be any extension of µ α to L. Then ν α is centered at x, since X is G-invariant and η -1 (m S ) = {x} by Proposition 2.10(i). We have:

ν α (z) = µ α (f i,Z )/i = δ α (h; u 1 , . . . , u n ; Z) ∈ Γ α ⊗ Z Q (2.20)
for each i, 1 ≤ i ≤ p such that F i,Z,α = 0. Since L|K is normal of degree p, the reduced ramification index e 0 of ν α |µ α is e 0 = 1 or e 0 = p. Assume that (charS/m S = p and i 0 (α) = p). Then in α h is in case (c) of (G) and we get (iii) from (2.20).

Assume that charS/m S = 0 or (charS/m S = p and i 0 (α) < p). Then h is in case (a) or (b). Since G = Z/p in these cases and X is G-invariant, G acts transitively on the roots of in α h. We have:

   Tot((gr α S)[Z]/(in α h)) = να QF (gr α S) if µ α splits; QF ((gr α S)[Z]/(in α h)) = QF (gr να S) otherwise,
and this proves (i). Statement (iii) follows from (2.20) if charS/m S = 0. Assume finally that (charS/m S = p and i 0 (α) < p). By (2.20), we have

pν α (z) = pµ α (f i 0 (α),Z )/i 0 (α) ∈ Γ α .
Since Γ α ≃ Z r for some r ≥ 1, this implies

δ α (h; u 1 , . . . , u n ; Z) = µ α (f i,Z )/i 0 (α) ∈ Γ α
which completes the proof of (ii).

Corollary 2.30. Assume that a normal crossings divisor

E = div(u 1 • • • u e ) ⊂ SpecS
is specified. We have pd j ∈ N, 1 ≤ j ≤ e, and ǫ(y) ∈ N for every y ∈ X .

Proof.

In
E = div(u 1 • • • u e ).
For y ∈ X , s := η(y), we define a principal ideal:

H(y) :=   div(u j )⊆Es u H j j   ⊆ S,
where H j := pd j ∈ N.

The discriminant assumption.

Discriminant theory has been used since Jung [START_REF] Jung | Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x, y in der Umgebung einer Stelle x = a, y = b[END_REF] in order to simplify singularities. Namely the fundamental group π 1 (C n \ {x 1 • • • x e = 0}) ≃ Z e classifies unramified coverings away from the normal crossing divisor D := {x 1 • • • x e = 0}. Jung's observation that any such covering can be described by a monomial mapping C n → C n allowed to resolve the singularities of surfaces n = 2 [55] [83][61]. This method extends to positive characteristics provided no wild ramification occurs and it is the content of Abhyankar's Lemma [START_REF] Grothendieck | Revêtements étales et groupe fondamental (SGA 1, Séminaire de Géométrie Algébrique du Bois-Marie 1960/61[END_REF] Appendice I. Even when wild ramification occurs, this method induces some simplification from the general case and is the starting point of several approaches [START_REF] Bravo | Singularities in positive characteristic, stratification and simplification of the singular locus[END_REF] [START_REF] Teissier | Overweight deformations of affine toric varieties and local uniformization[END_REF].

In any characteristic, getting Jung's situation of a discriminant with normal crossings in dimension n is a consequence of embedded resolution for the discriminant subscheme which has dimension n -1. For n = 3, this is possible as embedded resolution of surfaces is known [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF]. In our problem, this reduction is stated as corollary 4.19 below.

The main result in this section is Theorem 2.36 below which plays an important role in the proof of Theorem 1.5. Indeed, Theorem 2.36 basically reduces the proof of Theorem 1.5 to computations on purely inseparable or Artin-Schreier polynomials of degree p over fields of characteristic p > 0.

We now introduce the critical locus of the map η : X → SpecS together with its scheme structure given by the discriminant D := Disc X h ∈ S. We are interested in the case where D is a normal crossings divisor.

Note that D is by definition independent of the choice of regular parameters of S and invariant by those translations X ′ := Xφ, φ ∈ Ŝ used in minimizing polyhedra. If (S, h, E) is in case (c) of (G), then D = 0. Definition 2.32. Let S, h ∈ S[X] (2.1), X and E = div(u 1 • • • u e ) be specified. We say that (S, h, E) satisfies assumption (E) if char(S/m S ) = p > 0 and one of the following properties hold:

   (i) D = 0 and η(Sing p X ) ⊆ E, (ii) D = 0 and div(D) red ⊆ E ⊆ div(p) red . (2.21)
Assumption (E) is maintained up to the end of this chapter.

This assumption implies that Sing p X ⊆ η -1 (E) ⊂ X : (i) by definition; (ii) because η -1 (SpecS\E) is regular since SpecS\E is. In particular E = ∅ if Sing p X = ∅.

Example 2.33. Let us illustrate cases (i)(ii) by examples. As h is reduced, case (i) of Definition 2.32 cannot occur when char(S) = 0. When char(S) = p > 0, the following example fits into condition (i):

h = Z p + u a 1 u b 2 f, f ∈ S = k[[u 1 , u 2 , v]], a + b ≥ p, char(k) = p > 0, with V(f ) ⊂Spec(S) regular outside E =div(u 1 u 2 ).
The following is an example of condition (ii) with char(S) = 0. Let

A := Z p [π] π n(p-1) , n ∈ N -{0}, S := A[[u 2 , u 3 ]], E = div(π).
Let µ p be the group of p-th roots of unity. Note that

µ p ⊂ Z p [π n ] ⊂ A. Let h := X p -π ap (1 + f ), f ∈ m S , a ∈ N -{0}.
Note that (S, h, E) satisfies assumption (G) (section 2.3) since µ p acts on S[X]/(h) by x → ζx. We have:

Disc X (h) = ξ,h(ξ)=0 h ′ (ξ) = p p-1 ( ξ ξ) p-1 = p p-1 π ap(p-1) (1 + f ) p-1 .
Therefore assumption (E) is satisfied. Note that the coordinates (π, u 2 , u 3 ; X) are adapted but not well adapted (Definition 2.24). To minimize the polyhedron ∆(h; π, u 2 , u 3 ; X), we first make the translation: Z := Xπ a . This leads to:

h(Z) = Z p + 1≤i≤p-1 p i π ai Z p-i -π ap f.
The monomial pπ a(p-1) Z leads to the vertex (a + n, 0, 0) whenever

max λ∈A {ord π (f (0, 0) -λ p )} ≥ n.
Other vertices depend on the expansion of f .

Assumption (E) is also preserved by those base changes considered in the previous section: formal completion S ⊆ Ŝ, localization at a prime S ⊆ S s or regular local base change S ⊆ S, S excellent. For Hironaka-permissible blowing ups, we have: Proposition 2.34. Let π : X ′ → X be a Hironaka-permissible blowing up w.r.t. E at x ∈ X . Then, with notations as in Proposition 2.22, for every s ′ ∈ σ -1 (s), (S ′ , h ′ , E ′ ) satisfies again (E).

Proof. Any Hironaka-permissible center Y ⊂ X w.r.t. E at x is contained in E by the above remarks. Therefore the proposition is obvious in case (i) of Definition 2.32.

Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x and h(Z) ∈ S[Z] be the corresponding expansion. With notations as in Proposition 2.22 and (2.18), we have h ′ (X ′ ) = u -p j 0 h(X ′ u j 0 ) for some u j 0 ∈ I(W ). We deduce that

D ′ := Disc X ′ h ′ = u -p(p-1) j 0 Disc Z h = u -p(p-1) j 0 D, hence div(D ′ ) red ⊆ E ′ ⊆ div(p) red as required.
Remark 2.35. We call the next Theorem -Reduction to characteristic p > 0to emphasize the fact that once all the statements and proofs are phrased purely in terms of initial forms with respect to certain faces of the Newton polyhedron, for the computations of the invariants after a blowing up, there is no difference between the equal and the mixed characteristic cases and they will be treated uniformly. This allows us to adapt the techniques developed in [29] [30]. Cases (1) and (2) of Theorem 2.36 are called respectively purely inseparable and Artin-Schreier.

Theorem 2.36. (Reduction to characteristic p). Assume that (S, h, E) satisfies assumptions (G) and (E). Let x ∈ η -1 (m S ) be such that ǫ(x) > 0.

Then (X , x) is analytically irreducible. Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x and α ∈ R n >0 be a weight vector. Exactly one of the following properties holds.

(

1) i 0 (α) = p, i.e. in α h = Z p + F p,Z,α ; (2) i 0 (α) = p -1 i.e. in α h = Z p + F p-1,Z,α Z + F p,Z,α , F p-1,Z,α = 0.
Furthermore, we have

-f p-1,Z = γ p-1,Z e j=1 u A p-1,j j (2.

22)

with A p-1,j ∈ (p -1)N, 1 ≤ j ≤ e, and γ p-1,Z ∈ S a unit with residue γ p-1,Z ∈ (S/m S ) p-1 . In particular, -F p-1,Z,α = G p-1 for some nonzero G ∈ gr α S, and we have cl p(p-1)δα(h;u 1 ,...,un;Z) (Disc Z (h)) =< F p p-1,Z,α > .

Proof. We start with some comments about discriminants. Let

P := Z d + a 1 Z d-1 + • • • + a d ∈ S[a 1 , • • • , a d ][Z], d ≥ 1,
be the generic polynomial defined over the domain S, a Suppose that a specialization a i ❀ āi ∈ S is given, so P ❀ P , then

D P = Disc Z ( P ) = D P (ā 1 , • • • , ād ).
We apply this to P = h. Then, denoting D := D P , we have Suppose that h is analytically reducible. By Proposition 2.26 and Definition 2.11, ǫ(x) = δ(x) -e i=1 d j is determined by ∆ Ŝ (h; u 1 , . . . , u n ; Z), thus invariant by base change S ⊆ Ŝ. Therefore it can be assumed w.l.o.g. that S = Ŝ in order to prove the first statement, i.e. that h is in case (a) of property (G). Since h splits, there is a factorization

µ α (D) ≥ p(p -1)δ α (h; u 1 , . . . , u n ; Z), since µ α (f i,Z )/i ≥ δ α (h;
h = p i=1 (Z -ϕ j ) ∈ S[Z], ϕ 1 , . . . , ϕ p ∈ S.
Let z ∈ O X be the image of Z and g ∈ G = Z/p, g = 0. By property (G), we have g(z) ∈ O X and g(z) is a root of h(Z). Up to reindexing, it can therefore be assumed that

g i (z) = z -ϕ i+1 + ϕ 1 ∈ S, 1 ≤ i ≤ p -1.
In particular, we have g(z)z = ϕ 1ϕ 2 ∈ S and we deduce that

g i (z) -z = i-1 k=0 g k (g(z) -z) = i(g(z) -z), 1 ≤ i ≤ p -1.
Since (p -1)! is a unit in S, we get a formula 1) , γ 0 ∈ S, γ 0 a unit.

D = Disc Z (h) = γ 0 (ϕ 1 -ϕ 2 ) p(p-
By assumption, (u 1 , . . . , u n ) is adapted to E. Then Definition 2.32(ii) implies that ϕ 1ϕ 2 = γu a , with γ ∈ S a unit, and a j = 0, e + 1 ≤ j ≤ n. Take an expansion (2.5):

ϕ 1 = x∈S(ϕ 1 ) γ x u x , γ x ∈ S, γ x unit
with S(ϕ 1 ) ⊂ N n finite. If x j < a j for some x ∈ S(ϕ 1 ) and some j, 1 ≤ j ≤ e, then x is a vertex of ∆ S (h; u 1 , . . . , u n ; Z) with initial form

in x h = (Z -λU x ) p , λ ∈ S/m S , λ = 0.
This is a solvable vertex: a contradiction, since ∆ S (h; u 1 , . . . , u n ; Z) is minimal. Therefore ϕ 1 ∈ (u a ) and we get ǫ(x) = 0: a contradiction. Hence (X , x) is analytically irreducible as stated. It can be assumed that h is in case (b) of property (G) from now on.

Assume now that in α h is in cases (a) or (b) of property (G), i.e. i 0 (α) < p and Disc Z (in α h) = 0.

(2.24)

We now compute ord (u j ) D for 1 ≤ j ≤ e. Let s j := (u j ) ∈ SpecS, S j := S s j and y j ∈ η -1 (s j ).

To begin with, ∆ S j (h; u j , Z) is minimal by Proposition 2.12. We denote by G(s j ) = k(s j )[U j ] the graded ring of S j w.r.t. its valuation µ j := ord (u j ) and by in j the initial form map w.r.t. µ j . Let:

γ i,j U A i,j j := in j f i,Z ∈ G(s j ), 1 ≤ i ≤ p. (2.25)
By Definition 2.32(ii), we have charS/(u j ) = p. Therefore Proposition 2.27 and (2.23) apply to S j with α = 1 ∈ R. The corresponding integer i 0 (1) is denoted by i 0 (s j ) in order to avoid confusion and we have

µ j (D) ≥ p(p -1)δ(y j ) = (p -1)H j .
(2.26)

Case 1: i 0 (s j ) < p. Then equality holds in the former formula as remarked right after (2.23).

Case 2: i 0 (s j ) = p. Then inequality is strict in the former formula. Since ∆ S j (h; u j , Z) is minimal, we have γ p,j U A p,j j ∈ G(s j ) p and A p,j = H j . Let z ∈ L be the image of Z. The discrete valuation µ j of K has a unique extension to L, still denoted by µ j . There is an embedding G(s j ) ⊂ G j , where G j is the graded ring of the valuation ring O j := {f ∈ L : µ j (f ) ≥ 0}.

Case 2a: H j ∈ pN. We have

G j = k(s j )(γ 1 p p,j )[U j ], in j z = -γ 1 p p,j U H j p j ;
(2.27)

Case 2b: H j ∈ pN. We have

G j = k(s j )[γ l j p p,j U 1 p j ], in j z = -γ 1 p p,j U H j p j , (2.28)
where l j satisfies l j H j ≡ 1 modp, since the element t := z l j u -l j H j -1 p j is a regular parameter of O j with (in j t) p = -γ l j p,j U j .

Let g ∈ G = Gal(L|K) be nontrivial. We have

g(z) p -z p + p-1 i=1 f i,Z (g(z) p-i -z p-i ) = 0.
(2.29) Since µ j (g(z)z) > µ j (z) and µ j ((p -1)!) = 0, we deduce from (2.25) and (2.27)-(2.28) that

in j (f i,Z (g(z) p-i -z p-i )) = (-1) p-i iT j γ i,j γ (p-i-1)/p p,j U (p-i-1)
H j p +A i,j j (2.30) for 1 ≤ i ≤ p -1, where T j := in j (g(z)z). On the other hand, we have

g(z) p -z p = (g(z) -z) p + p-1 i=1 p i (g(z) -z) p-i z i . (2.31)
Computing µ j (D) by the Hilbert formula [START_REF] Zariski | Commutative Algebra I[END_REF] V.11. [START_REF] Bennett | On the characteristic functions of a local ring[END_REF] gives

µ j (D) = p(p -1)µ j (g(z) -z). (2.32)
Since equality is strict in (2.26), we have µ j (H(x) -(p-1) D) > 0 and we deduce that µ j (g(z) -z) > H j /p. Computing initial forms for each term on the right hand side of (2.31), we get for 1 ≤ i ≤ p -1: is a unit in S j .

in j ((g(z) -z) p-i z i ) = (-1) i T p-i j γ i p p,j U i H j p j . Since µ j (g(z) -z) > H j /p and µ j ( p i ) = µ j (p), 1 ≤ i ≤ p -1,
Let γ ∈ k(s j ) be its residue, so the family (γγ

p-1 p p,j , {γ p-i-1 p p,j } 1≤i≤p-1
) is a basis of the k(s j )-vector space k(s j )(γ 1/p p,j ). Tracing back to (2.29) an (2.31), the value of (g(z)z) p is the value of a sum of terms with linearly independent initial forms in G j . We deduce the formula µ j (g(z)-z) This is a contradiction, since it is assumed that ǫ(x) > 0.

We thus have pδ α (h; u 1 , . . . , u n ; Z) -e j=1 H j α j > 0. By (2.37), this implies i 0 (α) = p -1, since i 0 (α) ≤ p -1 was assumed in (2.24).

We may now sharpen (2.37) as follows, since it is an equality: equality holds in (2.36) and the minimum on the right hand side of (2.35) is achieved with i = i 0 (α) = p -1 for each j, 1 ≤ j ≤ e. These two properties are equivalent to the existence of an expansion (2.22) with γ p-1,Z ∈ S a unit.

By Proposition 2.29(i), G = Z/p acts on the roots of in α h. Let

z α ∈ (gr α S)[Z]/(in α h)
be the image of Z. Then (g(z α )z α ) p-1 + F p-1,Z,α = 0 for g ∈ G nontrivial, so the polynomial X p-1 +F p-1,Z,α is totally split over gr α S, i.e. -F p-1,Z,α is a (p -1) th in gr α S as required. The last formula in the theorem is obvious.

Adapted differential structure.

The use of differentials in the local study of singularities has a long history. We include a short summary of this exciting topic where the case of local rings of positive characteristic is put forward. The Jacobian criterion for regularity was formulated by O. Zariski in localizations of polynomial rings [START_REF] Zariski | The concept of a simple point of an abstract algebraic variety[END_REF] and by M. Nagata in localizations of formal power series rings [START_REF] Nagata | A Jacobian criterion of simple points[END_REF]. Differential operators are used for computing the HS-stratum in the works of B. Bennett [START_REF] Bennett | On the characteristic functions of a local ring[END_REF], H. Hironaka [START_REF] Hironaka | Additive groups associated with points of a projective space[END_REF], J. Giraud [START_REF]Contact maximal en caractéristique positive[END_REF] and M.J. Pomerol [START_REF] Pomerol | Sur la strate de Samuel du sommet d'un cne en caractŕistique positive[END_REF].

Hironaka's theory of maximal contact is differential in nature and its positive characteristic version was developed by J. Giraud [START_REF]Contact maximal en caractéristique positive[END_REF]. In a formal power series rings R = k[[X 0 , . . . , X n ]], chark = p > 0, let J ⊂ R define a subscheme X ⊂ Z = SpecR, x denote the closed point. There exists an associated scheme W ⊂ Z with "maximal contact" in the following sense [START_REF]Contact maximal en caractéristique positive[END_REF] Proposition 3.3, Theorem 5.2 and Corollary 5.4: 1-every permissible center Y ⊂ X is also permissible for W ; 2-this property is stable at every x ′ ∈ X ′ , x ′ near to x, where X ′ → X is the blowing up along Y .

Roughly speaking, the space W is constructed by taking a projection transverse to the tangent cone of J and applying transverse differential operators of appropriate order. The scheme W has a "simpler" singularity in the sense that its tangent cone C x W coincides with the ridge of C x X, viz. [39] 1.5. It is worth noting however that, unlike for chark = 0, W is not regular at x in general. Furthermore, W = X when the tangent cone C x X coincides with its ridge, i.e. its defining equations are additive polynomials:

λ 0 X p α 0 + • • • + λ n X p α n , λ 1 , . . . , λ n ∈ k, α ≥ 0.
New ideas were introduced with H. Hironaka's characteristic algebras [51] [52]. Hironaka and Giraud's ideas have been influential in the last decade. Introducing independent new ideas of their own, H. Kawanoue and K. Matsuki defined and studied idealistic filtrations [START_REF] Kawanoue | Toward resolution of singularities over a field of positive characteristic I. Foundation; the language of the idealistic filtration[END_REF] [57]. Giraud's result was extended to algebraic varieties over perfect fields by A. Benito, A. Bravo and O. Villamayor [10] [11] [14]. Furthermore, they introduced a differential Rees algebra encoding this extended Giraud space W and whose behavior by blowing up is traced by techniques of elimination theory developed by the authors. Both approaches have produced new proofs of resolution of singularities for surfaces [START_REF] Kawanoue | Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor[END_REF] [START_REF] Villamayor | Techniques for the study of singularities with applications to resolution of 2-dimensional schemes[END_REF]. C. Abad gave a relative version of Zariski's Jacobian criterion over regular rings of characteristic p > 0 with an absolute p-basis. He also used absolute differential operators for computing the singular locus of differential Rees algebras for varieties over a possibly non perfect field [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] Proposition 5.1 and Theorem 7.5.

In this section, we introduce the differential structure on the graded algebras gr α S. We will only consider here the case α = 1 ∈ R J >0 with notations as in Definition 2.3. These algebras appear naturally as blow up algebras of S along regular primes. Our construction uses formal coordinates and Nagata derivatives. For the reader's convenience, we remind the main concepts and classical results used here, refering to [START_REF] Matsumura | Commutative ring theory[END_REF] pp.201-205 and pp. 235-245 when necessary.

The final part of the section is devoted to practical computations. These can be performed without using formal coordinates when the exceptional divisor E is locally of finite type over some field, vid. Remark 2.48 below and following propositions.

To state the main goal of this section, assume for simplicity that some triple (S, h, E) is specified as in Definition 2.32. Assume furthermore that a permissible center Y ⊂ X at x w.r.t. E is specified, where x ∈ X is the closed point. Let W := η(Y) ⊂ SpecS. We will construct a certain O W -module V(F, E, W ) (Definition 2.42) and a certain O W -module J (F, E, W ) (Definition 2.45), where O W is the completion of O W . Factoring out a monomial part from these modules, we obtain V (F, E, W ) and J(F, E, W ) in (2.49).

The origin of these modules sits in Cossart's thesis [START_REF] Cossart | Sur le polyèdre caractéristique d'une singularité[END_REF] where resolution of singularities is proved for hypersurfaces with equation

h = y p -f (u 1 , u 2 , u 3 ) = 0
over fields of characteristic p > 0, see also [START_REF]Forme normale d'une fonction sur une surface de caractéristique positive[END_REF] for the case of surfaces. Starting with a point of multiplicity p of the hypersurface h = 0, making it drop by permissible blowing ups is very close to resolving the singularities of the form df . With notations as above, the ideal V(f, E) generated by the coefficients of df ∈ Ω S (log E) is a reasonable invariant for resolution and the goal is to get V(f, E) locally principal by blowing up regular centers W ⊂ SpecS. The transformation law for df ∈ Ω S (log E) involves a certain Jacobian ideal J (f, E, W ).

In our present -not necessarily equicharacteristic-setup, some initial form modules V(F, E, W ) and J (F, E, W ) can be defined from the algebra gr W S and initial form polynomial in W h, see Notation 2.37 below. Taking W = {m S }, we will define in forthcoming sections a numerical invariant ω(x) ∈ N (Definition 2.68) and a stratum Max(x) in the tangent cone. The corresponding transformation law is the forthcoming Blowup Formula Proposition 3.9(v).

We now proceed with formal definitions and constructions. As usual, (S, m S ) is an excellent regular local ring with residue field

k := S m S , chark = p > 0. A r.s.p. (u 1 , . . . , u n ) of S and a normal crossings divisor E = div(u 1 • • • u e ) are specified, 1 ≤ e ≤ n. We assume that char S (u j ) = p, 1 ≤ j ≤ e,
which is implied by assumption (E), Definition 2.32. We first adapt and simplify notations as much as possible in order to fit with the forthcoming computations.

Notation 2.37. Let W ⊆ E be a regular closed subset of SpecS having normal crossings with E. For some suitable r.s.p. (u 1 , . . . , u n ) adapted to E as above, we may write I(W ) := I J = ({u j } j∈J ) ⊂ S for some J ⊆ {1, . . . , n}.

Let J E := J ∩ {1, . . . , e}, J ′ := {1, . . . , n}\J, so (J ′ ) E = {1, . . . , e}\J E . Let O W := S/I(W ) and u j ′ ∈ O W be the image of u j ′ , j ′ ∈ J ′ , so

m S := m O W = ({u j ′ } j ′ ∈J ′ ).
The m S -adic completion of O W is denoted by O W . The algebra gr 1 S of Definition 2.3 is denoted by:

G(W ) := gr I(W ) S ≃ O W [{U j } j∈J ].
Since W ⊆ E, we have:

char G(W ) = chark = p > 0.
The initial form in 1 h w.r.t. the weight vector 1 ∈ R J >0 is now denoted

in W h = X p + p i=1 F i,X,W X p-i ∈ G(W )[X],
with

F i,X,W ∈ G(W ) iδ 1 (h;u 1 ,...,un;X) , 1 ≤ i ≤ p.
Any local equation of E has an initial form in G(W ), and we denote by E(W ) the associated divisor. Explicitly:

E(W ) := div   j∈J E U j j ′ ∈(J ′ ) E u j ′   ⊂ SpecG(W ).
(2.38)

We include in these definitions the case where W = div(u j ) is an irreducible component of E. This corresponds to (J ′ ) E = {1, . . . , e}\{j} and

G(W ) = S/(u j )[U j ], E(W ) = div   U j j ′ ∈(J ′ ) E u j ′   .
We now recall the notion of p-basis and its connection with differentials and derivatives. Definition 2.38. Let (λ l ) l∈Λ be a family of elements of k. A p-monomial on (λ l ) l∈Λ is any element of the form:

l∈Λ λ i l l , 0 ≤ i l ≤ p -1, i l = 0 for almost all l ∈ Λ.
The family (λ l ) l∈Λ is called an absolute p-basis of k if the family of all pmonomials on (λ l ) l∈Λ is a basis of the k p -vector space k.

This condition can be restated in terms of absolute differentials [START_REF] Matsumura | Commutative ring theory[END_REF] Theorem 26.5: Proposition 2.39. Let (λ l ) l∈Λ be a family of elements of k. The following properties are equivalent:

(1) (λ l ) l∈Λ is an absolute p-basis of k;

(2) (dλ l ) l∈Λ is a basis of the k-vector space of absolute differentials Ω k .

In particular, this proves that absolute p-bases of k do exist. The corresponding family of derivations is denoted by ( ∂ ∂λ l ) l∈Λ . They are defined by

∂λ l ′ ∂λ l = δ l,l ′ , l, l ′ ∈ Λ
where δ l,l ′ is the Kronecker symbol.

For A a ring and M an A-module, we denote by Der(A, M) the A-module of derivations of A with values in M. The module Der(A, A) is simply denoted by Der(A). For every k-vector space M, we have:

Der(k, M) = Hom k (Ω k , M) ≃ M Λ (2.
W ∼ -→k[[{X j ′ } j ′ ∈J ′ ]]
such that φ(γ l ) = λ l for l ∈ Λ, Φ(u j ′ ) = X j ′ for j ′ ∈ J ′ , and φ induces the identity map k = O W /m S -→ k on residue classes.

A slight abuse of notations allows us to write

∂ ∂u j ′ := φ -1 • ∂ ∂X j ′ • φ ∈ Der( O W ). Let D ∈ Der(k, k[[{X j ′ } j ′ ∈J ′ ]]) act coefficientwise on k[[{X j ′ } j ′ ∈J ′ ]], i.e. D •   a∈N J ′ µ a X a   = a∈N J ′ (D • µ a )X a .
The isomorphism φ then provides an inclusion

Der(k, k[[{X j ′ } j ′ ∈J ′ ]]) ⊆ Der( O W ), D → φ -1 • D • φ
and its image will be simply denoted by Der(k, O W ) (called "derivations w.r.t. to constants"). Collecting together, we have a decomposition

Der( O W ) = Der(k, O W ) ⊕ j ′ ∈J ′ O W ∂ ∂u j ′ . (2.40)
This is because derivations of O W are continuous for the m S -adic topology, so they are determined by their action on coefficients and variables. Let:

G(W ) := O W ⊗ O W G(W )≃ O W [{U j } j∈J ].
(2.41)

We now introduce the G(W )-module of absolute derivations of G(W ) which respect the logarithmic structure given by E(W ), viz. (2.38). Definition 2.41. With notations as above, let:

D(W ) := {D ∈ Der( G(W )) : D • I(E(W )) ⊆ I(E(W ))}. Once an isomorphism φ : O W ∼ -→k[[{X j ′ } j ′ ∈J ′ ]] has been chosen (Proposi-
B(W ) := {U j ∂ ∂U j } j∈J E , { ∂ ∂U j } j∈J\J E , {u j ′ ∂ ∂u j ′ } j ′ ∈(J ′ ) E , { ∂ ∂u j ′ } j ′ ∈J ′ \(J ′ ) E . (2.42)
Since S W is excellent and integrally closed, we have

( O W ) p ∩O W = (O W ) p . In particular, we get G(W ) p ∩ G(W ) = G(W ) p . Therefore for F ∈ G(W ),
there is an equivalence:

∀D ∈ D(W ), D • F = 0 ⇔ F ∈ G(W ) p .
(2.43)

Definition 2.42. Let F ∈ G(W ) d be homogeneous of degree d ≥ 1.
We define a homogeneous O W -submodule of G(W ) d-1 as follows:

V(F, E, W ) := j∈J\J E O W ∂F ∂U j ⊆ G(W ) d-1 . (2.44) Proposition 2.43. Let F ∈ G(W ) d . The O W -module V(F, E, W
) is independent of the choice of an isomorphism φ as in Proposition 2.40 and of an adapted r.s.p. (u 1 , . . . , u n ) of S such that I(W ) = ({u j } j∈J ). Furthermore, we have:

j∈J\J E O W ∂F ∂U j = O W ⊗ O W V(F, E, W ),
where the left hand side module is computed in G(W ) d .

Proof. Obvious from the definitions.

Definition 2.44. With notations as above, let:

D W := {D ∈ D(W ) : D • I(W ) I(W ) 2 ⊆ I(W ) I(W ) 2 } ⊆ D(W ).
Once an isomorphism φ :

O W ∼ -→k[[{X j ′ } j ′ ∈J ′ ]] has been chosen, D W
is generated as a G(W )-module by Der(k, G(W )) together with the finite family

B W := {U j ∂ ∂U j } j∈J E , {U j 1 ∂ ∂U j } j∈J\J E ,j 1 ∈J , {u j ′ ∂ ∂u j ′ } j ′ ∈(J ′ ) E , { ∂ ∂u j ′ } j ′ ∈J ′ \(J ′ ) E . (2.45)
Note that there is an equivalence

D W = D(W ) ⇔ W is an intersection of components of E. (2.46) Definition 2.45. Let F ∈ G(W ) d be homogeneous of degree d ≥ 1. We define a homogeneous O W -submodule of G(W ) d = O W ⊗ O W G(W ) d as follows: J (F, E, W ) := cl d (D W • F ) ⊆ G(W ) d .
(2.47)

The J (F, E, W )-version of Proposition 2.43 goes as follows:

Proposition 2.46. Let F ∈ G(W ) d . The O W -module J (F, E, W
) is independent of the choice of an isomorphism φ as in Proposition 2.40 and of an adapted r.s.p.

(u 1 , . . . , u n ) of S such that I(W ) = ({u j } j∈J ).
For any such choice of φ and (u 1 , . . . , u n ), there exists a finite subset

Λ F ⊆ Λ such that J (F, E, W ) = l∈Λ F O W ∂F ∂λ l + D∈B W O W (D • F ).
Proof. The first statement is trivial since D(W ) and D W do not depend on any choice of φ and (u 1 , . . . , u n ). To prove the second part of the proposition, we let:

J 0 (F, E, W ) = l∈Λ O W ∂F ∂λ l + D∈B W O W (D • F ) ⊆ J (F, E, W ). Since G(W ) d is a finite module over the Noetherian ring O W , it is sufficient to prove that J 0 (F, E, W ) = J (F, E, W ). Let n ∈ N be fixed. We expand F = |i|=d |a|≤n λ i,a u a U i + F n , with F n ∈ m n+1 S G(W ) d and λ i,a ∈ k (via the isomorphism φ). Since (λ l ) l∈Λ is an absolute p-basis of k, there exists a finite subset Λ 0 ⊂ Λ such that λ i,a = M ∈M 0 (λ i,a,M ) p M, λ i,a,M ∈ k,
where M 0 is the finite family of all p-monomials on (λ l 0 ) l 0 ∈Λ 0 . Let D ∈ Der(k, O W ). We deduce that

D • F ∈ J 0 (F, E, W ) + m n+1 S G(W ) d .
Since this holds for arbitrary n ≥ 0,

D • F belongs to the topological closure of J 0 (F, E, W ) in G(W ) d for the m S -adic topology of G(W ) d . Therefore D • F ∈ J 0 (F, E, W ) as required [65] Theorem 8.6.
Let H W be the initial form in G(W ) of the monomial ideal H(x) ⊆ S (Definition 2.31), where x ∈ η -1 (m S ), i.e.

H W :=   j∈J E U H j j j ′ ∈(J ′ ) E u H j ′ j ′   ⊆ G(W ) d W , (2.48)
where

d W := j∈J E H j . For F ∈ H W G(W ) d-d W , it follows from the above definitions that V(F, E, W ) ⊆ H W G(W ) d-d W -1 and J (F, E, W ) ⊆ H W G(W ) d-d W .
For such

F ∈ H W G(W ) d-d W , we denote:    V (F, E, W ) := H -1 W V(F, E, W ) ⊆ G(W ) d-d W -1 , J(F, E, W ) := H -1 W J (F, E, W ) ⊆ G(W ) d-d W .
(2.49)

For F = F p,X,W ∈ H W G(W ) d-d W , this defines the submodules V (F p,X,W , E, W ) ⊆ G(W ) d-d W -1 and J(F p,X,W , E, W ) ⊆ G(W ) d-d W .
We will continually apply this definition when the following properties (i) and (ii) hold:

(i) (u 1 , . . . , u n ; X) are well adapted coordinates at x ∈ η -1 (m S ) (Definition 2.24), and

(ii) d -d W = ǫ(y) with η -1 (s) = {y}, s the generic point of W . Note that F p,X,W ∈ H W G(W ) d-d W is then a consequence of Definition 2.25
and Proposition 2.26. Some considerations will require localizing S at some point s ∈ W . We then denote by W s the stalk of W at s. This notation is used jointly with Notation 2.23 sqq. about the stalk E s . The restriction of s is denoted by

s ∈ SpecO W = G(W ) 0 . We have G(W s ) = gr I(Ws) S s ≃ (O W ) s [{U j } j∈J ]. Example 2.47. S := k[u 1 , u 2 , u 3 ] (u 1 ,u 2 ,u 3 ) , char(k) = p > 0, E =div(u 1 u 2 ), W = {m S }, x ∈ SpecS[Z] is the point of parameters (Z, u 1 , u 2 , u 3 ). Let us study two different equations: h 1 := Z p + u a 1 u b 2 (u p+1 3 + φ) ∈ S[Z], a, b ∈ N >0 , φ ∈ m p+2 S , h 2 := Z p + u a 1 u b 2 (u p 3 + φ) ∈ S[Z], a, b ∈ N >0 , a or b = 0mod p, φ ∈ m p+1 S , in both cases, H W :=< U a 1 U b 2 >, d W = a+b, G(W ) = G(W ) = k[U 1 , U 2 , U 3 ].
In the first case,

in W h 1 = X p + U a 1 U b 2 U p+1 3 , d = a + b + p + 1, F = U a 1 U b 2 U p+1 3 ∈ k[U 1 , U 2 , U 3 ] d , V(F, E, W ) =< U a 1 U b 2 U p 3 >⊆ k[U 1 , U 2 , U 3 ] d-1 , V (F, E, W ) =< U p 3 >, J (F, E, W ) =< U a 1 U b 2 U p+1 3 >⊆ k[U 1 , U 2 , U 3 ] d , J(F, E, W ) =< U p+1 3 > .
In the second case,

in W h 2 = X p + U a 1 U b 2 U p 3 , d = a + b + p, F = U a 1 U b 2 U p 3 ∈ k[U 1 , U 2 , U 3 ] d , V(F, E, W ) =< 0 >⊆ k[U 1 , U 2 , U 3 ] d-1 , V (F, E, W ) =< 0 >, J (F, E, W ) =< U a 1 U b 2 U p 3 >⊆ k[U 1 , U 2 , U 3 ] d , J(F, E, W ) =< U p 3 > .
Remark 2.48. Formal coordinates and Nagata derivatives can be avoided if one assumes that

E = Spec(S/(u 1 • • • u e )) ⊂ SpecS (2.50)
is essentially of finite type over some field. We explain below how Zariski's Jacobian criterion [START_REF] Matsumura | Commutative ring theory[END_REF] Theorem 30.5 (2) can be used to avoid introducing formal coefficients in defining J (F, E, W ). We do not know any such alternative description for arbitrary excellent regular local rings of characteristic p > 0. We point out recent developments due to C. Abad [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] who extends the existence of p-basis and validity of the Jacobian criterion to affine neighborhoods of a regular point (instead of its local ring). The extra assumption (2.50) is satisfied for example when E is contained in the closed fiber of some previously performed blowing ups. In dimension three, this extra property is easily achieved from embedded resolution theorems in smaller dimensions, vid. Lemma 4.14.

For the remainder of this section, we consider a field k, chark = p > 0, and an absolute p-basis (λ l ) l∈Λ . Let S be a regular local ring which is essentially of finite type over k. This means that for some polynomial ring R := k[X 1 , . . . , X N ], we have

S := R IR P , I ⊂ R an ideal, P ∈ SpecR, P ∈ V (I). Let x i ∈ S denote the image of X i , 1 ≤ i ≤ N. Let:
k(P ) := S/P, t := tr.deg k k(P ), r := ht(IR P ), n := dimS = Nrt.

The following is the necessary condition in Zariski's Jacobian criterion for regularity applied to S [65] Theorem 30.5.

Proposition 2.49. (Zariski) With notations as above, there exists F 1 , . . . , F r ∈ I and a finite subset Φ ⊆ Λ such that the Jacobian matrix

J(F 1 , . . . , F r ; { ∂ ∂λ l } l∈Φ , ∂ ∂X 1 , . . . , ∂ ∂X n+r )
has a r × r minor with nonzero residue in k(P ).

The following proposition is merely a rewriting of Zariski's Jacobian criterion for regularity from the point of view of explicit computations. Its proof is elementary linear algebra. Proposition 2.50. With notations as above, there exists finite subsets

Φ ⊆ Λ, Ψ ⊆ {1, . . . , N}, | Ψ |= t+ | Φ |
with the following properties:

(1) let Λ ′ := (Λ\Φ) Ψ and define

λ ′ l ′ :=    λ l if l ′ ∈ Λ\Φ x l ′ if l ′ ∈ Ψ ; then the residue family (λ ′ l ′ ) l ′ ∈Λ ′ of (λ ′ l ′ ) l ′ ∈Λ ′ in k(P ) is an absolute p- basis of k(P ).
(2) for every r.s.p. (u 1 , . . . , u n ) of S, the family ((dλ ′ l ′ ) l ′ ∈Λ ′ , du 1 , . . . , du n ) is a basis of the free module Ω S of absolute differentials;

(3) the family of all p-monomials on

((λ ′ l ′ ) l ′ ∈Λ ′ , u 1 , . . . , u n ) is a basis of the free S p -module S.
Proof. First choose x i 1 , . . . , x it whose residues in k(P ) are a transcendence basis of k(P ) over k. We may replace k with = k(x i 1 , . . . , x it ), Λ with Λ {i 1 , . . . , i t } and {1, . . . , N} with {1, . . . , N}\{i 1 , . . . , i t } and thus assume that P is a maximal ideal.

We first prove the proposition when I = (0), so S = R P . Since (1) only refers to the residue field k(P ), we will only have to prove (2) and (3) for arbitrary I.

By elementary field theory, e.g. [START_REF] Matsumura | Commutative ring theory[END_REF] Theorem 5.1, P = (G 1 , . . . , G N ), where

G j = X m j j + m j i=1 G j,i (X 1 , . . . , X j-1 )X m j -i j , m j ≥ 1, G j,i ∈ k[X 1 , . . . , X j-1 ] for 1 ≤ j ≤ N. We have Ω R P = l∈Λ R P dλ l ⊕ N j=1 R P dX j .
(2.51)

We use induction on j,

1 ≤ j ≤ N, to construct finite subsets Φ j ⊆ Λ and Ψ j ⊆ {1, . . . , j}, | Ψ j |=| Φ j | such that {dλ l } l∈Λ\Φ j , {dX i } i∈Ψ j , dG 1 , . . . , dG j , dX j+1 , . . . , dX N (2.52)
is a basis of Ω R P , and the residue family

(λ l ) l∈Λ\Φ j , {dX i } i∈Ψ j (2.53)
form an absolute p-basis of k j := k[X 1 , . . . , X j ]. Take Φ 0 = ∅ to begin with and assume that Φ j-1 and Ψ j-1 have been constructed. Apply the following algorithm:

(A1) if

∂G j ∂X j = 0, take Φ j = Φ j-1 , Ψ j = Ψ j-1 ; otherwise go to (A2); (A2) pick i, 1 ≤ i ≤ m j such that µ j,i := G j,i (X 1 , . . . , X j-1 ) ∈ k p j-1 and go to (A3); (A3) choose any l ′ j ∈ (Λ\Φ j-1 ) Ψ j-1 such that ∂µ j,i ∂λ l ′ j = 0 or ∂µ j,i ∂X l ′ j = 0; take (Φ j = Φ j-1 ∪ {l ′ j }, Ψ j = Ψ j-1 ∪ {j}), or (Φ j = Φ j-1 , Ψ j = (Ψ j-1 \{l ′ j }) ∪ {j}) accordingly. The natural map k j ⊗ k j-1 Ω k j-1 -→ Ω k j is an isomorphism when step (A1) applies. When step (A2) applies, there is an exact sequence 0 -→ k j -→ k j ⊗ k j-1 Ω k j-1 -→ Ω k j -→ k j -→ 0.
Step (A3) then chooses a splitting 1 → dX j of the cokernel and a nonzero coefficient w.r.t to the basis (1⊗λ l ) l∈Λ\Φ j-1 , {1⊗dX i } i∈Ψ j-1 for the generator of the kernel m j i=1 X m j -i j ⊗ dG j,i (X 1 , . . . , X j-1 ). Applying (2.53) for j = N, this completes the proof of (1). Applying (2.51) together with (2.52) for j = N, we get (2) for I = (0). For I arbitrary, there is an exact sequence

0 -→ IR P (IR P ) 2 ⊗ S k(P ) -→ Ω R P ⊗ R P k(P ) -→ Ω S ⊗ S k(P ) -→ 0,
where exactness on the left holds because S is regular. Taking preimages of u 1 , . . . , u n in R P defines a splitting on the right and we get (2) for arbitrary I.

Finally, we consider the derivations

( ∂ ∂λ ′ l ′ ) l ′ ∈Λ ′ , ∂ ∂u 1 , . . . , ∂ ∂u n ∈ Der(S) (2.54)
corresponding to the basis of the free module Ω S given by (2). Let K be the quotient field of S. By Proposition 2.39, the family of all p-monomials on

((λ ′ l ′ ) l ′ ∈Λ ′ , u 1 , . . . , u n ) is a basis of the K p -vector space K. Let f ∈ S and f = a,b (f a,b ) p λ ′ a u b
be the corresponding expansion, where f a,b ∈ K. Applying the derivations in (2.54) and arguing by induction w.r.t. the graded lexicographical ordering, we get

f a,b ∈ S ∩ K p = S p .
This concludes the proof of (3).

Example 2.51. Let k 0 be a perfect field of characteristic p > 0,

k := k 0 (λ 1 , λ 2 ), λ 1 , λ 2 indeterminates. Take R = k[X 1 , X 2 ], I = (F ), with F := X p 1 + λ 1 X p 2 + λ 2 , P := (X p 1 -λ 1 λ 2 , X p 2 + λ 2 + λ 2 λ 1 ).
The above algorithm leads to:

λ ′ 1 = x 1 , λ ′ 2 = x 2 , u 1 = x p 1 -λ 1 λ 2 .
The reader may check that

k(P ) = k 0 (x 1 , x 2 )[λ 1 ], λ 2 1 + x 1 x 2 p λ 1 + x 1 x 2 p = 0,
and that the residue class map gives an isomorphism k(P ) ≃ k

1 p .
We now go back to the framework of the beginning of this section, see Notation 2.37, Definition 2.45 and Proposition 2.46.

Proposition 2.52. Assume that O W is essentially of finite type over some field. Let (b a ) a∈A be a family of elements of O W containing {u j ′ } j ′ ∈(J ′ ) E , and such that the family (db a ) a∈A forms a basis of the free

O W -module Ω O W . Write b a j ′ = u j ′ for j ′ ∈ (J ′ ) E . Let F ∈ G(W ) d and define: J ′ (F, E, W ) := {U j ∂F ∂U j } j∈J E , {U j 1 ∂F ∂U j } j∈J\J E ,j 1 ∈J , {u j ′ ∂F ∂u j ′ } j ′ ∈(J ′ ) E , { ∂F ∂ba } a∈A\{a j ′ } j ′ ∈(J ′ ) E ⊆ G(W ) d .
(2.55)

Then J (F, E, W ) = O W ⊗ O W J ′ (F, E, W ) ⊆ G(W ) d .
Proof. Applying Proposition 2.50(1)(2), we may assume that

A = Λ ′ J ′ , (b a ) a∈A = ((λ ′ l ′ ) l ′ ∈Λ ′ , (u j ′ ) j ′ ∈J ′ ).
Proposition 2.40 provides an associated isomorphism

φ : O W ∼ -→k[[{X j ′ } j ′ ∈J ′ ]]. Let f ∈ O W . By Proposition 2.50(3), there is a finite expansion f = a,b (f a,b ) p λ ′ a u b in terms of p-monomials, f a,b ∈ O W .
Applying φ to this equation, the current proposition follows directly from Definition 2.45 and Proposition 2.46.

Cones, ridge and directrix.

In this section, we recollect some facts about the directrix and Hilbert-Samuel stratum of a homogeneous ideal. These facts are then applied to extract numerical invariants from the vector spaces

V (F p,Z , E, m S ) ⊆ G(m S ) ǫ(x)-1 and J(F p,Z , E, m S ) ⊆ G(m S ) ǫ(x)
defined in the previous section (2.49) when (u 1 , . . . , u n ; Z) are well adapted coordinates at x ∈ η -1 (m S ). These considerations are based on elementary linear algebra. Theorem 2.36 distinguishes between two different cases for in m S h: (1) purely inseparable, (2) Artin-Schreier. Both vector spaces V (F p,Z , E, m S ) and J(F p,Z , E, m S ) are easily seen to be independent of the well adapted coordinates (u 1 , . . . , u n ; Z) in case (1) (Proposition 2.57(iii)). However, in case (2), they do depend on (u 1 , . . . , u n ; Z): see Example 2.60. To extract relevant information, we use a truncation map T (Definition 2.59) which kills all monomials in the expansion of F p,Z which may vary with (u 1 , . . . , u n ; Z). The relevant information is provided in Proposition 2.65.

Most difficulties in this section appear only for n ≥ 4, which will eventually lead us to define our main invariant ω(x) in a different way than in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 1 (for equicharacteristic S of dimension n = 3) in the next section.

Let k be a field, R 1 be a k-vector space of finite dimension n ≥ 1 and R := k[R 1 ] be the symmetric algebra. Let V := SpecR and I be a homogeneous ideal of R which defines a cone C = C(I) := Spec(R/I). With these notations, we define:

Definition 2.53. The directrix Vdir(I) of C = C(I) is the smallest k-vector subspace W of R 1 such that I = (I ∩ k[W ])R. We denote τ (I) := dim k Vdir(I), Dir(I) := Spec(R/(Vdir(I))).
Definition 2.54. Let C = C(F ) be a hypersurface cone, i.e. F is an homogeneous polynomial and I = (F ) is a nonzero principal ideal. We define a reduced subcone

Max(F ) := {x ∈ V : ord x F = ord 0 F } ⊆ C(F ),
where 0 is the origin (so ord 0 F = degF ).

Given a fixed degree d ≥ 1 and an ideal

I = (F 1 , . . . , F m ) ⊂ R defined by homogeneous polynomials F 1 , . . . , F m ∈ R, degF i = d for 1 ≤ i ≤ m, we let Max(I) := {x ∈ V : ord x F i = d, 1 ≤ i ≤ m} ⊆ C(I).
The cone Max(I) is the closed Hilbert-Samuel stratum of C(I). These two objects and the ridge are considered and connected by H. Hironaka in a more general context. See also [START_REF]Étude locale des singularités, Cours de 3 ème cycle[END_REF] [39] [START_REF] Pomerol | Sur la strate de Samuel du sommet d'un cne en caractŕistique positive[END_REF] for definition and computation of the ridge and Hilbert-Samuel stratum.

Proposition 2.55. (Hironaka) [START_REF] Hironaka | Additive groups associated with points of a projective space[END_REF] Let C = C(F ) be a hypersurface cone. There are inclusions

Dir(F ) ⊆ Max(F ) ⊆ C(F ).
If k is perfect or if dimR ≤ p + 1, the left hand side inclusion is an equality.

Remark 2.56. Counterexamples to the last statement exist for non-perfect k and dimR > p + 1. For dimR ≤ 4, such counterexamples exist only if dimR = 4 and p = 2. For applications to the proof of Theorem 1.5, we only have to deal with this difficulty for the initial form polynomial (dimR = 4) which is of the form

in m S h = Z 2 -λU 1 Z + F 2,Z , F 2,Z ∈ S/m S [U 1 , U 2 , U 3 ] 2 , λ ∈ S/m S .
By [START_REF] Hironaka | Additive groups associated with points of a projective space[END_REF], the polynomial in m S h is a counterexample to the last statement in Proposition 2.55 if and only if λ = 0 and, up to a linear change of variables,

in m S h = Z 2 + λ 2 U 2 1 + λ 1 U 2 2 + λ 1 λ 2 U 2 3 (2.56) with λ 1 , λ 2 2-independent, i.e. [(S/m S ) 2 (λ 1 , λ 2 ) : (S/m S ) 2 ] = 4.
This very special case is dealt with in Proposition 5.8.

Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x ∈ η -1 (m S ) (Definition 2.24). In case ǫ(x) > 0, we have η -1 (m S ) = {x}, k(x) = S/m S (Proposition 2.10) and the initial form polynomial has the form

in m S h = Z p -G p-1 Z + F p,Z ∈ G(m S )[Z] = S/m S [U 1 , . . . , U n ][Z] (2.57) by Theorem 2.36 applied to α = 1 ∈ R n >0 . There is an associated integer i 0 (x) = p -1 (resp. i 0 (x) = p) if G = 0 (resp. if G = 0). We denote by H ⊆ G(m S ) d the initial form vector space of the ideal H(x), d = e j=1 H j (Definition 2.31). If i 0 (x) = p -1, we have H -1 G p =< e j=1 U pB j j >, B j ∈ 1 p
N and e j=1 pB j = ǫ(x).

(2.58)

We can restate previous material as follows:

Proposition 2.57. Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x ∈ η -1 (m S ) and assume that ǫ(x) > 0. The following statements hold:

(i) the vector space V (F p,Z , E, m S ) ⊆ G(m S ) ǫ(x)-1 satisfies V (F p,Z , E, m S ) = 0 ⇔ F p,Z ∈ S/m S [U 1 , . . . , U e ][U p e+1 , . . . U p n ]; (ii) the vector space J(F p,Z , E, m S ) ⊆ G(m S ) ǫ(x) satisfies J(F p,Z , E, m S ) = 0 ⇔ F p,Z ∈ (S/m S [U 1 , . . . , U n ]) p ;
(iii) if i 0 (x) = p, the vector space V (F p,Z , E, m S ) is independent of the well adapted coordinates (u 1 , . . . , u n ; Z); if i 0 (x) = p and V (F p,Z , E, m S ) = 0, the vector space J(F p,Z , E, m S ) ǫ(x) is independent of the well adapted coordinates (u 1 , . . . , u n ; Z).

Proof. The first statement follows from (2.44) and (2.49), while (ii) follows from (2.43). Assume now that i 0 (x) = p, i.e. G = 0.

To begin with, the situation in (ii) does not occur because the polyhedron

∆ Ŝ (h; u 1 , . . . , u n ; Z) is minimal. If Z ′ = Z -θ, θ ∈ Ŝ with ord m S θ ≥ δ(x)/p, we have F p,Z ′ = F p,Z + Θ p for some Θ ∈ S/m S [U 1 , . . . , U n ] δ(x)/p (so Θ = 0 if δ(x) ∈ N). Hence D • F p,Z ′ = D • F p,Z for every D ∈ Der(G(m S )).
By elementary calculus, the vector space

V (F p,Z , E, m S ) = H -1 < ∂F p,Z ∂U j e+1≤j≤n >
is unchanged by adapted coordinate change (more generally by changes stabilizing the vector space < U 1 , . . . , U e >) and this proves the first statement in (iii). If V (F p,Z , E, m S ) = 0, the vector space

J(F p,Z , E, m S ) = H -1 < U j ∂F p,Z ∂U j 1≤j≤e , ∂F p,Z ∂λ l l∈Λ > .
is not affected either by changes of coordinates fixing each < U j >, j ≤ e.

We now turn to the version of Proposition 2.57(iii) for i 0 (x) = p -1. The problem is elementary, though more technical, and the remaining part of this section is devoted to it.

Let (e j ) 1≤j≤n be the standard basis of R n and let

E := {x ∈ R n : x e+1 = • • • = x n = 0} ≃ R e . Given d ∈ 1 p N and H ∈ N n ∩ E, we denote ∆ H (d) := {x = (x 1 , . . . , x n ) ∈ R n ≥0 :| x |= d and x j ≥ H j p , 1 ≤ j ≤ e} and V H (pd) := (U H ) ∩ G(m S ) pd ⊆ G(m S ) pd . (2.59)
We fix once and for all

b ∈ (N n ∩ ∆ H (d)) ∩ E. (2.60) Note that V H (pd) = (0) only if H 1 + • • • + H e ≤
pd and that such b as above exists only if d ∈ N. By convention, we take {b} = ∅ if d ∈ N in the following formulae. For applications, we will take d = δ(x 0 ), H as in Definition 2.31 and b will be defined by

< G >=:< U b 1 1 • • • U be e > . ( 2
F := x∈ 1 p N n ∩∆ H (d) λ(x)U px , λ(x) ∈ S/m S .
We denote

∆(F ) := Conv({x ∈ 1 p N n ∩ ∆ H (d) : λ(x) = 0} ∪ {b}) ⊆ ∆ H (d).
According to these conventions, we have ∆(0) = {b}.

Definition 2.59. With notations as above, let T : V H (pd) → V H (pd) be the S/m S -linear truncation operator defined as follows: let

A := {x ∈ 1 p N n ∩ ∆ H (d) : b + p(x -b) ∈ ∆ H (d)}. (2.62)
and

T F := x ∈A λ(x)U px ∈ V H (pd). ( 2

.63)

For d ∈ N, we have A = ∅ and T is the identity map.

The construction of the previous section associates to x ∈ η -1 (m S ) two vector spaces V (T F, E, m S ) and J(T F, E, m S ). Explicitly, we have:

V (T F, E, m S ) = U -H < ∂T F ∂U j , e + 1 ≤ j ≤ n >⊆ G(m S ) pd-1-|H| (2.64)
for the former one. If V (T F, E, m S ) = 0 (and only in this case), we will use the latter one, given explicitly by

J(T F, E, m S ) = U -H < {U j ∂T F ∂U j } 1≤j≤e , { ∂T F ∂λ l } l∈Λ >⊆ G(m S ) pd-|H| , (2.65)
with notations as in the previous section.

The S/m S -linear truncation operator T defined above is useful to give a definition of ω(x) (the adapted order defined below see Definition 2.68) independent of all possible choices of well adapted coordinates. The possible vanishing of V (T F, E, m S ) is essential in this definition. Let us point out the problem. For simplicity, we take E =div(u 1 u 2 ). The vector space V (F, E, m S ) depends on the choice of the pair (v, Z) where v = u 3 is a free variable and (u 1 , u 2 , v; Z) are well adapted coordinates. The truncation operator T is devised to suppress this dependence.

The following example shows the bad behavior of V (F, E, m S ) without truncating.

Example 2.60. char(k) = 2, k = F 2 , S = k[[u 1 , u 2 , v]], E = div(u 1 u 2 ), h = X 2 + u 3 1 u 2 2 X + u 1 u 2 [v 8 + u 4 2 v 4 + u 3 1 u 4 2 v] ∈ S[X]
. We get:

Discr(h) = u 6 1 u 4 2 , ǫ(x) = 8, δ(x) = 5.
We have

V (F Z,U 1 ,U 2 ,V , E, m S ) =< 0 > for any choice of Z such that (u 1 , u 2 , v; Z) are well adapted coordinates. Let w := v + λu 2 , λ ∈ k: (X, u 1 , u 2 , w) is a regular system of parameters at x, in m S h = X 2 +U 3 1 U 2 2 X +U 1 U 2 [W 8 +U 4 2 W 4 +U 3 1 U 4 2 W +λU 3 1 U 5 2 +[λ(λ+1)] 4 U 8 2 ].
As k = F 2 , we can choose λ such that λ(λ + 1) = 0. Then ∆ S (h; u 1 , u 2 , w; X) has three not solvable vertices of same modules δ x : M := (3, 2, 0) given by

U 3 1 U 2 2 X, N := ( 1 2 , 9 2 , 0) given by λ(λ + 1)U 1 U 2 U 8 2 , P := ( 1 2 , 1 2 , 4) given by U 1 U 2 W 8 . So (u 1 , u 2 , w; X) are well adapted parameters. The monomial λU 1 U 2 × U 3 1 U 4 2 W defines the point (2, 5/2, 1/2)
inside the first face and this monomial gives

V (F X,U 1 ,U 2 ,W , E, m S ) =< 0 > .
(2.66)

We make the change of variable: Z = X + u 1 u 3 2 w, as (1, 3, 1) is in the interior of the triangle MNP , ∆ S (h; u 1 , u 2 , w; Z) = ∆ S (h; u 1 , u 2 , w; X): the coordinates (u 1 , u 2 , w; Z) are well adapted. This gives:

in m S h = Z 2 +U 3 1 U 2 2 Z +U 1 U 2 [W 8 +U 4 2 W 4 +U 1 U 5 2 W 2 +λU 3 1 U 5 2 +[λ(λ+1)] 4 U 8 2 ].
With natural notations, we get

V (F Z,U 1 ,U 2 ,W , E, m S ) =< 0 > . (2.67)
By Lemma 2.61 below, V (T F, E, m S ) =< 0 > for all the well adapted coordinates we used above.

We can now state:

Lemma 2.61. Assume that d ∈ N. With notations as above, we have

KerT = U (p-1)b V ⌈ H p ⌉ (d),
where

⌈ H p ⌉ := (⌈ H 1 p ⌉, . . . , ⌈ He p ⌉, 0, . . . , 0). Let G := µU b , µ ∈ S/m S , Φ ∈ V ⌈ H p ⌉ (d) and F ∈ V H (pd). Then V (T (F + Φ p -G p-1 Φ), E, m S ) = V (T F, E, m S ). If V (T F, E, m S ) = 0, then J(T (F + Φ p -G p-1 Φ), E, m S ) = J(T F, E, m S ),
Proof. We analyze the definition of T in (2.63). The kernel of T is generated by those monomials U px ∈ V H (pd) such that

y := px -(p -1)b ∈ ∆ H (d). Since x ∈ 1 p N n , b ∈ N n , we have y ∈ N n for such y. Therefore KerT is generated by KerT =< {U (p-1)b U y : y ∈ N n , | y |= d and y j ≥ H j p , 1 ≤ j ≤ e} > .
This proves the first statement. For the second part, we have proved that

T (F + Φ p -G p-1 Φ) = T F + T Φ p . Hence D • T (F + Φ p -G p-1 Φ) = D • T F for every D ∈ Der(G(m S )).
We now study invariance properties of V (F, E, m S ) and J(F, E, m S ) under changes of adapted coordinates. Given two r.s.p.'s u = (u 1 , . . . , u n ) and

u ′ = (u ′ 1 , . . . , u ′ n ) adapted to E, there exists a matrix M ∈ M(S), M(S) := {(m ij ) ∈ GL(n, S) : m jj ′ = 0, (j, j ′ ) ∈ {1, . . . , e}×{1, . . . , n}, j = j ′ } such that u = Mu ′ . The set M(S)
is the set of S-points of an affine S-scheme M ⊂ GL(n, S). Denote by

GL(n, S) → GL(n, S/m S ), M → M the canonical surjection. Each such M induces a graded S/m S -automorphism of gr m S (S) ≃ S/m S [U 1 , . . . , U n ]
. By (2.59), this automorphism restricts to an automorphism of V H (pd) for each d ∈ 1 p N still denoted by M. Given a homogeneous polynomial F ∈ V H (pd) as above and a matrix M ∈ M(S/m S ), we denote for simplicity the transformed equation U → M U ′ by F ′ =:

x ′ ∈ 1 p N n ∩∆ H (d) λ ′ (x ′ )U ′ px ′ . (2.68) Let ∆(F ′ ) := Conv({x ′ ∈ 1 p N n ∩ ∆ H (d) : λ ′ (x ′ ) = 0} ∪ {b}) ⊆ ∆ H (d)
be the corresponding polytope and T ′ be the corresponding operator on V H (pd) with variable U ′ . The linear operator T obviously does not commute with M in general (i.e. (T F ) ′ = T ′ F ′ in general), but the lemma below extracts the relevant invariant data. We refer to Definition 2.54 for the Notation Max(I), I ⊂ G(m S ) generated by one homogeneous polynomial or homogeneous polynomials of the same degree. Notation 2.62. Recall (2.60) and (2.61). We denote by

B := {j, 1 ≤ j ≤ e : pb j -H j > 0} and U B := {U j , j ∈ B}.
(2.69)

We denote U B ′ := {U j , j ∈ B} and stick to our former conventions, i.e.

B ′ = {1, . . . , n}\B, (B ′ ) E = {1, . . . , e}\B.
Remark 2.63. The vector space U B is sometimes used to get a notion of maximal contact for our main invariant ω with the components div(u j ), j ∈ B: see chapter 6, κ(x) = 1.

Lemma 2.64. With notations as above, there is an equality of sets

Max(V (T F, E, m S ))∩{U B = 0} = Max(V (T ′ F ′ , E, m S ))∩{U ′ B = 0}. (2.70) If V (T F, E, m S ) = 0, then V (T ′ F ′ , E, m S ) = 0
and there is an equality of sets

Max(J(T F, E, m S )) ∩{U B = 0} = Max(J(T ′ F ′ , E, m S )) ∩{U ′ B = 0}. (2.71)
Proof. The operator T commutes with M when M stabilizes the vector space < U e+1 , . . . , U n >. In these cases, we have

V (T ′ F ′ , E, m S ) = V ((T F ) ′ , E, m S ). If V (T F, E, m S ) = 0, then V (T ′ F ′ , E, m S ) = 0 and J(T ′ F ′ , E, m S ) = J((T F ) ′ , E, m S ).
So the lemma is trivial in this case and we may therefore assume that

m jj ′ = 0, (j, j ′ ) ∈ {e+1, . . . , n}×{e+1, . . . , n}, j = j ′ and m jj = 1, 1 ≤ j ≤ n.
By elementary calculus, this new assumption implies for every Φ ∈ G(m S ):

∂Φ ′ ∂U ′ j = ∂Φ ∂U j ′ , e + 1 ≤ j ≤ n. (2.72) Let x ∈ 1 p N n ∩ ∆ H (d).
Since pb j = H j for j ∈ (B ′ ) E , we have by (2.62):

x ∈ A ⇔ ∀j ∈ B, px j ≥ (p -1)b j . Expand T F = y U y B F y (U B ′
), so we have:

V (T F, E, m S ) = U -H < { y U y B ∂F y (U B ′ ) ∂U j } e+1≤j≤n > .
For P ∈ SpecG(m S ) such that (U B ) ⊆ P , we get:

P ∈ Max(V (T F, E, m S )) ⇔ P ∈ y n j=e+1 Max(G y,j ), (2.73)
where G y,j :

= U -H ′ B ′ ∂Fy(U B ′ ) ∂U j , H ′ := (H j ′ ) j ′ ∈(B ′ ) E .
Suppose furthermore that M stabilizes the vector space < U B ′ >. Then T also commutes with M and each term G y,j in (2.73) is transformed into [START_REF] Pomerol | Sur la strate de Samuel du sommet d'un cne en caractŕistique positive[END_REF]) and (2.70) follows. Suppose furthermore that V (T F, E, m S ) = 0; then G y,j = 0 for each y, j in (2.70) and we get V (T ′ F ′ , E, m S ) = 0. For 1 ≤ j ≤ e and l ∈ Λ, we have

(G y,j ) ′ = U -H B ′ B ′ ∂F ′ y (U ′ B ′ ) ∂U ′ j by (2.
U j ∂T F ∂U j ′ = U ′ j ∂T ′ F ′ ∂U ′ j , ∂T F ∂λ l ′ = ∂T ′ F ′ ∂λ l , (2.74)
and (2.71) also follows. Hence we may furthermore assume that

m jj ′ = 0, (j, j ′ ) ∈ {e + 1, . . . , n} × (B ′ ) E .
In this situation, T does not commute any longer with M. However, for each term G y,j as above, we have

ord P (D • G y,j ) ≥ degG y,j -a (2.75)
for any differential operator D on S/m S [U B ′ ] of order not greater than a. Let

(G y,j ) ′ = |α|≤degG y,j (U ′ B ) α (D (α) • G y,j ), D (α) • G y,j ∈ S/m S [U ′ B ′ ] degG y,j -|α|
be the (characteristic free) Taylor expansion, where D (α) is a differential operator of order | α |. Take again P ∈ SpecG(m S ) such that (U B ) ⊆ P . By (2.75), we have

P ∈ Max(G y,j ) ⇒ P ∈ α Max(D (α) • G y,j ) ⇒ P ∈ Max((G y,j ) ′ ).
We deduce from (2.73) that

P ∈ Max(V (T F, E, m S )) ⇒ P ∈ Max(V ((T F ) ′ , E, m S )).
This proves (2.70). If V (T F, E, m S ) = 0, (2.71) follows from (2.74) as above.

This lemma is the key to our version of Proposition 2.57(iii) for i 0 (x) = p -1: Proposition 2.65. Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x ∈ η -1 (m S ) and assume that ǫ(x) > 0 and i 0 (x) = p -1. Let

d := δ(x), H := (H 1 , . . . , H e , 0, . . . , 0) and < U b 1 1 • • • U be e >:=< G >
be defined respectively by Definition 2.11, Definition 2.31 and (2) of Theorem 2.36. With notation as above, the following statements hold:

(i) the set Max(V (T F p,Z , E, m S )) ∩ {U B = 0} ⊆ SpecG(m S )
is independent of the well adapted coordinates (u 1 , . . . , u n ; Z);

(ii) the property V (T F p,Z , E, m S ) = 0 is independent of the well adapted coordinates (u 1 , . . . , u n ; Z); when it holds, the set

Max(J(T F p,Z , E, m S )) ∩ {U B = 0} ⊆ SpecG(m S )
is also independent of the well adapted coordinates (u 1 , . . . , u n ; Z).

Proof. For such (u 1 , . . . , u n ; Z), the corresponding initial form is

in m S h = Z p -G p-1 Z + F p,Z ∈ G(m S )[Z]. Since G = 0, we have d = δ(x) = degG ∈ N. If (u ′ 1 , . . . , u ′ n ) is an adapted r.s.p. of S, there exists M ∈ M(S) such that u = Mu ′ . Let (u ′ 1 , . . . , u ′ n ; Z ′ ) be well adapted coordinates at x. We have Z ′ = Z -φ for some φ ∈ S, with ord m S φ ≥ d. We deduce that in m S h = Z ′ p -G p-1 Z ′ + Φ p -G p-1 Φ + F p,Z ∈ G(m S )[Z ′ ] for Φ := cl d φ ∈ G(m S ) d . We deduce the formula F p,Z ′ = F p,Z + Φ p -G p-1 Φ. By Lemma 2.61, we have V (T F p,Z ′ , E, m S ) = V (T F p,Z , E, m S ); if moreover V (T F p,Z , E, m S ) = 0, then J(T F p,Z ′ , E, m S ) = J(T F p,Z , E, m S )
. By Lemma 2.64, we have an equality of sets [START_REF] Mc Quillan | Almost étale resolution of foliations[END_REF] and there is an equality of sets

Max(V (T F p,Z ′ , E, m S )) ∩ {U B = 0} = Max(V (T ′ F ′ p,Z ′ , E, m S )) ∩ {U ′ B = 0} and this proves (i). If V (T F p,Z ′ , E, m S ) = 0, then V (T ′ F ′ p,Z ′ , E, m S ) = 0 by Lemma 2.
Max(J(T F p,Z ′ , E, m S )) ∩ {U B = 0} = Max(J(T ′ F ′ p,Z ′ , E, m S )) ∩ {U ′ B = 0}.
This concludes the proof.

Remark 2.66. We consider Proposition 2.57(iii) as the special case B = ∅, T = id of Proposition 2.65.

Main invariants.

Let s ∈ SpecS and y ∈ η -1 (s). The purpose of this section is to attach to y a resolution complexity

ι(y) = (m(y), ω(y), κ(y)) ∈ {1, . . . , p} × N × {1, ≥ 2} (2.76)
with certain invariance properties. Auxiliary numbers

(τ (y), τ ′ (y)) ∈ {1, . . . , n + 1} × {1, . . . , n} (2.77)
are similarly attached to y.

The pair (m(y), τ (y)) are the standard multiplicity and Hironaka τ -number of X at y (Definition 2.67). The pair (ω(y), τ ′ (y)) plays the role of a differential multiplicity and differential τ -number attached to η : X → SpecS at y. The behavior of the function ι under blowing up is studied in Theorem 3.13 below.

In all definitions that follow it can be assumed without loss of generality that s = m S by localizing S at s, since our assumptions (G) and (E) are stable when changing (S, h, E) to (S s , h s , E s ) (Notation 2.23).

Definition 2.67. (Multiplicity). Let x ∈ η -1 (m S ). We have already defined

m(x) = ord m S[X]x h(X) ≤ p. Let M x ⊂ S[X] be the ideal of x, G x := Spec(gr Mx S[X] Mx ) and in x h(X) be the initial form of h in (G x ) m(x) . From Definition 2.53, we let τ (x) := τ (in x h(X)).
If m(x) < p, we let ι(x) := (m(x), ω(x) := 0, 1). Definition 2.68. (Adapted order). Assume that m(x) = p, where {x} = η -1 (m S ). Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x. We let

ω(x) = ǫ(x) -1 if V (T F p,Z , E, m S ) = 0 ǫ(x) if V (T F p,Z , E, m S ) = 0 .
We define:

κ(x) := 1 if (ω(x) = ǫ(x) and i 0 (x) = p -1) or if ω(x) = 0.
Otherwise, we simply let κ(x) ≥ 2.

Remark 2.69. Note that m(y) < p whenever s = η(y) ∈ E (Definition 2.32 and following comments). If m(y) = p, we have

s = η(y) ∈ E, η -1 (s) = {y} and k(y) = k(s) by Proposition 2.28. Applying Proposition 2.57(iii) (resp. Proposition 2.65(ii)) to S if i 0 (x) = p (resp. if i 0 (x) = p -1) proves that (ω(x), κ(x)
) is well-defined. We recall that T F p,Z = F p,Z whenever i 0 (x) = p (see Remark 2.66).

Remark 2.70. It is obvious from this definition that ω(x) is not determined by the characteristic polyhedra ∆ S (h; u 1 , . . . , u n ; Z), even for unspecified well adapted coordinates (u 1 , . . . , u n ; Z).

For example, take n = 3, p ≥ 3 for simplicity and k(x) algebraically closed of characteristic p > 0. Suppose:

in m S h = Z p + U 1 U 2 U p 3 + U p+2 1 + U p+2 2 + cU 3 U 2 U p 1 , E = div(u 1 u 2 ), where c ∈ k(x). Let (u ′ 1 , u ′ 2 , u ′ 3 ; Z ′ ) be well adapted coordinates such that div(u j ) = div(u ′ j ) for j = 1, 2. Then the corresponding initial face of ∆ S (h; u ′ 1 , u ′ 2 , u ′ 3 ; Z ′ ) is: Conv({v 1 , v 2 , v 3 }) ⊂ {x 1 + x 2 + x 3 = δ(x) = 1 + 2/p}
and is independent of c, where v 1 := ((p + 2)/p, 0, 0), v 2 := (0, (p + 2)/p, 0), v 3 := (1/p, 1/p, 1).

But ω(x) = p + 2 (resp. ω(x) = p + 1) for c = 0 (resp. for c = 0).

Remark 2.71. This definition is different from the one used in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 1, Definition II.4 when G = 0. Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x. There is an obvious implication

ω(x) = ǫ(x) -1 =⇒ V (F p,Z , E, m S ) = 0.
The converse is however false, even if it is assumed that V (F p,Z , E, m S ) = 0 for every possible choice of well adapted coordinates (u 1 , . . . , u n ; Z) at x and this is the reason for this difference. For n ≤ 3, this phenomenon is easily dealt with, vid. [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] In chapter 5, we define the projection number κ(x) ∈ {2, 3, 4} when n = 3 and state that ι(x) = (m(x), ω(x), κ(x)) can be decreased by Hironaka permissible blowing ups w.r.t. E (Projection Theorem 5.5 below).

We now turn to the definition of the adapted cone and directrix and the attached invariant τ ′ (x). Definition 2.72. (Adapted cone and directrix). Assume that m(x) = p and ω(x) > 0, where {x} = η -1 (m S ). Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x. We define a reduced subcone Max(x) ⊆ SpecG(m S ) by:

Max(x) := Max(V (T F p,Z , E, m S )) ∩ {U B = 0} if ω(x) = ǫ(x) -1 Max(J(T F p,Z , E, m S )) ∩ {U B = 0} if ω(x) = ǫ(x).
We define an affine subspace Dir(x) ⊆ SpecG(m S ) by

Dir(x) := Dir(V (T F p,Z , E, m S ), U B ) if ω(x) = ǫ(x) -1 Dir(J(T F p,Z , E, m S ), U B ) if ω(x) = ǫ(x).
We let Vdir(x) to be the underlying vector space of Dir(x) and

τ ′ (x) := dim k(x) Vdir(x).
Remark 2.73. Applying Proposition 2.57(iii) (resp. Proposition 2.65) if i 0 (x) = p (resp. if i 0 (x) = p -1) proves that Max(x), Dir(x) and τ ′ (x) are well defined. We will use the invariants Dir(x) and τ ′ (x) only when n = 3.

In this case, we have Dir(x) = Max(x) (last statement in Proposition 2.55 and remark following).

Let S ⊆ S be a regular local base change, S excellent. Recall Notation 2.14 and Notation 2.23. It has been explained when defining conditions (G) and (E) that they are stable by such base changes and by localization at a prime. Let s ∈ Spec S and ỹ ∈ η-1 (s). In order to relate ι(ỹ) and ι(y) (2.76), where y ∈ X is the image of ỹ, we may thus assume that s = m S , s = m S .

Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x ∈ η -1 (m S ). Then (u 1 , . . . , u n ) can be completed to a r.s.p. (u 1 , . . . , u ñ) of S which is adapted to Ẽ. There is an inclusion

G(m S ) = k(x)[U 1 , . . . , U n ] ⊆ G(m S ) = G(m S ) ⊗ k(x) S m S [U n+1 , . . . , U ñ].
(2.78)

Theorem 2.74. Let S ⊆ S be a local base change which is regular, S excellent. Let x ∈ η-1 (m S ) and x ∈ η -1 (m S ) be its image. The following holds:

(1) we have (m(x), ω(x)) = (m(x), ω(x));

(2) if m(x) = p, then (i) H(x) = H(x) S, i 0 (x) = i 0 (x), and (κ(x) = 1 ⇔ κ(x) = 1);

(ii) we have ǫ(x) ≥ ǫ(x), and ǫ(x) > ǫ(x) if and only if

in m S h = Z p + F p,Z , F p,Z ∈ (k(x)[U 1 , . . . , U n ]) p
where (u 1 , . . . , u n ; Z) are well prepared coordinates at x. When this holds, we have ñ > n, ǫ(x) = ǫ(x) + 1 and

in m S h = Zp + ñ j=n+1 U j Φ j (U 1 , . . . , U n )+Ψ(U 1 , . . . , U n ) ∈ G(m S )[ Z],
with Φ j = 0 for some j ≥ n + 1 and Φ j ∈ k(x)[U p 1 , . . . , U p n ] for every j ≥ n+1, where (u 1 , . . . , u ñ; Z) are well prepared coordinates at x. Example 2.75. Let us note that case (2)(ii) with ǫ(x) > ǫ(x) occurs in Example 2.16. We give another example involving a formal fiber. Let k be a field of characteristic p > 0,

R := k[u 1 , u 2 , u 3 , u 4 ] (u 1 ,u 2 ,u 3 ,u 4 ) , E = div(u 1 u 2 ), P := (u 1 , u 2 ), S := R P . Let φ ∈ k[[u 4 ]] be transcendental over k(u 4 ) with φ(0) = 0. Let P := (u 1 , u 2 , v := u 3 -φ p ) and S := R P . The local base change S ⊂ S is reg- ular as R is excellent. Let h := Z p + u p 1 u 3 + u p+1 2 ∈ S[Z].
We denote by x and x the closed points of Spec(S) and Spec( S). The coordinates (u 1 , u 2 ; Z) are well adapted at x. We have

ǫ(x) = p, in P (h) = Z p + u 3 U p 1 ∈ k(u 3 , u 4 )[U 1 , U 2 ].
Let Z = Z + u 1 φ, the coordinates (u 1 , u 2 , v; Z) are well adapted at x.

h = Zp + u p 1 v + u p+1 2 , ǫ(x) = p + 1, in P (h) = Zp + U p 1 V + U p+1 2 ∈ k((u 4 ))[U 1 , U 2 , V ]. Proof. The theorem is trivial if m(x) = 1: then m(x) = 1 because S ⊆ S is regular.
Assume that m(x) ≥ 2 and pick well prepared coordinates (u 1 , . . . , u n ; Z) at x, then complete (u 1 , . . . , u n ) to a r.s.p. (u 1 , . . . , u ñ) of S which is adapted to Ẽ. We have δ(x) > 0, so h ∈ (Z, u 1 , . . . , u n ), and k(x) = S/m S by Proposition 2.28. Applying (2.78) to the local base change S[Z] (m S ,Z) ⊆ S[Z] (m S ,Z) which is also regular gives

m(x) = ord x h(Z) = ord xh (Z) = m(x).
This concludes the proof when m(x) < p ( ω(x) = ω(x) = 0 in this case) and we assume from now on that m(x) = p. In particular we have

{x} = η-1 (m S ), k(x) = S/m S . Let in m S h = Z p + p i=1 F i,Z Z p-i ∈ G(m S )[Z],
be the corresponding initial form polynomial. Let x ∈ R n ≥0 be a vertex of the polyhedron ∆ S (h; u 1 , . . . , u n ; Z). We denote by

x := (x, 0, . . . , 0 ñ-n ) ∈ ∆ S (u 1 , . . . , u ñ; Z)
the corresponding vertex in ∆ S (h; u 1 , . . . , u ñ; Z). Note that x may be a solvable vertex of the latter polyhedron. We have:

x solvable ⇔ in xh ∈ ((gr α S)[Z]) p
with notations as in Definition 2.6. Therefore we have

x solvable ⇔ (in x h = Z p + F p,Z,x , x ∈ N n , F p,Z,x = λU px , λ ∈ k(x) p ).
We deduce for the initial form polynomial that

δ(x) > δ(x) ⇔ (i 0 (x) = p and F p,Z ∈ (k(x)[U 1 , . . . , U n ]) p ).
(2.79)

Since the fiber ring S/m S S is geometrically regular over k(x), the ring S[Y ]/(Y p -l) is regular for every unit l ∈ S with residue l ∈ k(x) p . Therefore if l ∈ k(x) p , we have ∀ l ∈ S, ṽ := lpl ∈ m S =⇒ ṽ is a regular parameter in S.

Such ṽ restricts to a regular parameter of S/m S S, so the previous formula is refined to: ṽ is a regular parameter transverse to div(u

1 • • • u n ) ⊂ Spec S.
(2.80)

This equation implies in particular that ñ > n. Let ξ ∈ Spec( S/m S S) be the generic point. Applying the above remarks to the regular local base change S ⊂ Sξ shows that k(ξ

) p ∩ k(x) = k(x) p .
Let s j := (u j ) ∈ SpecS, 1 ≤ j ≤ e, and apply this remark to the regular local base change S (u j ) ⊆ S(u j ) . This proves that the field inclusion QF (S/(u j )) ⊆ QF ( S/(u j )) is inseparably closed.

The polynomial in (s j ) h s j ∈ QF (S/(u j ))[U j ][Z] is not a p th -power by Proposition 2.12. Therefore in (s j ) h s j is not a p th -power in QF ( S/(u j ))[U j ][Z]. Turning back to Definition 2.25, we get

H(x) = H(x) S.
(2.81) Definition 2.25 now shows that ǫ(x) ≥ ǫ(x) and that

ǫ(x) > ǫ(x) ⇔ (i 0 (x) = p and F p,Z ∈ (k(x)[U 1 , . . . , U n ]) p ). (2.82)
This proves the first part of (2.ii). To go on with the proof, we consider two cases.

Case 1: assume that i 0 (x) < p. By (2.82), we have ǫ(x) = ǫ(x), so the proof of (2.ii) is already complete. Let φ ∈ S be such that ∆ S (u 1 , . . . , u ñ; Z) is minimal, with Z := Z -φ and ord m S φ ≥ δ(x). We have

in m S h = Zp + p i=i 0 F i, Z Zp-i ∈ G(m S )[ Z],
with F i 0 , Z = F i 0 ,Z by Proposition 2.27. Therefore i 0 (x) = i 0 (x) and it is sufficient to prove that ω(x) = ω(x) in order to complete the proof of (1) and (2.i) in the theorem (still under the assumption i 0 (x) < p). This is obvious if

ǫ(x) = 0, since 0 ≤ ω(x) ≤ ǫ(x) = ω(x) = 0. Assume that ǫ(x) > 0. We have i 0 (x) = p -1 and -F p-1,Z = G p-1 , with < G >=< U b > for some b ∈ N n ∩ E by Theorem 2.36(2) (in particular δ(x) ∈ N). We have V (T F p,Z , E, m S ) =< H -1 ∂T F p,Z ∂U j e+1≤j≤n > .
Note that the truncation maps T and T associated with the local rings S and S (Definition 2.59) commute with the inclusion G(m S ) ⊆ G(m S ) by (2.81).

Since

F p,Z ∈ G(m S ) = k(x)[U 1 , . . . , U n ], we have V ( T F p,Z , Ẽ, m S ) =< H -1 ∂ T F p,Z ∂U j ñ j=e+1 >= V (T F p,Z , E, m S ) ⊗ k(x) k(x)
with obvious notations, taking (2.81) into account. There exists Θ ∈ G(m S ) such that

F p, Z = F p,Z + Θp -G p-1 Θ.
By Lemma 2.61 applied to F p, Z ∈ G(m S ), we deduce that

V ( T F p, Z , Ẽ, m S ) = V (T F p,Z , E, m S ) ⊗ k(x) k(x). (2.83)
This completes the proof of the theorem when ω(x) = ǫ(x) -1, applying Definition 2.68. If ω(x) = ǫ(x), (1) and the last statement of (2.i) in the theorem also follow from (2.83) and the proof is complete.

Case 2: assume that i 0 (x) = p. The proof runs parallel to that of case 1 (with B = ∅, T = id, cf. Remark 2.66) provided that ǫ(x) = ǫ(x). Assume now that ǫ(x) > ǫ(x). To complete the proof, we have to show that

(i 0 (x), ω(x)) = (p, ω(x)),
as well as the last statement in (2.ii). By (2.82), we have ω(x) = ǫ(x), δ(x) ∈ N and there is an expansion

F p,Z = |x|=δ(x) λ(x)U px ∈ (k(x)[U 1 , . . . , U n ] δ(x) ) p , λ(x) ∈ k(x).
Note that this situation possibly occurs only if k(x) is not inseparably closed in k(x) (in particular ñ > n). We have x ∈ N n for every x such that λ(x) = 0. Without loss of generality, it can be assumed that λ(x) ∈ k(x) p for every x such that λ(x) = 0. Let l(x) ∈ S be a preimage of λ(x). By (2.80), we may pick for every such x a unit l(x) ∈ T such that ṽ(x) := l(x) pl(x) is a regular parameter of S transverse to div(u

1 • • • u n ). Expand h = Z p + p i=1 f i,Z Z p-i ∈ S[Z], ord m S f i,Z ≥ iδ(x).
For 1 ≤ i ≤ p -1, the above inequality is strict, since i 0 (x) = p. On the other hand, we have δ(x) ∈ N, so we deduce that

ord m S f i,Z i ≥ δ(x) + 1 i > δ(x) + 1 p , 1 ≤ i ≤ p -1. (2.84) Let Z := Z + |x|=δ(x) l(x)u x .
By (2.84), there is an expansion

f p, Z = - |x|=δ(x) ṽ(x)u px + g + g, (2.85)
with g ∈ S, ord m S g ≥ pδ(x) + 1 and g ∈ S, ord m S g > pδ(x) + 1 . We deduce that δ(h; u 1 , . . . , u ñ; Z) = δ(x) + 1 p .

Since δ(x) + 1 p ∈ N, ∆ S (h; u 1 , . . . , u ñ; Z) has no solvable vertex within its initial face {x ∈ R ñ ≥0 :| x |= δ(x) + 1 p }. Let (u 1 , . . . , u ñ; Z1 ) be well adapted coordinates at x. Without loss of generality, it can be assumed that Z1 = Z -θ1 with ord m S θ1 ≥ δ(x) + 1. By (2.85), we get

in m S h = Zp 1 - |x|=δ(x) Ṽ (x)U px + G(U 1 , . . . , U n ) ∈ G(m S )[ Z1 ] (2.86)
and (2.ii) is proved. We have i 0 (x) = p, δ(x) = δ(x) + 1 p and ǫ(x) = ǫ(x) + 1. Finally, we have

∂F p, Z1 ∂U j = |x|=δ(x) ∂ Ṽ (x) ∂ Ṽj U px ∈ k(x)[U 1 , . . . , U n ], n + 1 ≤ j ≤ ñ, so V (F p, Z1 , Ẽ, m S ) = 0 and ω(x) = ǫ(x) -1 = ω(x).
Remark 2.76. Theorem 2.74 reduces computations of ω(x) to the case where S is strict Henselian, i.e. Henselian with separably algebraically closed residue field S/m S by changing S to its strict Henselianization S, dim S = n = dimS.

Applying the theorem to a tower S of smooth local base changes of the form S ⊆ S[Y ] (m S ,Y p -l) with l ∈ S a unit with residue l ∈ (S/m S ) p also reduces computations of ω(x) to the case of an algebraically closed residue field for some S with dim S > n = dimS.

The cone Max(x) and directrix Dir(x) have no such good behavior w.r.t. regular local base changes.

2.8 Resolution when ω(x) = 0.

In this section, we prove that the multiplicity of X can be reduced at any point x such that (m(x), ω(x)) = (p, 0). This is achieved by combinatorial blowing ups in a way which is similar to the equal characteristic zero situation.

Up to the end of this article, we will define a resolution algorithm which picks up local blowing ups centers in a way which is independent on the choice of a valuation. The word independent is defined below.

The total quotient ring L = Tot(S[X]/(h)) is a direct product of fields. By a valuation of L, we mean a valuation of one of these fields. Definition 2.77. Let (S, h, E) be as before, x ∈ X and L = Tot(S[X]/(h)). Suppose that to every valuation µ of L centered at x, a composition of local Hironaka-permissible blowing ups (Definition 2.20)

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ) (2.87)
is associated, where x i ∈ X i is the center of µ, 0 ≤ i ≤ r. The sequence (2.87) is said to be independent if the blowing up center Y i ⊂ (X i , x i ) does not depend on the chosen valuation µ having center

x i in X I , 0 ≤ i ≤ r -1.
Let (u 1 , . . . , u n ; Z) be well adapted coordinates at

x ∈ η -1 (m S ). If ǫ(x) > 0, recall that η -1 (m S ) = {x}, k(x) = S/m S ,

and that in

m S h = Z p -G p-1 Z + F p,Z ∈ G(m S )[Z] = k(x)[U 1 , . . . , U n ][Z] by (2.57). The initial form of H(x) in G(m S ) is denoted H as before. Lemma 2.78. Assume that m(x) = p and ǫ(x) = 1, where {x} = η -1 (m S ). Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x ∈ η -1 (m S ). If H -1 F p,Z < U 1 , . . . , U e >, then ω(x) = 0.
Proof. According to Definition 2.68, we must show that V (T F p,Z , E, m S ) = 0. Expand

H -1 F p,Z =< n j=1 α j U j >⊆ G(m S ) 1 , α j ∈ k(x).
By assumption, we have α j 0 = 0 for some j 0 , e + 1 ≤ j 0 ≤ n, so

0 = H -1 ∂F p,Z ∂U j 0 ⊆ V (F p,Z , E, m S ). (2.88) If i 0 (x) = p, we have T F p,Z = F p,Z . If i 0 (x) = p -1, then H -1 G p =< U j 1 > for some j 1 , 1 ≤ j 1 ≤ e,
x ∈ A =⇒ px j 1 > H j 1 , therefore F p,Z -T F p,Z ∈ HU j 1 . So (2.88) implies that V (T F p,Z , E, m S ) = 0. Proposition 2.79. Assume that (m(x), ω(x)) = (p, 0), {x} := η -1 (m S ). Let Y ⊂ (X , x) be a Hironaka-permissible center w.r.t. E, π : X ′ → (X , x) be the blowing up along Y and x ′ ∈ π -1 (x). If W := η(Y) is an intersection of components of E or if ǫ(y) = ǫ(x), then (m(x ′ ), ω(x ′ )) ≤ (p, 0).
Proof. According to Definition 2.68, there are two different cases to consider:

(1) ǫ(x) = 0; (2) ǫ(x) = 1, V (T F p,Z , E, m S ) = (0).
To begin with, we have δ(x) ≥ 1 by Proposition 2.10(ii). Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x with I(W ) = ({u j } j∈J ) for some subset J ⊆ {1, . . . , n}. By Definition 2.25, we have:

ǫ(x) = min 1≤i≤p ord m S (H(x) -i f p i,Z ) i .
(2.89)

Case 1: ǫ(x) = 0. By (2.89), we have    H(x) -i f p i,Z ⊆ m S , 1 ≤ i < i 0 (x) H(x) -i 0 (x) f p i 0 (x),Z = S, H(x) -i f p i,Z ⊆ S, i 0 (x) < i ≤ p.
(2.90) By Proposition 2.22, there exists a commutative diagram

X π ←-X ′ ↓ ↓ SpecS σ ←-S ′
where σ : S ′ → SpecS is the blowing up along W . Let

η ′ : X ′ → S ′ , s ′ := η ′ (x ′ ), S ′ := O S ′ ,s ′ , E ′ := (σ -1 (E) red ) s ′ .
Since W ⊆ E, it can be assumed after possibly reordering coordinates that

(J ′ ) E := {2, . . . , e 0 }, J = {1, e 0 + 1, . . . , n 0 }, 1 ≤ e 0 ≤ e ≤ n 0 . Furthermore, it can be assumed that s ′ ∈ Spec(S[u e 0 +1 /u 1 , . . . , u n 0 /u 1 ]) or that s ′ ∈ Spec(S[u 1 /u n 0 , u e 0 +1 /u n 0 , . . . , u n 0 -1 /u n 0 ]) with n 0 > e 0 .
We first prove the proposition when

s ′ ∈ Spec(S[u e 0 +1 /u 1 , . . . , u n 0 /u 1 ]). Let h ′ := u -p 1 h = Z ′ p + f 1,Z ′ Z ′ p-1 + • • • + f p,Z ′ ∈ S ′ [Z ′ ],
where

Z ′ := Z/u 1 , f i,Z ′ := u -i 1 f i,Z ∈ S ′ for 1 ≤ i ≤ p.
We have

E ′ = div(u 1 • • • u e 0 u e 0 +1 u 1 • • • u e u 1 ) (2.91)
and (S ′ , h ′ , E ′ ) satisfies both conditions (G) and (E) by Propositions 2.28 and 2.34. There exists an adapted r.s.p. of S ′ of the form

(u ′ 1 := u 1 , . . . , u ′ e 0 := u e 0 , u ′ e 0 +1 , . . . , u ′ n ′ 0 , u ′ n 0 +1 := u n 0 +1 , . . . , u ′ n := u n ). Since we do not assume that x ′ is a closed point, we have e 0 ≤ n ′ 0 ≤ n 0 in general, with n ′ := dimS ′ = n -(n 0 -n ′ 0
). We emphasize that the number of irreducible components e ′ of E ′ satisfies e 0 ≤ e ′ ≤ e and that e ′ = e in general because some of the u j /u 1 in (2.91) may be units. After reordering coordinates, we may also assume that

E ′ = div(u ′ 1 • • • u ′ e ′ ) and u ′ j := u j /u 1 , e 0 + 1 ≤ e ′ ≤ e. Since Y is Hironaka-permissible at x, we have (see Definition 2.31): ord W H(x) = p j∈J d j ≥ p.
Therefore I ′ := u -p 1 H(x) ⊆ S ′ and this ideal is monomial in (u ′ 1 , . . . , u ′ e ′ ), i.e.

I ′ =: (u ′ 1 H ′ 1 • • • u ′ e ′
H ′ e ′ ). We let:

x ′ := (H ′ 1 /p, . . . , H ′ e ′ /p, 0, . . . , 0) ∈ 1 p N n ′ ,
where

H ′ 1 = p( j∈J d j -1) and H ′ j = H j = pd j , 2 ≤ j ≤ e ′ .
(2.92)

Then (2.90) gives:      I ′ -i f p i,Z ′ ⊆ m S S ′ 1 ≤ i < i 0 (x) I ′ -i 0 (x) f p i 0 (x),Z ′ = S ′ I ′ -i f p i,Z ′ ⊆ S ′ i 0 (x) < i ≤ p.
(2.93)

This shows that

∆ Ŝ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ) = x ′ + R n ′ ≥0 . (2.94) If i 0 (x) < p, or if j∈J E d j ∈ N or if d j ′ ∈ N for some j ′ , 2 ≤ j ′ ≤ e ′ , then x ′ is not solvable (Definition 2.6) by (2.94), hence ∆ Ŝ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′
) is minimal. Therefore we may compute ǫ(x ′ ) from (2.94) and get ǫ(x ′ ) = 0, so the proposition is proved in this case.

If (i 0 (x) = p, j∈J E d j ∈ N and d j ′ ∈ N for all j ′ , 2 ≤ j ′ ≤ e ′ ), write f p,Z = γu px , γ ∈ S a unit and x := (d 1 , . . . , d e , 0, . . . , 0) ∈ 1 p N n . We have

in x ′ h ′ = Z ′ p + λ( e j=e ′ +1 λ H j j )U ′ px ′ , (2.95)
where λ ∈ k(x) (resp. λ j ∈ k(x ′ )) is the residue of γ (resp. of u j /u 1 ). We let:

λ ′ := λ e j=e ′ +1 λ H j j ∈ k(x ′ ), λ ′ = 0. If λ ′ ∈ k(x ′ ) p , then x ′ is not solvable and we also have ǫ(x ′ ) = 0. If λ ′ ∈ k(x ′ ) p , let C ′ := Spec k(x)[Z, U 1 , U e 0 +1 , . . . , U e ] (H) , H := in m S h = Z p + λ e j=e ′ +1 U H j j .
We claim that the affine cone C ′ is regular away from the torus

T := A e-e 0 +2 k(x) \V (Z j∈J E U j ).
To see this, let (λ l ) l∈Λ be an absolute p-basis of k(x). By Zariski's Jacobian criterion [START_REF] Matsumura | Commutative ring theory[END_REF] Theorem 30.5, the ideal of the singular locus of C ′ is:

I(SingC ′ ) = H, { ∂H ∂λ l } l∈Λ , { ∂H ∂U j } e ′ +1≤j≤e .
If d j ∈ N for some j, e ′ + 1 ≤ j ≤ e, then ∂H ∂U j does not vanish on T. Otherwise, we have λ ∈ k(x) p because x is a vertex of ∆ S (u 1 , . . . , u n ; Z) and is not solvable. Therefore ∂H ∂λ l does not vanish on T for any l ∈ Λ such that ∂λ ∂λ l = 0 and the claim is proved. We deduce that there exists a unit l ′ ∈ S ′ such that

v ′ := l ′ p + γ e j=e ′ +1 u j u 1 H j
is a regular parameter of S ′ transverse to

E ′ 1 := div(u ′ 1 • • • u ′ e ′ u ′ n 0 +1 • • • u ′ n ′ ), E ′ 1 ⊇ E ′ .
We may thus take u ′ e ′ +1 := v ′ in our r.s.p. of S ′ adapted to

E ′ . Let Z ′ 1 := Z ′ -l ′ u ′ px ′ , so the polyhedron ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ 1 ) has a vertex x ′ 1 := (H ′ 1 /p, . . . , H ′ e ′ /p, 1/p, 0, . . . , 0) ∈ 1 p N n ′ (2.96) which is not solvable, since x ′ 1 ∈ N n ′ . Let Z ′ 2 := Z ′ 1 -θ ′ , θ ′ ∈ S ′ , be such that ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′
2 ) is minimal. We deduce from (2.93) and (2.96) that

H(x ′ ) = (u ′ px ′ ), ǫ(x ′ ) = 1 and H ′ -1 F p,Z ′ 2 < U ′ 1 , . . . , U ′ e ′ > .
We get m(x ′ ) = 1 if x ′ = 0, and (m(x ′ ), ω(x ′ )) = (p, 0) otherwise by Lemma 2.78 as required.

If

s ′ ∈ Spec(S[u 1 /u n 0 , u e 0 +1 /u n 0 , . . . , u n 0 -1 /u n 0 ]
), it can be furthermore assumed that s ′ ∈ Spec(S[u e 0 +1 /u 1 , . . . , u n 0 /u 1 ]), i.e. u j /u n 0 is not a unit in S ′ for j ∈ J E . The proof is now a simpler variation of the above one: (2.91) is replaced by

E ′ = div( u 1 u n 0 u 2 • • • u e 0 u e 0 +1 u n 0 • • • u e u n 0 u n 0 ). The polyhedron ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ) in (2.94) is minimal except if (d j ∈ N for each j, 1 ≤ j ≤ e,
and λ ∈ k(x ′ ) p ) with notations as above. We have ǫ(x ′ ) = 0 (resp. ǫ(x ′ ) = 1) in the former (resp. in the latter) situation. This concludes the proof in case 1.

Case 2: ǫ(x) = 1. The proof runs parallel to that in case 1 and we only indicate the necessary changes. By assumption, W is an intersection of components of E (case 2a) or ǫ(y) = ǫ(x) = 1 (case 2b).

To begin with, let v ∈ S be such that H(x) -1 f p,Z = (v). By assumption, we have V (T F p,Z , E, m S ) = (0), so v is transverse to E.

In case 2a, we may assume that (u 1 , . . . , u e , v, u e+2 , . . . , u n ) is an adapted r.s.p. of S after renumbering variables. Since x 0 := (d 1 , . . . , d e , 1 p , . . . , 0) ∈ N n is the unique vertex of ∆ S (h; u 1 , . . . , u e , v, u e+2 , . . . , u n ; Z) induced by f p,Z , this polyhedron has no solvable vertex. In other terms, it can be assumed that v = u e+1 .

In case 2b, Proposition 2.12 implies that v ∈ I(W ), so (u 1 , . . . , u e , v) can be completed to an adapted r.s.p. of S such that I(W ) = ({u j } j∈J ) for some subset J ⊆ {1, . . . , n}. The polyhedron ∆ S (h; u 1 , . . . , u e , v, u e+2 , . . . , u n ; Z) has no solvable vertex either and it can also be assumed that v = u e+1 .

We remark in both cases 2a and 2b that, if ∆ S (h; u 1 , . . . , u n ; Z) has a vertex distinct from x 0 , then it has exactly two vertices: this follows from Theorem 2.36 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF], the other vertex being then given by

x 1 := ( D 1 p(p -1) , . . . , D e p(p -1) , 0, . . . , 0), (Disc Z (h)) =: (u D 1 1 • • • u De e ). (2.97)
After blowing up, we obtain a (S ′ , h ′ , E ′ ) again satisfying conditions (G) and (E).

In case 2a, there exists an adapted r.s.p. of S ′ of the form

(u ′ 1 := u 1 , . . . , u ′ e 0 := u e 0 , u ′ e 0 +1 , . . . , u ′ e 1 , u ′ e+1 := u e+1 , . . . , u ′ n := u n ), with J = {1, e 0 + 1, . . . , e} and E ′ = div(u ′ 1 • • • u ′ e ′ ) after reordering variables, 1 ≤ e 0 ≤ e ′ ≤ e 1 ≤ e. Then ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ) has again a vertex x ′ := (H ′
1 /p, . . . , H ′ e ′ /p, 0, . . . , 0, 1/p, 0, . . . , 0) ∈ N n-(e-e 1 ) , thus x ′ is not solvable. We deduce that ǫ(x ′ ) ≤ 1 and ω(x ′ ) = 0 follows from Lemma 2.78 if (m(x ′ ), ǫ(x ′ )) = (p, 1).

In case 2b, it can be assumed after reordering variables that (J ′ ) E := {2, . . . , e 0 }, J = {1, e 0 + 1, . . . , n 0 }, 1 ≤ e 0 ≤ e, e + 1 ≤ n 0 .

We let u ′ j ′ := u j ′ for j ′ ∈ J ′ and consider three distinct situations depending on x ′ , up to reordering coordinates:

(1) s ′ ∈ Spec(S[u e 0 +1 /u 1 , . . . , u n 0 /u 1 ]) and u e+1 /u 1 ∈ m S ′ . We may complete the family ({u j ′ } j ′ ∈J ′ ) to an adapted r.s.p. of S ′ by adding

(u ′ 1 := u 1 , u ′ e 0 +1 , . . . , u ′ e 1 , u ′ e 1 +1 := u e+1 /u 1 ), n ′ := dimS ′ = n-(n 0 -e 1 ). Then ∆ Ŝ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ) has a vertex x ′ := (H ′ 1 /p, . . . , H ′ e ′ /p, 1/p, 0, . . . , 0) ∈ N n ′ , thus x ′ is not solvable. We conclude that ǫ(x ′ ) ≤ 1 and that ω(x ′ ) = 0 if (m(x ′ ), ǫ(x ′ )) = (p, 1) by Lemma 2.78. (2) s ′ ∈ Spec(S[u 1 /u n 0 , u e 0 +1 /u n 0 , . . . , u n 0 -1 /u n 0 ]) and u e+1 /u n 0 ∈ m S ′ ,
where n 0 > e + 1. After dealing with (1), we may assume furthermore that u j /u n 0 ∈ m S ′ , j ∈ J E . We complete the family ({u j ′ } j ′ ∈J ′ ) to an adapted r.s.p. of S ′ by adding

(u ′ e 0 +1 := u e 0 +1 /u n 0 , . . . , u ′ e+1 := u e+1 /u n 0 , u ′ n 1 , . . . , u ′ n 0 -1 , u ′ n 0 := u n 0 ), with n ′ := dimS ′ = n -(n 1 -e -2)
. We conclude as in (1).

(3) I(W )S ′ = (u e+1 ). We complete the family ({u j ′ } j ′ ∈J ′ ) to an adapted r.s.p. of S ′ by adding

(u ′ 1 := u e+1 , u ′ e 0 +1 , . . . , u ′ n 1 ), n ′ := dimS ′ = n -(n 0 -n 1 )
.

Let E ′ =: div(u ′ 1 • • • u ′ e ′
) and consider two situations as in case 1:

If 1 p + j∈J E d j ∈ N or if d j ′ ∈ N for some j ′ , 2 ≤ j ′ ≤ e ′ , then the polyhedron ∆ Ŝ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ) is minimal and we have ǫ(x ′ ) = 0. If ( 1 p + j∈J E d j ∈ N and d j ′ ∈ N for every j ′ , 2 ≤ j ′ ≤ e ′ ), the initial form polynomial in x ′ h ′ has the form in x ′ h ′ = Z ′ p -µ p-1 U ′ (p-1)x ′ Z ′ + λ( e j=e ′ +1 λ H j j )U ′ px ′ ,
where λ ∈ k(x) (resp. λ j ∈ k(x ′ )) is the residue of γ (resp. of u j /u e+1 ), vid. (2.95). We have µ = 0 in the above formula precisely if

U p(x 1 -x 0 ) = U j 0 /U e+1 , u j 0 /u e+1 ∈ S ′ a unit
for some j 0 , e 0 + 1 ≤ j 0 ≤ e with notations as in (2.97). Then µ p-1 is the residue in k(x ′ ) of

γ p-1,Z e j=e ′ +1 u j u e+1 A p-1 ,j
with notations as in Theorem 2.36 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. The end of the proof goes along as in case 1.

This completes the proof of (3), hence the proof of the proposition in case 2.

Remark 2.80. This proposition is a lighter version of Theorem 3.13 where it is assumed that ω(x) > 0 and that the blowing up centers are permissible of the first or second kind (Definitions 3.1 and 3.5 below).

Theorem 2.81. Assume that (m(x), ω(x)) = (p, 0), where {x} = η -1 (m S ).

For every valuation µ of L = Tot(S[X]/(h)) centered at x, there exists a finite and independent composition of local Hironaka-permissible blowing ups (2.87) such that m(x r ) < p.

Proof. We will produce a Hironaka-permissible center Y ⊂ (X , x) w.r.t. E satisfying the assumptions of Proposition 2.79 and such that the following holds:

(*) let π : X ′ → (X , x) be the blowing up along Y and x ′ ∈ π -1 (x). Then

δ(x ′ ) < δ(x).
Applying Proposition 2.79, the center x 1 ∈ X ′ of a given valuation µ again satisfies the assumptions of the theorem if m(x 1 ) = p. Iterating, any finite sequence (2.87) induces a sequence

δ(x r ) < δ(x r-1 ) < • • • < δ(x) provided that m(x i ) = p, 1 ≤ i ≤ r -1. Since δ(x i ) ∈ 1
p N, we have δ(x r ) < 1 for some r ≥ 1, hence m(x r ) < p by Proposition 2.10(2), so the theorem follows from claim (*). In order to construct Y with the required properties, we consider two cases as in the proof of Proposition 2.79.

Case 1: ǫ(x) = 0. We have δ(x) = e j=1 d j ≥ 1. Therefore there exists a (not necessarily unique) subset

J ⊆ {1, . . . , e}, j∈J d j ≥ 1,
with smaller possible number of elements among all subsets of {1, . . . , e} with this property. Let W := V ({u j } j∈J ) ⊂ SpecS and remark that

ord W H(x) = p j∈J d j ≥ p.
Hence Y := η -1 (W ) = V (Z, {u j } j∈J ) is Hironaka-permissible w.r.t. E and W is an intersection of components of E. By (2.92), we have

ord m S ′ H(x ′ ) ≤ p(δ(x) + j∈J\{j 0 } d j -1), (2.98)
where I(W )S ′ = (u j 0 ). The minimality property required of J implies that

j∈J\{j 1 } d j < 1 for every j 1 ∈ J (so j∈J d j < 2 if | J |≥ 2). (2.99) If ǫ(x ′ ) = 0, we deduce from (2.98) that pδ(x ′ ) = ord m S ′ H(x ′ ) < pδ(x) as required in (*). Note that if | J |= 1, we have λ = λ ′ in (2.95) and S = S ′ , hence λ ′ ∈ k(x ′ ) p = k(x) p . Since ǫ(x ′ ) = 0 in this situation, we may now assume that | J |≥ 2. If ǫ(x ′ ) = 1,
we are in the situation discussed in (2.96). We may then take

j 0 = 1, E ′ = div(u ′ 1 • • • u ′ e ′ ) and have j∈J d j ∈ N, d j ∈ N for 2 ≤ j ≤ e ′ .
By (2.99), we have j∈J d j = 1, d j = 0 for 2 ≤ j ≤ e ′ , so H(x ′ ) = (1) and m(x ′ ) = 1. This concludes the proof in case 1.

Case 2: ǫ(x) = 1. We have δ(x) = 1 p + e j=1 d j ≥ 1. If δ(x) > 1, there exists a subset J ⊆ {1, . . . , e}, j∈J d j ≥ 1,
with smaller possible number of elements among all subsets of {1, . . . , e} with this property as in case 1 and we also let W := V ({u j } j∈J ) ⊂ SpecS. The proof goes along as in case 1, with

pδ(x ′ ) -pδ(x) ≤ ord m S ′ H(x ′ ) -ord m S H(x) < 0.
If δ(x) = 1, we may assume that H(x) -1 f p,Z = (u e+1 ) and that (2.97) holds if ∆ S (h; u 1 , . . . , u n ; Z) has more than one vertex. In this case, this polyhedron has exactly two vertices and we have H(x) -(p-1) f p p-1,Z = (u j 0 ) p-1 for some j 0 , 1 ≤ j 0 ≤ e by Theorem 2.36 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. We deduce that

H(x) -i f p i,Z ⊆ (u j 0 , u e+1 ) i , 1 ≤ i ≤ p (2.100)
by definition of ∆ S (h; u 1 , . . . , u n ; Z). We let J := {j : d j > 0} ∪ {e + 1} and 3 Permissible blowing ups.

W := V ({u j } j∈J ) ⊂ SpecS, Y := η -1 (W ) = V (Z, {u j } j∈J ). We have ord W H(x) = p, so Y is Hironaka-permissible w.r.t. E. Since H(x) -1 f p,Z = (u e+1 ), we have ǫ(y) = ǫ(x) =
3.1 Blowing ups of the first and second kind.

In this section, we introduce a notion of permissible blowing up which is well behaved w.r.t. our main resolution invariant y → ι(y) on X . We assume that m (ii) ǫ(y) = ǫ(x).

(x) = p, {x} = η -1 (m S )
If y ∈ X satisfies m(y) = p, it follows from the definition that Y := {y} is permissible of the first kind at y. It also follows from (ii) that a permissible center of the first kind has codimension at least two in X .

The main result of this chapter (Theorem 3.13 below) will require comparing the initial form polynomials in W h and in m S h. We keep notations as in section 2.4: given well adapted coordinates (u 1 , . . . , u n ; Z) at x, we let

W := η(Y), I(W ) = ({u j } j∈J ). (3.1)
We denote:

in W h = Z p + p i=1 F i,Z,W Z p-i ∈ G(W )[Z]
and (Theorem 2.36 since ǫ(x) > 0)

in m S h = Z p -G p-1 Z + F p,Z ∈ G(m S )[Z].
There are associated homogeneous submodules

H W ⊆ G(W ) d W (resp. H := H m S ⊆ G(W ) d ) (3.2)
by (2.48), with

d W := j∈J E H j , d = e j=1 H j ,
where J E := J ∩ {1, . . . , e}.

A word of caution is required at this point: formula (2.48) defines the monomial ideal H W which is the initial form of H(x) in G(W ) and is different in general from the ideal H(Ξ) associated to the triple

(G(W ) Ξ , in W h, E W ), Ξ := ({U j } j∈J ) + m O W .

For an example, let h

:= Z p + u a 1 (u b 2 + u b+1 1 ), ab > 0, W = m S , E =div(u 1 u 2 ), we have H W = U a 1 , H(Ξ) = U a 1 U b 2 .
Corresponding to the above choice for H W (resp. to H), there are associated O W -submodules

V (F p,Z,W , E, W ) ⊆ G(W ) ǫ(y)-1 , J(F p,Z,W , E, W ) ⊆ G(W ) ǫ(y) (resp. k(x)-vector subspaces V (F p,Z , E, m S ) ⊆ G(m S ) ǫ(x)-1 , J(F p,Z , E, m S ) ⊆ G(m S ) ǫ(x) )
given by (2.49). Notation 3.2. We first recall notations and definitions from section 2.4. We denote

J E := J ∩ {1, . . . , e}, J ′ := {1, . . . , n}\J and (J ′ ) E := {1, . . . , e}\J E .
The image m S of m S in O W has regular parameters (u j ′ ) j ′ ∈J ′ , the respective residues of the corresponding parameters of S.

Let now d ∈ N be fixed and

F = |a|=d fa U a ∈ G(W ) d = O W [{U j } j∈J ] d . Note that gr m S G(W ) d ≃ gr m S G(W ) d and
F := |a|=d (cl d 0 fa )U a ∈ (gr m S G(W ) d ) d 0 . (3.3)
This notation requires specifying d 0 to avoid ambiguity. We extend the notation to homogeneous submodules M ⊆ G(W ) d as follows:

M :=< F , F ∈ M >⊆ (gr m S G(W ) d ) d 0
for fixed d 0 ≤ min{d 0 (F ), F ∈ M} with obvious notations. For fixed d, d 0 , there is an inclusion of S/m S -vector spaces:

(gr m S G(W ) d ) d 0 ⊂ G(m S ) d+d 0 < ({U j } j∈J ) d+1 ∩ G(m S ) d+d 0 > . ( 3
I(W ) = ({u j } j∈J ), the initial form in m S h ∈ G(m S )[Z] satisfies H -1 < G p , F p,Z >⊆ k(x)[{U j } j∈J ] ǫ(x) ,
with notation as in (3.2).

Proof. The existence of well adapted coordinates (u 1 , . . . , u n ; Z) such that I(W ) = ({u j } j∈J ) follows from Proposition 2.12. This theorem furthermore implies that the polyhedron

∆ Ŝ (h; {u j } j∈J ; Z) = pr J (∆ S (h; u 1 , . . . , u n ; Z)) is minimal, (3.5)
where pr J : R n → R J denotes the projection on the (u j ) j∈J -space. By (ii) of Definition 3.1, we have ǫ(x) = ǫ(y). Therefore

H -i F p i,Z = cl 0 (H -i W F p i,Z,W ) ⊆ G(m S ) iǫ(x) = k(x)[U 1 , . . . , U n ] iǫ(x)
is simply the reduction of

H -i W F p i,Z,W modulo m S for 1 ≤ i ≤ p, i.e. taking d 0 = 0 in Notation 3.2, via the inclusion (3.4) k(x)[{U j } j∈J ] iǫ(y) ≃ (gr m S G(W ) iǫ(y) ) 0 ⊂ G(m S ) iǫ(y) ≃ k(x)[U 1 , . . . , U n ] iǫ(x) .
We get respectively (H -1 G p ) p-1 , (H -1 F p,Z ) p for i = p -1, p and this completes the proof.

The following corollary will be required in the proof of the blowing up theorem below. The adapted cone Max(x) ⊆ G(m S ) is defined in Definition 2.72.

Corollary 3.4. With notations as above, let Y be permissible of the first kind at x. The defining ideal IMax(x) ⊆ G(m S ) of Max(x) satisfies

IMax(x) = (IMax(x) ∩ k(x)[{U j } j∈J ])G(m S ).
Proof. This follows from Proposition 3.3 and Definition 2.72. Note that the truncation operator T used in the Definition of Max(x) does not affect the conclusion of the corollary since it is obvious from the definitions that:

V (F p,Z , E, m S ) ⊆ k(x)[{U j } j∈J ] ǫ(x)-1 ⇒ V (T F p,Z , E, m S ) ⊆ k(x)[{U j } j∈J ] ǫ(x)-1 .
The same implication holds for J(F p,Z , E, m S ) and J(T F p,Z , E, m S ).

We now define a second kind of permissible blowing up. Definition 3.5. Let Y ⊂ X be an integral closed subscheme with generic point y. We say that Y is permissible of the second kind at x if m(y) = m(x) = p and the following conditions hold:

(i) Y is Hironaka-permissible w.r.t. E at x (Definition 2.21);

(ii) ǫ(y) = ǫ(x) -1 and i 0 (y) ≤ i 0 (x);

(iii) J(F p,Z,W , E, W ) := cl 0 J(F p,Z,W , E, W ) = 0.

The following important example constructs a threefold X such that every resolution of singularities X → X which is a composition of Hironakapermissible blowing ups does actually involve blowing up a permissible curve of the second kind.

Example 3.6. Let k be a perfect field of characteristic p > 0,

A := k[u 1 , u 2 , u 3 ], P ∈ k[T ]\k[T p ] and take h := Z p + P (u 3 )u p 2 + u p+1 1 ∈ A[Z], E := div(u 1 )
. For x ∈ U, there exist well adapted coordinates

Let Y := V (Z, u 1 , u 2 ) ⊆ Sing p X
(u 1 , u 2 , v x ; Z x := Z -γ x u 2 ) at x, γ x ∈ A η(x) a unit such that h = Z p x + v x u p 2 + u p+1 1 ∈ A η(x) [Z x ].
Then Y is permissible of the second kind at every x ∈ U since

J(F p,Zx,W , E, W ) = ∂F p,Zx,W ∂v x = U p 2 = 0, F p,Zx,W = v x U p 2 ∈ G(W ) p
with notations as in Definition 3.5(iii). This is dealt with in the course of the proof of Theorem 1.5 in Proposition 7.21 when applying Lemma 7.20 (κ(x) = 2 in this example, cf. Definition 5.1).

When n = 3, permissible blowing ups of the second kind only occur in Propositions 7.21 and 7.29 (κ(x) = 2). Proposition 3.7. Let Y be permissible of the second kind at x. For any well adapted coordinates (u 1 , . . . , u n ; Z) at x such that

I(W ) = ({u j } j∈J ), the initial form in m S h ∈ G(m S )[Z] satisfies    H -1 G p ⊆ U j 0 k(x)[{U j } j∈J ] ǫ(y) for some j 0 ∈ (J ′ ) E H -1 F p,Z = < j∈J ′ U j ′ Φ j ′ ({U j } j∈J ) + Ψ({U j } j∈J ) >⊆ G(m S ) ǫ(x)
.

(3.6) with Φ j ′ = 0 for some j ′ ∈ J ′ \(J ′ ) E . In particular ǫ(y) = ω(x).

Proof. We argue as in the proof of Proposition 3.3 and build up from (3.5). By (ii) of Definition 3.5, we have ǫ(x) = ǫ(y) + 1. Therefore

cl 0 (H -i W F p i,Z,W ) = 0, 1 ≤ i ≤ p. This shows that H -i W F p i,Z,W ⊆ m S O W [{U j } j∈J E ] iǫ(y) . We have ǫ(y) > 0, so F i,Z,W = 0, 1 ≤ i ≤ p -2 by Theorem 2.36. For i = p -1, we have -F p-1,Z,W = G p-1 W for some G W ∈ G(W ) δ(y) (so G W = 0 if δ(y) ∈ N). We deduce that H -1 W (G p W , F p,Z,W ) ⊆ m S O W [{U j } j∈J E ] ǫ(y) . (3.7)
If i 0 (x) = p, we have H -1 G p = 0 so the first part of (3.6) is trivial. If i 0 (x) = p -1, we have i 0 (y) = p -1 by Definition 3.5(ii), so G W = 0. The first part of (3.6) then follows from (3.7), i.e.

H -1 G p = cl 1 (H -1 W G p W ) ⊆ U j 0 k(x)[{U j } j∈J ] ǫ(y) , for some j 0 ∈ (J ′ ) E .
We deduce from (3.7) that

J (F p,Z,W , E, W ) =< cl 0 (H -1 W ∂F p,Z,W ∂u j ′ ), j ′ ∈ J ′ \(J ′ ) E >⊆ k(x)[{U j } j∈J ] ǫ(y) .
Taking classes as in (3.3) with d 0 = 1, we get

cl 1 (H -1 W F p,Z,W ) ⊆ j ′ ∈J ′ U j ′ k(x)[{U j } j∈J ] ǫ(y) . Since cl 1 (H -1 W F p,Z,W ) is a homomorphic image of H -1 F p,Z ∈ G(m S ) ǫ(x)
as described in (3.4), there exists an expansion (3.6). For j ′ ∈ J ′ \(J ′ ) E , we have

H -1 ∂F p,Z ∂U j ′ = cl 0 (H -1 W ∂F p,Z,W ∂u j ′ ).
Collecting together for all j ′ ∈ J ′ \(J ′ ) E , we get

J(F p,Z,W , E, W ) =< H -1 ∂F p,Z ∂U j ′ , j ′ ∈ J ′ \(J ′ ) E >⊆ k(x)[{U j } j∈J ] ǫ(y)
and the second part of (3.6) follows from Definition 3.5(iii).

Note that ǫ(y) = ω(x) is an immediate consequence of Definition 2.68 if i 0 (m S ) = p. If i 0 (m S ) = p -1, we must introduce a truncation operator T : G(m S ) δ(x) → G(m S ) δ(x) in order to compute ω(x). The first part of (3.6) now shows that there exists j 0 ∈ (J ′ ) E such that

H -1 (F p,Z -T F p,Z ) ∈ U j 0 k(x)[{U j } j∈J ] ǫ(y) . Since J (F p,Z,W , E, W ) ⊆ k(x)[{U j } j∈J ] ǫ(y)
, we thus have:

H -1 ∂F p,Z ∂U j ′ = H -1 ∂T F p,Z ∂U j ′
for every j ′ ∈ J ′ \(J ′ ) E . This proves that ω(x) = ǫ(y).

Note that it follows from the above proposition that a permissible center of the second kind has codimension at least two in X , since ǫ(y) > 0.

Permissible blowing ups of the second kind appear naturally from permissible blowing ups of the first kind if one requires stability by regular base change: Theorem 3.8. Let S ⊆ S be a local base change which is regular, S excellent. Let x ∈ η-1 (m S ) and x ∈ η -1 (m S ) be its image.

If Y ⊂ X is a permissible center (of the first or second kind) at x, then

Ỹ := Y × S Spec S ⊆ X = X × S Spec S
is permissible (of the first or second kind) at x.

Proof. We denote ( S, h, Ẽ) and (u 1 , . . . , u ñ) as in Notations 2. [START_REF] Bravo | Singularities in positive characteristic, stratification and simplification of the singular locus[END_REF]

ǫ(ỹ) = ǫ(y), ǫ(x) ≥ ǫ(x), i 0 (ỹ) = i 0 (y), i 0 (x) = i 0 (x)
Cases of inequality ǫ(x) > ǫ(x) are classified in ibid.(2.ii).

Suppose that ǫ(x) > ǫ(x). Then,

F p,Z ∈ k(x)[U p 1 , . . . , U p n ] and i 0 (m S ) = i 0 (m S ) = p. Then Y is permissible of the first kind since F p,Z ∈ k(x)[U p 1 , . . . , U p n ]
is incompatible with the conclusion of Proposition 3.7. Note that

ǫ(y) = ǫ(x) = ǫ(x) -1 = ǫ(ỹ).
We claim that Ỹ is permissible of the second kind at x.

To prove the claim, note that Definition 3.5(i) and (ii) are already checked. We have

H -1 ∂F p, Z ∂U j ′ = H -1 Φ j ′ (U 1 , . . . , U n ) = 0, (3.8)
with notations as in Theorem 2.74(2.ii) for some j ′ , n + 1 ≤ j ′ ≤ ñ. Since H(x) = H(x) S by Theorem 2.74(2.i), and H -1 F p,Z ⊆ k(x)[{U j } j∈J ] ǫ(x) by Proposition 3.3, we have

H -1 F p, Z ⊆ ñ j=1 U j k(x)[{U j } j∈J ] ǫ(x) .
This proves that Definition 3.5(iii) holds and Ỹ is permissible of the second kind at x.

Assume now that ǫ(x) = ǫ(x). If Y is permissible of the first kind at x, we have ǫ(ỹ) = ǫ(x), so Ỹ is also permissible of the first kind at x.

If Y is permissible of the second kind at x, Definition 3.5(ii) is checked. Finally by Proposition 3.7, the polyhedron ∆ S (h; u 1 , . . . , u n ; Z) has a vertex x such that x j ′ ∈ N for some j ′ ∈ J ′ \(J ′ ) E . The corresponding vertex

x := (x, 0, . . . , 0 ñ-n ) ∈ ∆ S (h; u 1 , . . . , u ñ; Z)
is thus not solvable. We hence get x ∈ ∆ S (h; u 1 , . . . , u ñ; Z) and Definition 3.5(iii) is checked. Hence Ỹ is permissible of the second kind at x as required, since H(x) = H(x) S.

Blowing up Theorem.

Let π : X ′ → X be the blowing up along a permissible center Y (of the first or second kind) at x ∈ Y, {x} = η -1 (m S ). Our objective is to relate ω(x ′ ) to ω(x) for points x ′ ∈ π -1 (x). The main result is Theorem 3.13 which states that the pair (m(x), ω(x)) does not increase and studies the equality case. In contrast, we may have ǫ(x ′ ) > ǫ(x), see 3. [START_REF] Bierstone | Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant[END_REF](2) about this jumping phenomenon.

We keep notations as in Proposition 2.22 and Proposition 2.28. Then σ : S ′ → SpecS denotes the blowing up along W and there is a commutative diagram (2.17). Let

η ′ : X ′ → S ′ , s ′ := η ′ (x ′ ) ∈ σ -1 (m S ), S ′ := O S ′ ,s ′ .
We denote by W ′ := σ -1 (W ) and E ′ := σ -1 (E) red . We do not change notations to denote stalks at s ′ , i.e. we will write η ′ : X s ′ → SpecS ′ for the stalk at s ′ of the above map η ′ , and W ′ , E ′ for the stalks at s ′ of the corresponding divisors. By Proposition 2.28, we have η

′ -1 (s ′ ) = {x ′ } if x ′ is not a regular point of X ′ .
For the purpose of computations, we shall pick well adapted coordinates (u 1 , . . . , u n ; Z) such that

I(W ) = ({u j } j∈J ), Y = V (Z, {u j } j∈J ).
with notations as in (3.1). We denote by u ∈ S ′ a local equation for W ′ , which can be taken to be some u j 1 , where j 1 ∈ J depends on s ′ . We have

X ′ = Spec(S ′ [X ′ ]/(h ′ )), where h ′ := u -p h = X ′ p + f 1,X ′ X ′ p-1 + • • • + f p,X ′ ∈ S ′ [X ′ ], (3.9)
and

X ′ := Z/u, f i,X ′ := u -i f i,Z ∈ S ′ for 1 ≤ i ≤ p. (3.10)
Since Y is permissible, we have ǫ(y) > 0 so the initial form in W h reduces to :

in

W h = Z p -G p-1 W Z + F p,Z,W ∈ G(W )[Z], (3.11)
with

G W ∈ G(W ) δ(y) and F p,Z,W ∈ G(W ) pδ(y) (in particular G W = 0 if δ(y) ∈ N). Since σ -1 (W ) = ProjG(W ), the restriction map
gives an inclusion

U -d G(W ) d = O W [{U j /U} j∈J ] ≤d ⊂ O W ′ ,s ′ = S ′ /(u) (3.12)
for each d ≥ 0. There is an identification:

U -d G(W ′ ) d = (O W [{U j /U} j∈J ]) s ′ = S ′ /(u). (3.13)
Finally, we note that

D W ′ = D(W ′ ) by (2.46) since W ′ is a component of E ′ .
These remarks are essential for stating the blow up formula in Proposition 3.9(v) below.

Proposition 3.9. (Blow up formula) Let π : X ′ → X be the blowing up along a permissible center Y at x, {x} = η -1 (m S ) and x ′ ∈ π -1 (x) be a closed point. With notations as above, the following holds:

(i) there exists a r.s.p.

(u ′ 1 , . . . , u ′ n ) of S ′ which is adapted to (S ′ , h ′ , E ′ ); (ii) in W ′ h ′ = X ′ p -G p-1 W ′ X ′ + F p,X ′ ,W ′ ∈ G(W ′ )[X ′ ]
and is given by

G W ′ = U -1 G W ∈ G(W ′ ) δ(y)-1 , F p,X ′ ,W ′ = U -p F p,Z,W ∈ G(W ′ ) p(δ(y)-1) ;
(iii) the polyhedron ∆ S ′ (h ′ ; u; X ′ ) is minimal;

(iv) we have H(x ′ ) = u ǫ(y)-p H(x) ⊆ S ′ ;

(v) there is an equality of ideals of

O W ′ ,s ′ :    H -1 W ′ G p W ′ = (U -ǫ(y) H -1 W G p W ) s ′ , J(F p,X ′ ,W ′ , E ′ , W ′ ) = (U -ǫ(y) J(F p,Z,W , E, W )) O W ′ ,s ′ .
Proof. Statement (i) is proved in Proposition 2.22. The formula in (ii) is obvious from (3.9), (3.10) and (3.11).

If i 0 (W ) = p -1, i.e. G W = 0 in (3.11), we have G W ′ = 0 by (ii), so ∆ S ′ (h ′ ; u; X ′ ) ⊆ R ≥0 is minimal. If i 0 (W ) = p, then F p,Z,W ∈ G(W ) p , i.e. δ(y) ∈ pN or U -δ(y) F p,Z,W ∈ k(W ′ ) p . Note that G(W ) p = (k(W ′ )[U, U -1 ]) p ∩G(W ) since G(W ) is integrally closed. By (ii), F p,X ′ ,W ′ = U -p F p,Z,W so F p,X ′ ,W ′ ∈ G(W ′
) p and this proves (iii).

To prove (iv), first consider those irreducible components W j = div(u j ) of E, 1 ≤ j ≤ e, whose strict transform W ′ j passes through s ′ . We may pick a r.s.p. (u ′ 1 , . . . , u ′ n ) of S ′ which is adapted to (S ′ , h ′ , E ′ ), containing u, u ′ j := u j /u for j ∈ J E and u ′ j := u j for j ∈ J ′ . Let

in W j h(Z) = Z p + F 1,Z,W j Z p-1 + • • • + F p,Z,W j ∈ S/(u j )[U j ][Z].
We have in

W ′ j h ′ = in W j u -p h(uX ′ ) ∈ S ′ /(u ′ j )[U ′ j ][X ′ ], since u is a unit in S ′ (u ′ j ) = S (u j ) . Since ∆ S (h; u 1 , . . . , u n ; Z) is minimal, we have ∆ S (u j ) (h; u j ; Z) = ∆ S ′ (u ′ j ) (h ′ ; u ′ j ; X ′ )
minimal as well by Proposition 2.12, hence ord (u ′ j ) H(x ′ ) = ord (u j ) H(x). By (ii) and (iii), we have ord (u) H(x ′ ) = p(δ(y) -1). Therefore

ord (u) H(x ′ ) -ord (u) H(x) = p(δ(y) -1) -ord W H(x) = ǫ(y) -p
and the conclusion follows.

We now prove (v). The first part of the statement follows immediately from (ii) and (iv). With notations as in (2.47), we have

     J(F p,Z,W , E, W ) = H -1 W J (F p,Z,W , E, W ) ⊆ G(W ) ǫ(y) , J(F p,X ′ ,W ′ , E ′ , W ′ ) = H -1 W ′ J (F p,X ′ ,W ′ , E ′ , W ′ ) ⊆ G(W ′ ) 0 .
We now define and make explicit the required inclusion

G(W ) ǫ(y) ⊂ G(W ′ ) 0 .
Let us first complete the O W -algebra G(W ) for the m S -adic topology. There is an induced inclusion

G(W ) ⊂ lim ← G(W ) m n S . The inclusions G(W ) ⊂ G(W ′ ) ⊂ G(W ′ ) lead to the following inclusions G(W ) ⊂ lim ← G(W ) m n S ⊂ lim ← G(W ′ ) m n S ⊂ lim ← G(W ′ ) m n S .
(3.14)

Explicitly, we pick an isomorphism

O W ≃ k(x)[[{u j ′ } j ′ ∈J ′ ]]
given by Proposition 2.40. The last three terms in (3.14) are formal power series rings in variables {u j ′ } j ′ ∈J ′ with respective coefficient rings:

A := k(x)[{U j } j∈J ] ⊂ A ′ [U] := k(x)[{V j } j∈J\{j 1 } ] s ′ [U] = O σ -1 (m S ),s ′ [U] ⊆ Â′ [U],
where

V j := U j /U ∈ G(W ′ ) 0 , j ∈ J\{j 1 }. Finally, we have lim ← G(W ′ ) m n S ≃ Â′ [U][[{u j ′ } j ′ ∈J ′ ]], G(W ′ ) ≃ Â′ [[{u j ′ } j ′ ∈J ′ ]][U].
(3.15)

The required map

G(W ) ǫ(y) ⊂ G(W ′ ) 0 in (v) is given by F → U -ǫ(y) F .
Applying (ii) and (iv), we get:

F p,X ′ ,W ′ = U -p F p,Z,W , H W ′ = H W U ǫ(y)-p G(W ′ ).
Since D • U p = 0 for every D ∈ D W ′ , (v) can be written in the following form:

U -degF p,Z,W J (F p,Z,W , E ′ , W ′ ) = (U -degF p,Z,W J (F p,Z,W , E, W )) S ′ /(u). (3.16) Any D ∈ {u j ′ ∂ ∂u j ′ } j ′ ∈(J ′ ) E ∪ { ∂ ∂u j ′ } j ′ ∈J ′ \(J ′ ) E
extends in the obvious way to the right hand side of this diagram, so it commutes with the inclusion A ⊂ Â′ [U]. In other terms, we are reduced to a statement on the coefficients of the power series (3.15) expliciting (3.14). This reduces the proof of (v) to the special case where W = {m S } is the closed point (J = {1, . . . , n}).

By (2.45), the G(m S )-module D m S is generated by the family

{U j ∂ ∂U j } 1≤j≤e , {U j 1 ∂ ∂U j } 1≤j 1 ≤n,e+1≤j , { ∂ ∂λ l } l∈Λ .
The A ′ -module of absolute differentials

Ω 1 A ′ log(U e j=1 V j )
has a basis obtained by collecting together (dλ l ⊗ 1) l∈Λ , dU/U and the {dV j /V j } 1≤j≤e , {dV j } e+1≤j≤n with j = j 1 . For F ∈ A, we deduce the following standard formulae in

A ′ :              U ∂F ∂U = n j=1 U j ∂F ∂U j V j ∂F ∂V j = U j ∂F ∂U j 1 ≤ j ≤ e, j = j 1 ∂F ∂V j = U ∂F ∂U j e + 1 ≤ j ≤ n, j = j 1 . (3.17)
Taking F ∈ A d , d ∈ N, we have for j ≥ e + 1:

(U -d {U j 1 ∂F ∂U j } 1≤j 1 ≤n )A ′ = (U -d U ∂F ∂U j )A ′ .
Collecting together this equation with (3.17), we get

U -d J ′ (F, E ′ , W ′ ) = (U -d J (F, E, m S ))A ′ ,
where

J ′ (F, E ′ , W ′ ) = {D ′ ∈ Der(G(W ′ )) : D ′ • I(E ′ (W ′ )) ⊆ I(E ′ (W ′ ))},
notations as in Proposition 2.52. Since S ′ is essentially of finite type over k(x), this proposition implies that J (F, E ′ , W ′ ) = J ′ (F, E ′ , W ′ ) Â′ . This concludes the proof.

We now state the main theorem of this section. Recall that the function y → ω(y) and κ(y) ∈ {1, ≥ 2} have been defined for given (S, h, E) and y ∈ X (Definition 2.67 and Definition 2.68). By Proposition 2.34, (S ′ , h ′ , E ′ ) satisfies again conditions (G) and (E). The values of ǫ(x ′ ), ι(x ′ ) are computed w.r.t. the adapted structure (S ′ , h ′ , E ′ ). Notation 3.10. Choice of coordinates: by Proposition 3.9(i), there exists a r.s.p. (u ′ 1 , . . . , u ′ n ′ ) which is adapted to (S ′ , h ′ , E ′ ) for some n ′ ≤ n. We take u ′ 1 := u. Let

u ′ i := u j i u , 2 ≤ i ≤ e ′ 0 ,
where {j 2 , . . . , j e ′ 0 } := {j ∈ J E :

u j u ∈ m S ′ }.
Let {j e ′ 0 +1 , . . . , j e ′ } := (J ′ ) E , {j e ′ +1 , . . . , j n ′ 0 } =: J ′ \(J ′ ) E . We take

u ′ i := u j i , e ′ 0 + 1 ≤ i ≤ n ′ 0 .
Let

u ′ i := u j i u , n ′ 0 + 1 ≤ i ≤ n ′ 1 , where {j n ′ 0 +1 , . . . , j n ′ 1 } := {j ∈ J\J E : u j u ∈ m S ′ } and complete (u ′ 1 , . . . , u ′ n ′ 1 ) to a r.s.p. (u ′ 1 , . . . , u ′ n ′ ) of S ′ . Notation 3.11. Let S ′ := Ôσ -1 (m S ),s ′ = Ŝ′ /(u, {u j ′ } j ′ ∈J ′ ) = k(x)[{U j /U} j∈J ] m ′ ,
where m ′ denotes the ideal of the restriction of s ′ to σ -1 (m S ):

m ′ := ({u ′ i } i∈F ), F := {2, . . . , e ′ 0 } ∪ {n ′ 0 + 1, . . . , n ′ }.
For I ′ ⊆ Ŝ′ /(u) an ideal, we denote by

ordI ′ := ord m Ŝ′ /(u) I ′ = min ϕ ′ ∈I ′ {ord m Ŝ′ /(u) ϕ ′ }, ordI ′ := ord m ′ I ′ S ′ .
For every I ′ ⊆ Ŝ′ /(u), we have ordI ′ ≤ ordI ′ ≤ +∞. If furthermore d ′ is given, d ′ ≤ ordI ′ , we write

I ′ ⊆ gr m ′ S ′ d ′ = k(x ′ )[{U ′ i } i∈F ] d ′
for the initial part of degree d ′ of the ideal I ′ S ′ .

We now introduce the adapted cone associated to a permissible blowing up. Recall the definition of B from (2.69) (cf. also Definition 2.68). We have B = ∅ if i 0 (m S ) = p, and

B = {j : U j divides H -1 G p } if i 0 (m S ) = p -1.
Definition 3.12. Let Y ⊂ X , with generic point y, be a permissible center at x. We define a subcone

C(x, Y) ⊂ Spec(k(x)[{U j } j∈J ])
as follows: if Y is of the first kind, we let:

C(x, Y) := Spec k(x)[{U j } j∈J ] (IMax(x) ∩ k(x)[{U j } j∈J ]) ;
if Y is of the second kind, we let B J := B\{j 0 } with notations as in Proposition 3.7 and define:

C(x, Y) := Max(J(F p,Z,W , E, W )) ∩ {U B J = 0}.
In both cases, we denote the associated projective cone by

P C(x, Y) ֒→ σ -1 (m S ) ≃ P |J|-1 k(x) .
Theorem 3.13. Assume that m(x) = p, ω(x) > 0, where {x} = η -1 (m S ).

Let π : X ′ → X be the blowing up along a permissible center Y (of the first kind or second kind) at x, x ′ ∈ π -1 (x) and η ′ : X ′ → SpecS ′ be with notations as above, where

s ′ = η ′ (x ′ ). Then (m(x ′ ), ω(x ′ ), κ(x ′ )) ≤ (m(x), ω(x), κ(x)). (3.18)
If equality holds in (3.18), then s ′ ∈ P C(x, Y).

If ǫ(x ′ ) > ǫ(x), the following holds:

(1) we have i 0 (m S ) = p, ǫ(y) = ǫ(x) = ω(x), δ(y) ∈ N, H j ′ ∈ pN for every j ′ ∈ (J ′ ) E and

F p,Z ∈ (k(x ′ )[U 1 , . . . , U n ]) p [{U j } j∈J E \{j 2 ,...,j e ′ 0 } ];
(2) let (u ′ 1 , . . . , u ′ n ′ ; Z ′ ) be well adapted coordinates at x ′ . Then

H ′ -1 F p,Z ′ k(x ′ )[U ′ 1 , . . . , U ′ n ′ 1 ] ǫ(x ′ ) ⊕ ({U ′ i } i ∈F ) ∩ G(m S ′ ) ǫ(x ′ ) (3.19)
and there exists

Φ ′ ∈ k(x ′ )[U ′ 1 p , . . . , U ′ n ′ 1 p ][U ′ n ′ 1 +1 , . . . , U ′ n ′ ] pδ(x ′ ) such that H ′ -1 (F p,Z ′ -Φ ′ ) ⊆ ({U ′ i } i ∈F ) ∩ G(m S ′ ) ǫ(x ′ ) . (3.20)
Proof. Since Y is permissible, Y is Hironaka-permissible at x and this implies that m(x ′ ) ≤ m(x) = p in any case. We are done unless equality holds, so assume that m(x ′ ) = p. The polyhedron ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ′ ; X ′ ) need not be minimal. We must take

Z ′ = X ′ -θ ′ , θ ′ ∈ S ′ such that the polyhedron ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ′ ; Z ′ ) is minimal in order to read off ǫ(x ′ ) and ω(x ′ ) from in m S ′ h ′ .
By Proposition 3.9(iii), we have ord

(u) H(x ′ ) = p(δ(y) -1). The initial form H W ′ of H(x ′ ) in G(W ′
) is given by Proposition 3.9(iv):

H W ′ =< U p(δ(y)-1) e ′ i=2 u ′ i H j i > . (3.21) We have θ ′ p ∈ H(x ′ ) since f p,X ′ ∈ H(x ′ ). Let Θ ′ ∈ G(W ′ ) δ(y)-1 be the initial form of θ ′ (in particular Θ ′ = 0 if δ(y) ∈ N). Then in W ′ h ′ = Z ′ p -G p-1 W ′ Z ′ + F p,X ′ ,W ′ + Θ ′ p -G p-1 W ′ Θ ′ ∈ G(W ′ )[Z ′ ] (3.22)
where

G W ′ = U -1 G W , F p,X ′ ,W ′ = U -p F p,
Z,W by Proposition 3.9(ii). According to our notations, we have:

F p,Z ′ ,W ′ = F p,X ′ ,W ′ + Θ ′ p -G p-1 W ′ Θ ′ .
Note that derivatives in D W ′ decrease orders by at most one. Since H W ′ is the initial form of H(x ′ ) in G(W ′ ), we have:

ǫ(x ′ ) ≤ min{ord m S ′ /(u) (H -1 W ′ G p W ′ ), 1 + ord m S ′ /(u) J(F p,Z ′ ,W ′ , E ′ , W ′ )}. (3.23)
Inequality may be strict, since the H(x ′ ) -i f p i,Z ′ , 1 ≤ i ≤ p may acquire terms of lower order not coming from in W h. Moreover, some derivatives in D W ′ do not decrease orders and may give a sharper bound in (3.23).

Recall that if M ⊆ G(W ) d , d ∈ N is a submodule, and d 0 is given, there are associated initial forms

M ⊆ gr m S G(W ) d d 0 ⊂ G(m S ) d+d 0 < ({U j } j∈J ) d+1 ∩ G(m S ) d+d 0 >
under the conditions described in (3.3) and (3.4). Note that (3.22). We have Θ ′ = 0 or δ(y) ∈ N and

gr m S G(W ) d 0 = Γ(σ -1 (m S ), O σ -1 (m S ) (d)) = k(x)[{U j } j∈J ] d for d 0 = 0. Since θ ′ p ∈ H(x ′ ), we have Θ ′ p ∈ H W ′ in
G p-1 W ′ Θ ′ ∈ G p-1 W ′ H 1 p W ′ , H 1 p W ′ :=< U δ(y)-1 e ′ i=2 u ′ i H j i p > . Since D • Θ ′ p = 0 for every D ∈ D W ′ , we deduce from (3.22) that J(F p,Z ′ ,W ′ , E ′ , W ′ ) ≡ J(F p,X ′ ,W ′ , E ′ , W ′ ) modH -1 W ′ G p-1 W ′ H 1 p W ′ . (3.24)
Note that if i 0 (m S ) = p, or if H j ′ ∈ pN for some j ′ ∈ (J ′ ) E , we have

G W = 0 or ord (u j ′ ) (H -1 W G p W ) > 0 for some j ′ ∈ (J ′ ) E (3.25)
by applying Proposition 2.29(iii) in the latter case. In this case, we obtain the following from Proposition 3.9(v) and (3.24):

(H -1 W ′ G p W ′ )S ′ = 0, J(F p,Z ′ ,W ′ , E ′ , W ′ )S ′ = J(F p,X ′ ,W ′ , E ′ , W ′ )S ′ . (3.26)
Case 1: i 0 (m S ) = p and Y is of the first kind. In order to get an estimate of ǫ(x ′ ) from (3.23), we take:

M = J(F p,Z,W , E, W ), d = ǫ(y) = ǫ(x), d 0 = 0.
Remark 3.14. By Proposition 3.3, there is an equality

H -1 F p,Z = cl ǫ(x) H -1 W F p,Z,W ⊆ k(x)[{U j } j∈J ] ǫ(x) ,
but we emphasize that the induced inclusion

J(F p,Z , E, m S ) ⊆ cl ǫ(x) J(F p,Z,W , E, W ). (3.27)
is strict in general: this is because elements of the form

cl ǫ(x) (H -1 W ∂F p,Z,W ∂u j ′ ), j ′ ∈ J ′ \(J ′ ) E may be nonzero.
By Proposition 2.57(ii) and the remark, we have

0 = J(F p,Z , E, m S ) ⊆ M ⊆ k(x)[{U j } j∈J ] ǫ(x) . Let I ′ = J(F p,X ′ ,W ′ , E ′ , W ′ ) ⊆ S ′ /(u), d ′ = ordI ′ . By Proposition 3.9(v), we have U -ǫ(x) J(F p,Z , E, m S ) m ′ ⊆ I ′ S ′ .
Since i 0 (m S ) = p, we obtain from (3.26) that:

U -ǫ(x) J(F p,Z , E, m S ) m ′ ⊆ I ′ S ′ = J(F p,Z ′ ,W ′ , E W ′ , W ′ )S ′ . (3.28) If ω(x) = ǫ(x), Definition 2.72 gives Max(x) = Max(J(F p,Z , E, m S )).
We deduce that ordI ′ ≤ ω(x) and

s ′ ∈ P C(x, Y) =⇒ ordI ′ < ω(x). (3.29) If ω(x) = ǫ(x) -1, Definition 2.72 gives Max(x) = Max(V (F p,Z , E, m S )).
Since U j 1 V (F p,Z , E, m S ) ⊆ J(F p,Z , E, m S ) (recall that u = u j 1 ), we also deduce that ordI ′ ≤ ω(x) and (3.29) holds. We have:

ǫ(x ′ ) ≤ 1 + ordI ′ = 1 + d ′ ≤ 1 + ordI ′ ,
by (3.23). We have proved that

ǫ(x ′ ) ≤ 1 + ordI ′ ≤ 1 + ω(x) (3.30)
with strict inequality on the right hand side under the assumption of (3.29).

The proof is now an easy consequence of the following claim:

ǫ(x ′ ) = 1 + ordI ′ =⇒ ω(x ′ ) = ǫ(x ′ ) -1.
Namely, assuming the claim, we have ω(x ′ ) ≤ ω(x) and this inequality is strict under the assumption of (3.29). The first part of the proof is complete since i 0 (m S ) = p implies κ(x) ≥ 2. To prove the claim, let

in m S ′ h = Z ′ p -G ′ p-1 Z ′ + F p,Z ′ ∈ G(m S ′ )[Z ′ ]
be the initial form polynomial. Since it is assumed that ǫ(x ′ ) = 1 + ordI ′ , we have I ′ = 0 and:

I ′ =< H ′ -1 ∂F p,Z ′ ∂U ′ j n ′ j=n ′ 0 +1 > mod({U ′ j ′ } j ′ ∈F ) ∩ G(m S ′ ) d ′ . (3.31)
To compute ω(x ′ ), we must introduce a truncation operator

T ′ : G(m S ′ ) pδ(x ′ ) → G(m S ′ ) pδ(x ′ )
as in Definition 2.68. By (3.21), we have

H ′ := cl pδ(x ′ )-ǫ(x ′ ) H(x ′ ) =< U p(δ(y)-1) e ′ i=2 U ′ i H j i >∈ G(m S ′ ).
Going back to Definition 2.59, we have

F p,Z ′ -T ′ F p,Z ′ ∈< G ′ p-1 U δ(y)-1 e ′ i=2 U ′ i H j i p > .
Since i 0 (m S ) = p, (3.26) applies and implies that

H ′ -1 (F p,Z ′ -T ′ F p,Z ′ ) ⊆ ({U ′ i } i ∈F ) ∩ G(m S ′ ) ǫ(x ′ ) . (3.32)
Comparing with (3.31), there exists i,

n ′ 0 + 1 ≤ i ≤ n ′ such that H ′ -1 ∂T ′ F p,Z ′ ∂U ′ i = 0, (3.33)
since I ′ = 0. This proves that ω(x ′ ) = ǫ(x ′ ) -1 as claimed.

To conclude the proof in case 1, assume that ǫ(x ′ ) > ǫ(x). If some inequality is strict in (3.29), we have ǫ(x ′ ) ≤ ω(x) ≤ ǫ(x): a contradiction. So ω(x ′ ) = ω(x) and by the above claim, we get

ǫ(x) = ω(x) = ω(x ′ ) = ǫ(x ′ ) -1 = ordI ′ = ordI ′ . (3.34)
We use notations as in (2.42). Suppose that there exists j ′ ∈ (J ′ ) E such that H j ′ ∈ pN. By Proposition 3.3, we have

H -1 U j ′ ∂F p,Z ∂U j ′ = 0.
Going back to (3.28), we have

φ j ′ := U -ǫ(x) H -1 U j ′ ∂F p,Z ∂U j ′ m ′ ⊆ J(F p,Z ′ ,W ′ , E ′ , W ′ )S ′ .
Applying the transformation rule in Proposition 3.9(v), we have

φ j ′ = (H -1 W ′ u j ′ ∂F p,Z ′ ,W ′ ∂u j ′ )S ′ .
Since ordφ j ′ ≤ ǫ(x), we deduce that

ǫ(x ′ ) ≤ ord(H -1 W ′ F p,Z ′ ,W ′ ) ≤ ord(H -1 W ′ u j ′ ∂F p,Z ′ ,W ′ ∂u j ′ ) ≤ ǫ(x).
This is a contradiction with (3.34). Hence H j ′ ∈ pN for every j ′ ∈ (J ′ ) E . Suppose that δ(y) ∈ N. Similarly, by Proposition 3.3, we have:

H -1 D • F p,Z = 0, D := j∈J U j ∂ ∂U j ∈ Der(G(W )).
Note that we have Θ ′ = 0 in (3.22) since δ(y) ∈ N. By (3.17):

φ D := U -ǫ(x) H -1 D • F p,Z S ′ /(u) = H -1 W ′ U ∂F p,Z ′ ,W ′ ∂U .
Arguing as above, we get a contradiction from:

ǫ(x ′ ) ≤ ord(H -1 W ′ F p,Z ′ ,W ′ ) ≤ ord(H -1 W ′ U ∂F p,Z ′ ,W ′ ∂U ) ≤ ǫ(x).
Let now i ∈ {2, . . . , e ′ 0 }. By (3.28), we have

φ i := U -ǫ(x) H -1 U j i ∂F p,Z ∂U j i m ′ ⊆ J(F p,Z ′ ,W ′ , E W ′ , W ′ )S ′ .
Applying once again (3.17) and since ǫ(x ′ ) > ǫ(x) = ω(x), we get

cl ǫ(x) ({H -1 W ′ u i ∂F p,Z,W ′ ∂u i } 2≤i≤e ′ 0 ) ≡ cl ǫ(x) ({φ i } 2≤i≤e ′ 0 ) mod({U ′ i ′ } i ′ ∈F )∩G(m S ′ ) ǫ(x)
.

If φ i = 0 for some i, 2 ≤ i ≤ e ′ 0 , we get ǫ(x ′ ) ≤ ord(H -1 W ′ F p,Z ′ ,W ′ ) ≤ ord(H -1 W ′ u i ∂F p,Z,W ′ ∂u i ) ≤ ǫ(x),
again a contradiction. Since ǫ(x) = ω(x), we have

∂F p,Z ∂U j = 0 for every j ∈ J\J E .
Finally, assume that

F p,Z ∈ k(x ′ ) p [U 1 , . . . , U n ]. Let (dλ ′ l ′ ) l ′ ∈Λ ′ be a basis of Ω 1 k(x ′ )
. By assumption, there exists l ∈ Λ such that ∂F p,Z ∂λ l = 0. We may assume w.l.o.g. that λ l = λ ′ l ′ for some l ′ ∈ Λ ′ . Arguing as above, we get

cl ǫ(x) (H -1 W ′ ∂F p,Z ′ ,W ′ ∂λ l ) ≡ cl ǫ(x) U -ǫ(x) H -1 ∂F p,Z ∂λ l m ′ mod({U ′ i ′ } i ′ ∈F )∩G(m S ′ ) ǫ(x) ,
a contradiction and the proof of (1) in the theorem is complete.

We now proceed to prove [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. By Proposition 3.9(i), we have

H -1 W ′ F p,X ′ ,W ′ S ′ = (U -ǫ(x) H -1 W F p,Z,W ) m ′ = (U -ǫ(x) H -1 F p,Z ) m ′ .
By [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] in the theorem and Proposition 3.3, there is an expansion

F p,Z =   e ′ i=e ′ 0 +1 U H j i j i   a∈A F p,Z,a ({U j } j∈J ′ 1 ) j∈J 1 U pa j j , A ⊂ N J 1 , with J 1 := {j 2 , . . . , j e ′ 0 , j n ′ 0 +1 , . . . , j n ′ 1 }, J ′ 1 := J\J 1 , F p,Z,a ∈ k(x ′ ) p [{U j } j∈J ′ 1 ]. We deduce that (U -ǫ(x) H -1 F p,Z ) m ′ = H ′ -1 a∈A F p,Z,a ({ U j U } j∈J ′ 1 ) j∈J 1 ( U j U ) pa j , (3.35)
with

H ′ := ( e ′ 0 i=2 U j i U H j i ) ⊆ S ′ . Since (H -1 W ′ G p W ′ )S ′ = 0 by (3.26), there exists θ ′ ∈ S ′ /(u) such that H -1 W ′ F p,Z ′ ,W ′ S ′ = H -1 W ′ (F p,X ′ ,W ′ + θ ′ p )S ′ . (3.36)
We deduce from (3.35) that there exists a finite subset

A ′ ⊂ N J 1 , A ⊆ A ′ and elements θ ′ a ∈ k(x)[{ U j U } j∈J ′ 1 ] for every a ∈ A ′ such that (letting F p,Z,a ({ U j U } j∈J ′ 1 ) = 0 for a ∈ A ′ \A)
we have:

H -1 W ′ F p,Z ′ ,W ′ S ′ = H ′ -1 a∈A ′ (F p,Z,a ({ U j U } j∈J ′ 1 ) + θ ′ a p ) j∈J 1 ( U j U ) pa j . Let d a := ǫ(x ′ ) + e ′ 0 i=2 H j i -p | a | for a ∈ A ′ . Since ord(H -1 W ′ F p,Z ′ ,W ′ ) = ǫ(x ′ ) we have ord(F p,Z,a ({ U j U } j∈J ′ 1 ) + θ ′ a p ) ≥ d a
for every a ∈ A ′ . Taking classes in G(m ′ ), we define:

Φ ′ a := cl da (F p,Z,a ({ U j U } j∈J ′ 1 ) + θ ′ p a ) ∈ k(x ′ )[U ′ n ′ 1 +1 , . . . , U ′ n ′ ] da .
To conclude the proof, let I 1 := {2, . . . , e ′ 0 , n ′ 0 + 1, . . . , n ′ 1 }. We take

Φ ′ := U ′ 1 p(δ(y)-1)   e ′ i=e ′ 0 +1 U ′ i H j i   a∈A ′ Φ ′ a i∈I 1 U ′ i pa j i
and claim that Φ ′ satisfies (2) in the theorem. By the above definition and (1) in the theorem, we have

Φ ′ ∈ k(x ′ )[U ′ 1 p , . . . , U ′ n ′ 1 p ][U ′ n ′ 1 , . . . , U ′ n ′ ] pδ(x ′ )
. Also (3.20) follows immediately from (3.36). With notations as in the above proof of (1), we have

J(F p,Z , E, m S ) = H -1 < {U j ∂F p,Z ∂U j } j∈J E \{j 2 ,...,j e ′ 0 +1 } , { ∂F p,Z ∂λ l } l∈Λ > .
Applying once more (3.17), we get

cl ǫ(x) ({H -1 W ′ ∂F p,Z,W ′ ∂u ′ i } n ′ 1 ≤i≤n ′ ) ≡ cl ǫ(x) (U -ǫ(x) J(F p,Z , E, m S )) m ′ mod({U ′ i ′ } i ′ ∈F ) ∩ G(m S ′ ) ǫ(x) .
Since J(F p,Z , E, m S ) = 0, we obtain that

H ′ -1 ∂F p,Z ′ ∂U ′ i ∈ ({U ′ i ′ } i ′ ∈F ) ∩ G(m S ′ ) ǫ(x)
for some i, n ′ 1 ≤ i ≤ n ′ , and the conclusion follows. This concludes the proof of (2).

Case 2: i 0 (m S ) = p -1 and Y is of the first kind. We first take d = ǫ(y) and

M := H -1 W G p W , d 0 = 0.
By Proposition 3.3, there is an expansion

H -1 G p =< j∈J U pB j j
>. With notations as in Definition 2.68, we have pb j -H j = pB j , j ∈ J and B = {j ∈ J : B j > 0}.

(3.37)

We deduce:

(0) = M = ( j∈B U pB j j ) ⊆ k(x)[{U j } j∈J ] ǫ(x) . Let I ′ 0 = H -1 W ′ G p W ′ , d ′ 0 = ordI ′ 0 .
We have: .38) This proves that ǫ(x ′ ) ≤ ordI ′ 0 ≤ ǫ(x) and equality holds only if

I ′ 0 S ′ = U -ǫ(x) j∈B U pB j j m ′ . ( 3
s ′ ∈ Proj k(x)[{U j } j∈J ] (U B ) . (3.39)
Suppose that ǫ(x ′ ) < ǫ(x). Then :

ω(x ′ ) ≤ ǫ(x ′ ) ≤ ǫ(x) -1 ≤ ω(x).
If ω(x ′ ) = ω(x), then ω(x) = ǫ(x) -1, so κ(x) ≥ 2. On the other hand, we have ω(x ′ ) = ǫ(x ′ ) and therefore κ(x ′ ) = 1 by Definition 2.68. Hence inequality is strict in (3.18). In other terms, it can be assumed from now on that (3.39) holds and that ǫ(x ′ ) = ǫ(x).

(3.40)

We now resume the argument used in case 1 by taking

M = J(F p,X,W , E W , W ), d = ǫ(y) = ǫ(x), d 0 = 0.
To begin with, (3.28) holds whenever (3.26) applies, i.e. if H j ′ ∈ pN for some j ′ ∈ (J ′ ) E or if δ(y) ∈ N. Suppose that δ(y) ∈ N and H j ′ ∈ pN for every j ′ ∈ (J ′ ) E . In this case, (3.24) reduces to

J(F p,Z ′ ,W ′ , E ′ , W ′ ) ≡ J(F p,X ′ ,W ′ , E ′ , W ′ ) modK ′ Ŝ′ (u) , (3.41)
K ′ := ( e ′ 0 i=2 u ′ i (p-1)b j i -H j i + H j i p ) ⊆ S ′
with notations as in (3.37). We let :

k ′ := j∈J (p -1)b j -H j + H j p = ord m S ′ K ′ .
Going back to Definition 2.68, we have

F p,Z -T F p,Z ∈ ( j∈J U (p-1)b j + H j p j G(m S )) pδ(x)
and we deduce now from (3.41) that

J(F p,Z ′ ,W ′ , E W ′ , W ′ )S ′ ≡ U -ǫ(x) J(T F p,Z , E, m S ) m ′ modK ′ S ′ . (3.42)
Note that the previous equation remains valid when H j ′ ∈ pN for some j ′ ∈ (J ′ ) E or when δ(y) ∈ N. The proof now goes on as in case 1 and we deduce that ordI ′ ≤ ω(x); joining (3.39) and (3.42), we obtain that (3.29) holds, i.e.

s ′ ∈ Proj k(x)[{U j } j∈J ] (IMax(x) ∩ k(x)[{U j } j∈J ]) =⇒ ordI ′ < ω(x).
Equation (3.30) now follows, while (3.31) gets replaced by

I ′ =< H ′ -1 ∂F p,Z ′ ∂U ′ j n ′ j=n ′ 0 +1 > mod(({U ′ j ′ } j ′ ∈F ) + (cl k ′ K ′ )) ∩ G(m S ′ ) d ′ .
(3.43) Finally, we obtain that

H ′ -1 (F p,Z ′ -T ′ F p,Z ′ ) ⊆ (({U ′ i } i ∈F ) + (cl k ′ K ′ )) ∩ G(m S ′ ) ǫ(x ′ )
and this concludes the proof of the claim, hence of the theorem, as in case 1.

Case 3: Y is of the second kind. First recall from Proposition 3.7 that ǫ(x) -1 = ω(x), so κ(x) ≥ 2 in particular. Let

I ′ 0 := H -1 W ′ G p W ′ , d ′ 1 = ordI ′ 0 .
Suppose that i 0 (m S ) = p -1. By Proposition 3.7, there exists an expansion

H -1 G p =< U j 1 j∈B J U pB j j >, j 1 ∈ (J ′ ) E , B j > 0 for j ∈ B J ,
with notations as in Definition 3.12. By Proposition 3.9(v), we have:

I ′ 0 S ′ /(u) = u j 1 U -ǫ(y) j∈B U pB j j m S ′ /(u) . (3.44)
This proves that ǫ(x ′ ) ≤ ordI ′ 0 ≤ ǫ(x) and equality holds only if

s ′ ∈ Proj k(x)[{U j } j∈J ] (U B J ) . (3.45)
Suppose furthermore that ǫ(x ′ ) < ǫ(x). We have:

ω(x ′ ) ≤ ǫ(x ′ ) ≤ ǫ(x) -1 = ω(x).
If ω(x ′ ) = ω(x), then ω(x ′ ) = ǫ(x ′ ) and therefore κ(x ′ ) = 1 by Definition 2.68, so inequality is strict in (3.18). Therefore if i 0 (m S ) = p -1, it can be assumed that ǫ(x ′ ) = ǫ(x) and in particular that (3.45) holds.

Going back to the general situation of case 3, we now take

M = J(F p,X,W , E W , W ), d = ǫ(y), d 0 = 0.
Note that (3.26) is always valid in this case 3: we either have i 0 (m S ) = p or (3.25) holds for j ′ = j 0 . Applying Proposition 3.9(v) gives:

J(F p,Z ′ ,W ′ , E W ′ , W ′ )S ′ = U -ǫ(y) J(F p,Z,W , E W , W ) m ′ .
With notations as in Proposition 3.7, we have

(0) = J(F p,Z,W , E W , W ) =< {Φ j ′ ({U j } j∈J )} j ′ ∈J ′ \(J ′ ) E > .
We deduce that

J(F p,Z ′ ,W ′ , E W ′ , W ′ )S ′ =< { U -ǫ(y) Φ j ′ ({U j } j∈J ) m ′ } j ′ ∈J ′ \(J ′ ) E > . (3.46)
Since Definition 3.12 gives

C(x, Y) := Max(J(F p,Z,W , E, W )) ∩ {U B J = 0},
we deduce that ordJ(F p,Z ′ ,W ′ , E W ′ , W ′ ) ≤ ω(x) and equality holds only if s ′ ∈ P C(x, Y). We obtain:

ǫ(x ′ ) ≤ 1 + ordJ(F p,Z ′ ,W ′ , E W ′ , W ′ ) ≤ 1 + ordJ(F p,Z ′ ,W ′ , E W ′ , W ′ ) ≤ ǫ(x).
(3.47) Suppose that s ′ ∈ P C(x, Y) and ω(x ′ ) ≥ ω(x). Formula (3.47) shows that ǫ(x ′ ) = ω(x ′ ) = ω(x). If i 0 (m S ′ ) = p -1, we get κ(x ′ ) = 1 so inequality is strict in (3.18). If i 0 (m S ′ ) = p, we may pick

j ′ = j i ∈ J ′ \(J ′ ) E , e ′ +1 ≤ i ≤ n ′ 0 , such that ord U -ǫ(y) Φ j ′ ({U j } j∈J ) m ′ < ω(x).

By (3.46), we have H

′ -1 ∂F p,Z ′ ∂U ′ i = 0
. This is a contradiction with the assumption ǫ(x ′ ) = ω(x ′ ). Thus it can be assumed that s ′ ∈ P C(x, Y).

We get ω(x ′ ) ≤ ǫ(x ′ ) ≤ ω(x) unless all inequalities in (3.47) are equalities. In this case, we claim that ω(x ′ ) = ǫ(x ′ ) -1 and this will conclude the proof. To prove the claim, we may pick j i ∈ J ′ \(J ′ ) E , e ′ + 1 ≤ i ≤ n ′ 0 , such that Φ j i ({U j } j∈J ) = 0 by Proposition 3.7. Arguing as above, we have

H ′ -1 ∂F p,Z ′ ∂U ′ i ≡< cl ω(x) U -ǫ(y) Φ j i ({U j } j∈J ) m ′ > mod(({U ′ j ′ } j ′ ∈F )∩G(m S ′ ) ω(x) , (3.48)
and this proves that

H ′ -1 ∂F p,Z ′ ∂U ′ i = 0. If i 0 (m S ′ ) = p, we get ω(x ′ ) = ω(x). If i 0 (m S ′ ) = p -1,
we must introduce a truncation operator

T ′ : G(m S ′ ) pδ(x ′ ) → G(m S ′ ) pδ(x ′ )
as in Definition 2.68 in order to compute ω(x ′ ). In any case, we have

H ′ -1 G ′ p ⊆ (U ′ i ∈F ) ∩ G(m S ′ ) ǫ(x ′ ) , (3.49)
which follows from the identity I ′ 0 S ′ /(u) = 0 (resp. from (3.44)) if i 0 (m S ) = p (resp. if i 0 (m S ) = p -1), cf. beginning of the proof of case 3.

Going back to Definition 2.59, we have

H ′ -1 (F p,Z ′ -T ′ F p,Z ′ ) ⊆ ({U ′ i } i ∈F ) ∩ G(m S ′ ) ǫ(x ′ ) .
It now follows from (3.48) that

H ′ -1 ∂T ′ F p,Z ′ ∂U ′ i ≡< cl ω(x) U -ǫ(y) Φ j i ({U j } j∈J ) m ′ > mod(({U ′ j ′ } j ′ ∈F )∩G(m S ′ ) ω(x) .
This proves at last that H ′ -1 ∂T ′ F p,Z ′ ∂U ′ i = 0, so ω(x ′ ) = ǫ(x ′ ) -1 and this concludes the proof of the claim, hence of the theorem.

Consequences of the Blowing up Theorem and constructibility.

In this section, we prove some basic properties of our main invariant y → (m(y), ω(y), κ(y))

and of our notion of permissibility. The following theorem expresses the persistence of permissibility under permissible blowing ups.

Theorem 3.15. Assume that m(x) = p, ω(x) > 0, where {x} = η -1 (m S ).

Let Y 0 ⊃ Y 1 with respective generic point y 0 , y 1 be permissible centers (of the first or second kind) at x and π : X ′ → X be the blowing up along Y 1 .

The strict transform Y ′ 0 of Y 0 is permissible at every x ′ ∈ π -1 (x).

Proof. By definition of permissibility, we have m(y 0 ) = m(y 1 ) = p. Let W i = η(Y i ), i = 0, 1 be with notations as in the previous theorem. There exist associated subsets J 0 ⊂ J 1 ⊆ {1, . . . , n} such that I(W i ) = ({u j } j∈J i ) for a certain choice of an adapted r.s.p. (u 1 , . . . , u n ) of S. Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x. By Proposition 2.12, the polyhedron

∆ Ŝ (h; {u j } j∈J i ; Z) = pr J i (∆ S (h; u 1 , . . . , u n ; Z)) is minimal,
where pr J i : R n → R J i denotes the projection on the (u j ) j∈J i -space, i = 0, 1.

In particular, we have

Y i = V (Z, {u j } j∈J i ), i = 0, 1. The strict transform W ′ 0 of W 0 at s ′ has normal crossings with E ′ := σ -1 (E) red . Since m(x ′ ) ≥ m(y 0 ) for every x ′ ∈ Y ′ 0 , this proves that Y ′ 0 is Hironaka-permissible w.r.t. E ′ .
For convenience of the reader, we include at the end of the proof a table summing up the possible kinds for the permissible centers Y 0 , Y 1 and Y ′ 0 . The current proof requires discussing the first two columns of the table in cases 1 and 2 below.

Applying again Proposition 2.12 and the definitions of permissible centers, we have

ǫ(y 0 ) ≤ ǫ(y 1 ) ≤ ǫ(x) ≤ 1 + ǫ(y 0 ), ǫ(y 0 ) ≤ ǫ(x ′ ). (3.50)
On the other hand, Theorem 3.13 applied to π gives ǫ(x ′ ) ≤ ǫ(x)+1 while classifying equality cases in (1) and [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. Thus Y ′ 0 is permissible of the first kind except possibly in the following two cases: Case 1: Y 1 is of the first kind and ǫ(x ′ ) = ǫ(x) + 1; Case 2: Y 0 is of the second kind and ǫ(x ′ ) = ǫ(x).

Since x ′ ∈ Y ′ 0 , we have, with notations as in Theorem 3.13 (cf. Notation 3.10):

(J 0 ) E ⊆ {j i , 2 ≤ i ≤ e ′ 0 }, J 0 \(J 0 ) E ⊆ {j i , n ′ 0 + 1 ≤ i ≤ n ′ 1 }. (3.51)
Also, letting F 0 := {2, . . . , e ′ 0 } ∪{n ′ 0 + 1, . . . , n ′ 1 }, we have (cf. Notation 3.11):

J 0 ⊆ F 0 ⊆ F = F 0 ∪ {n ′ 1 + 1, . . . , n ′ }. (3.52)
Proof in case 1: an immediate consequence of Theorem 3.13(1) is that :

i 0 (m S ) = p, ∂F p,Z ∂U j = 0, j ∈ J 0 or j ≥ e + 1.
This is incompatible with Definition 3.7(iii) applied to Y 0 , so Y 0 is also of the first kind. By Proposition 3.3 we deduce that

H -1 G p = 0, H -1 F p,Z ⊆ k(x)[{U j } j∈J 0 ] ǫ(x) . (3.53)
Since ǫ(y 0 ) = ǫ(x ′ ) -1, we also have

H ′ -1 < G ′ p , F p,Z ′ >⊆ ({U ′ i } j i ∈J 0 ) ǫ(y 0 ) ∩ G(m S ′ ) ǫ(x ′ ) . (3.54)
We claim that Y ′ 0 is permissible of the second kind at x ′ . To prove the claim, note that (3.53) implies that

H -1 W 1 G p W 1 ⊆ (u j ′ )G(W 1 ) ǫ(x) for some j ′ ∈ (J ′ 1 ) E .
Since Y 0 is permissible of the first kind at x, we actually have

H -1 W 1 G p W 1 ⊆ (u j ′ )S/({u j } j∈J 1 )[{U j } j∈J 0 ] ǫ(x) .
Letting j ′ =: j i ′ , e ′ 0 + 1 ≤ i ′ ≤ e, Proposition 3.9(ii) then shows that

H -1 W ′ 1 G p W ′ 1 ⊆ (u i ′ )S ′ /(u ′ 1 )[{U ′ i } j i ∈J 0 ] ǫ(x) , W ′ 1 := σ -1 (W 1 ).
In other terms, we have

H ′ -1 G ′ p ⊆ (U ′ 1 , U i ′ )k(x ′ )[{U ′ i } j i ∈J 0 ],
and this proves that Y ′ 0 satisfies property (ii) of Definition 3.5. Finally, applying (3.54) gives an expansion

H ′ -1 F p,Z ′ =< n ′ i=1 U ′ i Φ i ({U ′ i ′ } j i ′ ∈J 0 ) > .
Then Definition 3.5(iii) is equivalent to:

∃i ∈ J ′ 0 ∩ {e ′ + 1, . . . , n ′ } : Φ i = 0.
By equation (3.19) in Theorem 3.13(2), there exists i

≥ n ′ 1 + 1 (hence i ∈ J ′ 0 ) such that Φ i = 0, since j i ′ ∈ J 0 =⇒ i ′ ≤ n ′
1 by (3.51) and this completes the proof in case 1.

Proof in case 2. Since Y 0 is permissible of the second kind, the initial form in m S h ∈ G(m S )[Z] satisfies (3.6). The corresponding integer j 0 satisfies j 0 ∈ J ′ 0 and the corresponding family (Φ j ′ ({U j } j∈J 0 )) j ′ ∈J ′ 0 is such that Φ j ′ = 0 for some j ′ ∈ J ′ 0 \(J ′ 0 ) E . In order to prove that Y ′ 0 is of the second kind at x ′ , we consider two subcases: Case 2a: Y 1 is of the second kind at x. Then j 0 ∈ J ′ 1 and Φ j ′ = 0 for some

j ′ ∈ J ′ 1 \(J ′ 1 ) E .
By assumption ǫ(x ′ ) = ǫ(x), and we deduce from (3.44) (resp. from (3.49)

) if i 0 (m S ) = p -1 (resp. if i 0 (m S ) = p) that the initial form in m S ′ h ′ ∈ G(m S ′ )[Z ′ ] satisfies H ′ -1 G ′ p ⊆ U j ′ 0 k(x ′ )[{U ′ i } j i ∈J 0 ] ǫ(y 0 ) for some j ′ 0 ∈ {1, e ′ 0 + 1, . . . , e ′ } (3.55)
and Definition 3.5(ii) is checked for Y ′ 0 at x ′ . Similarly, Definition 3.5(iii) is checked from (3.48): we have

H ′ -1 ∂F p,Z ′ ∂U ′ i = 0 for any i, e ′ + 1 ≤ i ≤ n ′ 0 such that j i ∈ J ′ 1 \(J ′
1 ) E and Φ j i = 0; take j i = j ′ with notations as above.

Case 2b: Y 1 is of the first kind at x. Then j 0 ∈ J 1 and Φ j ′ = 0 for any j ′ ∈ J ′ 1 . By Proposition 3.7 and our assumption ǫ(x ′ ) = ǫ(x), we have

ω(x) = ǫ(y 0 ) = ǫ(x) -1 = ǫ(x ′ ) -1 ≤ ω(x ′ ).
Therefore Theorem 3.13 implies that ω(x ′ ) = ω(x). We have κ(x), κ(x ′ ) ≥ 2 since ω(x) = ǫ(x) -1, ω(x ′ ) = ǫ(x ′ ) -1. This is the equality case (m(x ′ ), ω(x ′ ), κ(x ′ )) = (m(x), ω(x), κ(x)) discussed in Theorem 3.13. If i 0 (m S ) = p, we are in the equality case of (3.30). Then (3.55) holds and there exists i, n

′ 1 + 1 ≤ i ≤ n ′ or (n ′ 0 + 1 ≤ i ≤ n ′ 1 and Φ j i = 0) such that H ′ -1 ∂F p,Z ′ ∂U ′ i = 0 (3.56)
by (3.33). We may take here j i := j ′ ∈ J ′ 0 \(J ′ 0 ) E . This checks Definition 3.5(ii) and (iii) respectively.

If i 0 (m S ) = p -1, the initial form in m S ′ h ′ ∈ G(m S ′ )[Z ′ ] satisfies H ′ -1 G ′ p ⊆ U ′ i 1 k(x ′ )[{U ′ i } j i ∈J 0 ] ǫ(y 0 ) ,
where j i 1 := j 0 ∈ J ′ 0 , 2 ≤ i 1 ≤ e ′ 0 and Definition 3.5(ii) is checked. Equation (3.56) also remains valid for some i, n ′ 0 + 1 ≤ i ≤ n ′ , in this case: this follows from (3.33) which is still valid (end of the proof of case 2 of Theorem 3.13 where (3.43) replaces (3.31). This checks Definition 3.5(iii) and the proof is complete.

The following table sums up the different cases occurring in the proof. The proof is immediate for the last two columns: ǫ(x ′ ) = ǫ(y 0 ) in these cases.

kinds kinds kinds kinds

Y 0 1 2 2 1 Y 1 1 1 or 2 1 or 2 1 Y ′ 0 2 2 1 1 case in proof case 1

case 2 trivial trivial

We now turn to formal arcs on X and their image. Recall that it is assumed all along this chapter that m(x) = p, ω(x) > 0 and {x} = η -1 (m S ). Definition 3.16. A formal arc on (X , x) is a local morphism ϕ : SpecO → (X, x), where (O, N, l) is a complete discrete valuation ring. We denote the closed (resp. generic) point of SpecO by O (resp. ξ) and call support of ϕ the subscheme Z(ϕ) := {ϕ(ξ)} ⊆ (X , x).

The arc ϕ is said to be well parametrized if the inclusion

O ξ := O ∩ k(ϕ(ξ)) ⊆ O induces an isomorphism O ξ ≃ O. The arc ϕ is said to be nonconstant if ϕ(ξ) = x = ϕ(O).
Let us note that, up to replacing O by O ξ , the arc ϕ becomes well parametrized.

Given a nonconstant formal arc on (X , x), and π : X ′ → X a blowing up along a permissible center Y ⊂ X at x such that Y Z(ϕ), there exists a unique lifting ϕ ′ : SpecO → X ′ . Let

x ′ := ϕ ′ (O), (X 1 , x 1 ) := (X ′ , x ′ ) and ϕ 1 : SpecO → (X 1 , x 1 )
be the induced morphism. The arc ϕ 1 is again nonconstant, so the process can be iterated. Let If m(x) = p and ω(x) > 0, Theorem 3.13 implies that (m(x 1 ), ω(x 1 ), κ(x 1 )) ≤ (m(x), ω(x), κ(x)).

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ) ← • • • (3.
If m(x r ) = p and ω(x r ) > 0 for every r ≥ 0, we let m(ϕ) := p, ω(ϕ) := min r≥0 {ω(x r )} > 0.

Proposition 3.17. With notations as above, let ϕ : SpecO → (X , x) be a nonconstant well parametrized formal arc on (X , x) whose quadratic sequence is such that m(ϕ) = p and ω(ϕ) > 0. Then l|k(x r ) is algebraic for r >> 0. Assume that l|k(x r ) is algebraic with finite inseparability degree for some r ≥ 0. Then there exists r 0 ≥ 0 such that the following holds: the support Z r (ϕ) is Hironaka-permissible at x r and ǫ(x r ) = ǫ(x r 0 ) for every r ≥ r 0 ; furthermore exactly one of the following conditions is satisfied:

(1) Z r (ϕ) is permissible of the first kind at x r for every r ≥ r 0 ;

(2) there exists a finite sequence (3.57):

(X r 0 , x r 0 ) =: (X ′ , x ′ ) ← (X ′ 1 , x ′ 1 ) ← • • • ← (X ′ r 1 , x ′ r 1 ) =: ( X , x)
of local blowing ups with permissible centers of the first kind contained in and of codimension one in the successive strict transforms of Z r 0 (ϕ), such that the quadratic sequence along ϕ:

( X , x) =: ( X0 , x0 ) ← ( X1 , x1 ) ← • • • ← ( Xr , xr ) ← • • •
has the following properties for every r ≥ 0:

(a) ǫ(x r ) = ǫ(x r 0 );

(b) dimO Zr(ϕ),xr = dimO Zr 0 (ϕ),xr 0 ≥ 2;

(c) Zr (ϕ) is permissible of the second kind at xr (resp. ω(x r ) = 0) if ǫ(x r 0 ) ≥ 2 (resp. if ǫ(x r 0 ) = 1).

Proof. It can be assumed without loss of generality that

d(ϕ) = dimO X ,x , m(x) = p and ω(x) = ω(ϕ) > 0.
Since m(ϕ) = p and ω(ϕ) > 0, we let η r : (X r , x r ) → SpecS r be the corresponding projection, I r (ϕ) ⊆ S r be the ideal of W r (ϕ) := η r (Z r (ϕ)). We drop the reference to ϕ in what follows in order to avoid cumbersome notations.

For f ∈ m S 0 , f ∈ I 0 we denote by f ∈ O, f = 0 its image by ϕ ♯ . Let v be the discrete valuation associated with O and let

M r := {v(f ), f ∈ S r \I r }
be the semigroup of values of S r w.r.t. v. The group generated by M r is the value group of the restriction v |K to K = QF (S/I 0 ), hence independent of r ≥ 0, and is denoted by aZ ⊆ v(N)Z, a ∈ N.

Suppose that M 0 = aN. Let α ≥ 2, β ∈ N\αN be defined by: aα := min{M 0 \(0)}, aβ := min{M 0 \aαN}.

(3.59)

We pick u, w ∈ m S 0 such that v(u) = aα, v(w) = aβ. Obviously u is a regular parameter of S and wu -1 ∈ m S 1 . Suppose M 1 = aN. There are associated integers α 1 , β 1 as in (3.59) which satisfy (α 1 , β 1 ) < (α, β) for the lexicographical ordering. This can repeat only finitely many times so we get M r = aN for some r ≥ 0. W.l.o.g. it can be assumed that M 0 = aN.

Let (u 1 , . . . , u n ) be a r.s.p. of S = S 0 which is adapted to E = div(u 1 • • • u e ). Without loss of generality, it can be assumed that v(u e ) = a. Up to renumbering coordinates, there exists e(ϕ), 0 ≤ e(ϕ) < e such that (u 1 , . . . , u e(ϕ) ) ⊆ I := I 0 , u j ∈ I for e(ϕ) + 1 ≤ j ≤ e.

For j, e(ϕ) + 1 ≤ j ≤ e -1, let v(u j ) =: aα j , α j ≥ 1. Note that u j u -α j e is a unit in S α j ; in other terms, replacing S by S max{α j } , it can be assumed that e(ϕ) = e -1.

Let f ∈ m S 0 \I 0 and write f = u αr(f ) e f r ∈ S r , where u e does not divide f r in S r and note that

f r ∈ m Sr =⇒ v(f ) > α r (f )v(u e ) ≥ ar.
Since M 0 = aN, there exists r ≥ 0 such that f r is a unit. This implies that for every ideal J ⊆ S 0 /I 0 , JS r /I r is a principal ideal for r >> 0. This is a well known characterization of valuation rings, i.e.

O v |K = r≥0 S r /I r .
(3.60)

Let l 0 be the residue field of the valuation v |K . Then l|l 0 is algebraic (of degree at most p) and l 0 |k(x r ) is algebraic for r >> 0 by (3.60). This proves the first statement in the theorem. We thus may assume from now on, again by (3.60), that l 0 |k(x 0 ) is separable algebraic.

Let S sh be the strict Henselization of S, so l sh := S sh /m S sh is the separable algebraic closure of l. The residue action induces an isomorphism Gal(S sh |S h ) ≃ Gal(l sh |k(x))

where S h is the Henselization of S. Let S be the fixed subring of S sh by the inverse image of Gal(l sh |l 0 ) under the previous group morphism. Then S ⊂ S is a local ind-étale map such that l 0 = S/m S . In particular S ⊂ S is regular [START_REF] Illusie | Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasiexcellents[END_REF] Theorem I.8.1(iv). Since O is Henselian and l 0 ⊆ l = O/N, the morphism ϕ factors through S.

Recall Notation 2.14 and Notation 2.23 for the regular local base change S ⊂ S. We apply Theorem 2.74 with s := m S and get:

m(x) = m(x) = p, ω(x) = ω(ϕ) > 0 and ǫ(x) = ǫ(x) > 0,
the right hand side equality holding because ñ = n. Applying Theorem 2.36, X = Spec( S[X]/( h)) is irreducible, so in the separable case (case (b) of assumption (G)), the G = Z/p-action extends uniquely to X and (G) holds for ( S, h, Ẽ). This proves that ( S, h, Ẽ) satisfies the assumption of the Proposition, all other assumptions being trivially satisfied. 123 Now W 0 × k(x 0 ) Specl 0 may be reducible, but W r × k(xr) Specl 0 is irreducible for r >> 0. After possibly changing indices, it can be assumed that W := W 0 × k(x 0 ) Specl 0 is irreducible. Then W has normal crossings with E at x if and only if W := W × S Spec S has normal crossings with Ẽ at x. Let Z := Z × S Spec S and z be the generic point of a component of Z. By Theorem 2.74, we have m(z) = m(z), so Z is Hironaka-permissible at x w.r.t. Ẽ if and only if Z is Hironaka-permissible at x w.r.t. E. In other terms, we may replace S by S and thus assume that l 0 = k(x 0 ) in order to prove the second statement. Let Îr be the kernel of φr , so we have I r Ŝr ⊆ Îr and I r = Îr ∩ S r .

(3.62)

After possibly replacing S 0 by S r for some r ≥ 0, it can be assumed that the curve Spec( Ŝ0 / Î0 ) is transverse to Ê = div(u 1 • • • u e ) ⊂ Spec Ŝ0 . We claim that I 0 = (u 1 , . . . , u e-1 , u e+1 , . . . , u e+t 0 ).

(3.63)

To prove the claim, suppose that I 0 = J 0 := (u 1 , . . . , u e-1 , u e+1 , . . . , u e+t 0 ). We let ûj := u j , 1 ≤ j ≤ e + t 0 and pick a basis Î0 = J 0 + (û e+t 0 +1 , . . . , ûn ) (3.64) of Î0 . Since S 0 is excellent, the ring ( Ŝ0 /I 0 ) Î0 is regular, hence reduced. By assumption, I 0 = J 0 , so there exists f ∈ I 0 \J 0 such that f restricts to a regular parameter f in S := ( Ŝ0 /J 0 ) Î0 :

ord Î0 f = 1, ord m S f = 1. (3.65)
Let F ∈ gr Î0 ( Ŝ0 ) ≃ Ŝ0 / Î0 [{ Ûj } j =e ] be the initial form of f . There is an expansion

F = j =e F j Ûj , F j ∈ Ŝ0 / Î0 .
By (3.65) we have F j = 0 for some j, 1 ≤ j ≤ e + t 0 . Suppose that

∃j 0 , 1 ≤ j 0 ≤ e + t 0 | m := min j =e {ord (ue) F j } = ord (ue) F j 0 .
Replacing f with fγ j 0 u j 0 u m e for some unit γ j 0 ∈ S 0 preserves (3.65) while increasing ord (ue) F j 0 . Applying finitely many times this procedure, it can be assumed that Furthermore the last statement in ibid. shows that in Îr g r ∈ (gr Îr Ŝr ) 1 is transverse to the initial forms u -r e U j , 1 ≤ j ≤ e+t 0 , j = e by (3.66). Since g r ∈ I r , this implies that e r > e 0 : a contradiction, so claim (3.63) is proved. Since (3.63) is stable by further blowing ups, this proves that W r is transverse to the reduced preimage of div(u 1 • • • u e ) for every r >> 0.

m := min j =e {ord (ue) F j } < min j 0 ≤e+t 0 {ord (ue) F j 0 }. ( 3
Let (û 1 , . . . , ûn ; Z) be well adapted coordinates at x. There is an associated expansion

h = Z p + f 1,Z Z p-1 + • • • + f p,Z , f 1,Z , . . . , f p,Z ∈ Ŝ0 .
We factor out f i,Z = u m i e g i,Z , 1 ≤ i ≤ p, with g i,Z = 0 or (u e does not divide g i,Z , m i ∈ N). The formal completion Ŝ1 of the local blowing up S 1 has a r.s.p. (û ′ 1 , . . . , û′ n ) given by û′ e = ûe = u e and û′ j = ûj /u e , j = e.

Let Z ′ := Z/u e , h ′ := u -p e h ∈ S 1 [Z ′ ] define the strict transform (X 1 , x 1 ), since m(ϕ) = p. We thus have Let Ẑ0 := V (Z ′ , Î0 ) ⊂ ( X0 , x) and ẑ be its generic point. Suppose that δ(ẑ) < 1 and let i 0 such that i 0 δ(ẑ) = ord Î0 f i 0 ,Z < i 0 . Applying (3.67) gives

f i,Z ′ = u -i e f i,Z , 1 ≤ i ≤ p. ( 3
ord m Ŝ1 f i 0 ,Z ′ = m i 0 + i 0 (δ(ẑ) -1) < m i 0 .
This can repeat only finitely many times, a contradiction with m(ϕ) = p. Hence δ(ẑ) ≥ 1, i.e. m(ẑ) = p. By excellence, this implies that m(z) = p. Therefore Z r is Hironaka-permissible at x r for every r >> 0.

Similarly, replacing S 0 by S r for some r ≥ 0 and arguing as above, it can be assumed that

ǫ(ẑ) = min 1≤i≤p ord Î0 (H(x) -i f p i,Z ) i = ǫ(x).
This proves that Ẑ0 is permissible of the first kind at x. Note that this furthermore implies that ǫ(x r ) = ǫ(ẑ) for every r ≥ 0 and the second statement of the proposition is proved.

In order to prove that alternative (1) in the last statement holds, we may also replace S by S as above and thus assume that l 0 = k(x 0 ). If ǫ(z) = ǫ(ẑ), then Z r is permissible of the first kind at x r (Definition 3.1(ii)). This proves that alternative (1) in the proposition is fulfilled or ǫ(ẑ) > ǫ(z) which we may assume from now on. By Theorem 2.74(2.ii), we have dimZ r ≥ 2 (statement ñ > n of ibid. applied under the assumption l 0 = k(x 0 )) and

ǫ(ẑ) -1 = ω(z) = ǫ(z) = ǫ(x) -1 = ǫ(x) -1, i 0 (ẑ) = i 0 (z) = p. (3.69)
We pick again well adapted coordinates (û 1 , . . . , ûn ; Ẑ) at x. Since Ẑ0 is permissible of the first kind at x, Proposition 3.3 (with notations as therein) gives the following property for the initial form in m Ŝ0 h ∈ G(m Ŝ0 )[ Ẑ]:

H -1 0 G p 0 ∈ k(x)[ Û1 , . . . , Ûe-1 , Ûe+1 , . . . Ûn ] ǫ(x) .
Since i 0 (ẑ) = p, we have G 0 = 0, i.e. i 0 (x) = p. This proves that Definition 3.5(ii) is satisfied in any case.

To prove that alternative (2) in the proposition is fulfilled, we first assume that l 0 = k(x 0 ) as before, then push down the result from S to S. Let (u 1 , . . . , u n ; Z) be well adapted coordinates at x and consider the initial form in

W 0 h = Z p + F p,Z,W 0 ∈ G(W 0 )[Z]. Let J := {1, . . . , e -1, e + 1, . . . , e + t 0 }. Since ǫ(ẑ) > ǫ(z), we have δ(z) ∈ N and G(W 0 ) = S 0 /I 0 [{U j } j∈J ], F p,Z,W 0 ∈ ( Ŝ0 / Î0 [{U j } j∈J ] δ(z) ) p (3.70)
by Theorem 2.74(2.ii). By Proposition 2.12, the polyhedron

∆ Ŝ0 (h; {u j } j∈J ; Z) = pr J (∆ Ŝ (h; u 1 , . . . , u n ; Z)) is minimal,
where pr J : R n → R J denotes the projection on the (u j ) j∈J -space. Let

Φ j := H -1 W 0 ∂F p,Z,W 0 ∂u j ⊆ G(W 0 ) ǫ(z) , cl 0 Φ j = 0, j ∈ J, j = e, (3.71)
since ǫ(x) = ǫ(z) + 1. The local blowing up S 1 has a r.s.p.

(u ′ 1 , . . . , u ′ n ) given by    u ′ j = u j /u e if j ∈ J u ′ e =
u e u ′ j = u j /u eδ j if j ∈ J, j = e where δ j ∈ S 0 is a unit or zero since we are assuming that l 0 = k(x 0 ). Let

Z ′ := Z/u e -θ, θ ∈ S 1 , h ′ := u -p e h ∈ S 1 [Z ′ ] define the strict transform (X 1 , x 1 ), with ∆ S 1 (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ )

minimal and consider the initial form in

W 1 h = Z ′ p + F p,Z ′ ,W 1 ∈ G(W 1 )[Z ′ ], G(W 1 ) = S 1 /I 1 [{U ′ j } j∈J ].
It is easily derived from (3.70)(3.71) that

Φ ′ j := H -1 W 1 ∂F p,Z ′ ,W 1 ∂u ′ j = u -ǫ(x) e Φ j ⊆ G(W 1 ) ǫ(z) , j ∈ J, j = e.
Applying again Lemma 3.21 below, it can be assumed w.l.o.g. that (Φ j = u m j e Ψ j , cl 0 Ψ j = 0) or Φ j = 0, j ∈ J, j = e. (3.72) This equation is valid when l 0 = k(x 0 ) and holds for S if and only if it holds for S. We may therefore replace S by S as before. Let x = (x 1 , . . . , x n ) ∈ N n be a vertex of ∆ S 0 (h; u 1 , . . . , u n ; Z) mapping to a vertex of ∆ S 0 (h; {u j } j∈J ; Z) with j∈J x j = δ(y). By (3.70) we have x j ∈ N for j ∈ J. Suppose that x j ∈ N for every j = e. Since Ŝ0 / Î0 ≃ k(x)[[u e ]], (3.70) implies that x is solvable: a contradiction. Taking j such that x j ∈ N, there exists j ∈ J, j = e such that Φ j = 0. This proves that r 1 := min{m j , j ∈ J, j = e : Φ j = 0} is well defined and that we have

Φ p,Z,W 0 := u -r 1 e H -1 W 0 F p,Z,W 0 ⊆ G(W 0 ) ǫ(z) , cl 1 Φ p,Z,W 0 ∈ (u e )G(W 0 ) ǫ(z) . (3.73) If r 1 = 0, then alternative (2) is fulfilled (Definition 3.5(iii)) since J (F p,Z,W 0 , E, W 0 ) =< {cl 0 Φ j } j ∈J,j =e > = 0.
by (3.73). Note that this situation does not occur if ǫ(x r 0 ) = 1, since ω(ϕ) > 0.

Otherwise, we define V 0 := V (u e , I 0 ) and Y 0 := η -1 0 (V 0 ) ⊂ Z 0 . Then Y 0 is Hironaka-permissible at x 0 and its generic point y 0 has ǫ(y 0 ) = ǫ(x) by (3.73). Let X1 be the blowing up of X 0 along Y 0 and note that ϕ lifts to the point x1 on the strict transform Z1 of Z 0 . Let h := u -p e h ∈ S1 [ Z] define the strict transform ( X1 , x1 ) of (X , x), W1 := η1 ( Z1 ). By Proposition 2.18, the initial form (3.74) By Proposition 3.7, we now have ω(x r 1 ) = ǫ(z) = ǫ(x r 0 ) -1 ≥ 0. Thus ω(x r 1 ) > 0 if ǫ(x r 0 ) ≥ 2 and we are done by the former case r 1 = 0. Otherwise, ǫ(x r 0 ) = 1 and ω(x r 1 ) = 0 and the conclusion follows.

in W1 h = Zp + F p, Z, W1 ∈ G( W1 )[ Z], G( W1 ) = S1 / Ĩ1 [{ Ũj } j∈J ]
Example 3.18. Take S = k[u 1 , u 2 , u 3 , u 4 ] (u 1 ,u 2 ,u 3 ,u 4 ) with k a field of characteristic p > 0. We let:

h = Z p + u p 2 u 4 u p 3 + u 3 u p 1 ∈ S[Z].
Then (u 1 , u 2 , u 3 , u 4 ) are adapted to (S, h, E), E := div(u 1 u 2 ) (Definition 2.19) and (u 1 , u 2 , u 3 , u 4 ; Z) are well adapted coordinates at the closed point x = (Z, u 1 , u 2 , u 3 , u 4 ) of X = Spec(S[Z]/(h)) (Definition 2.24). Indeed, it is easily seen that:

Sing p X := {y ∈ X : m(y) = p} = V (Z, u 1 , u 2 ) ∪ V (Z, u 1 , u 3 ), ω(x) = p. Let ϑ(t) := i≥1 λ i t i ∈ k[[t]
] be a power series which is transcendental over k(t). We define a nonconstant well-parametrized k-linear formal arc on (X , x) by:

ϕ(Z) = ϕ(u 1 ) = ϕ(u 3 ) = 0, ϕ(u 2 ) = t, ϕ(u 4 ) = ϑ(t) p . Let u (0) j := u j , 1 ≤ j ≤ 4. For r ≥ 1, well adapted coordinates at x r are u (r) j := u (r-1) j /u 2 , j = 1, 3, u (r) 2 := u 2 and v (r) 4 := u -r 2 (u 4 - ip≤r λ p i u ip 2 ), T r := u -r 2 (Z + (u (r) 3 ) p ip≤r λ p i u ip 2 ).
Then ϕ lifts through

(X r , x r ) = Spec(S r [T r ]/(h r ), x r ), S r = S[u (r) 1 , u (r) 3 , u (r) 4
] (u (r) 1 ,...,v (r)
4 ) , and the strict transform h r of h is given by

h r := T p r + (u (r) 2 ) r (u (r) 2 ) p v (r) 4 (u (r)
3

) p + u (r) 3 (u (r) 1 ) p .
We have Z r := V (T r , u

3 ) for every r ≥ 1. Note that Z r is not permissible at x r . Therefore ϕ fulfills alternative (2) of Proposition 3.17.

Remark 3.19. We do not know if the conclusion of Proposition 3.17 is still valid for n ≥ 4 when removing the assumption "l|k(x r ) is algebraic with finite inseparable degree for some r ≥ 0".

When n = 3, it can be proved that the above assumption is actually implied by "m(ϕ) = p and ω(ϕ) > 0". This is a (very) special case of the proof of Theorem 5.5. The following elementary corollary will be used repeatedly.

Corollary 3.20. Assume that n = 3. Let (S, h, E) be as before and x ∈ X . Let (X , x) =:

(X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ) ← • • • (3.75)
be a (possibly infinite) composition of local blowing ups at closed points with (m(x r ) = p, ω(x r ) > 0 and k(x r ) = k(x)) for every r ≥ 0. With notations as in Proposition 2.22 and Notation 2.23, assume that (S r , E r , h r ) is such that E r is irreducible for every r ≥ 0. Then (3.75) is finite.

Proof. Let E = div(u 1 ) and (u 1 , u

(0) 2 , u (0)
3 ; Z (0) ) be well adapted coordinates at x. Since k(x r ) = k(x) and E r is irreducible for every r ≥ 1, S r has well adapted coordinates

(u 1 , u (r) 2 := u (r-1) 2 /u 1 -γ (r) 2 , u (r) 3 := u (r-1) 3 /u 1 -γ (r) 3 ; Z (r) := Z (r-1) /u 1 -φ (r) ) where γ (r) 2 , γ (r) 3 , φ (r) ∈ S. Suppose that (3.75) is infinite. We let ûj := u 2 - r≥1 γ (r) j u (r) 1 ∈ Ŝ, j = 2, 3, and Ẑ := Z -φ, φ := r≥1 φ (r) u (r) 1 ∈ Ŝ. The induced morphism ϕ : Spec( Ŝ[Z]/(û 2 , û3 , Ẑ)) -→ (X , x)
is a nonconstant well parametrized formal arc on (X , x) with l = k(x) and whose associated quadratic sequence is (3.75). By Proposition 3.17, Z r (ϕ) is Hironaka-permissible for some r ≥ 0: a contradiction with (E), since Z r (ϕ) E r .

The following lemma is elementary and well-known. Lemma 3.21. Let S be a regular local ring of dimension n ≥ 1 with r.s.p. (u 1 , . . . , u n ) and

C := V (u 1 , . . . , u n-1 ) ⊂ (S 0 , s 0 ) := SpecS be a regular curve. Let (S 0 , s 0 ) ← (S 1 , s 1 ) ← • • • ← (S i , s i ) ← • • •
be the composition of local blowing ups such that S i is the blowing up of S i-1 along s i-1 and s i ∈ S i is the point on the strict transform C i of C for i ≥ 1.

Let f ∈ S, f = 0 and denote d := ord C f . There exists m, i 0 ∈ N such that for every i ≥ i 0 , there is a decomposition

f = u m+di n g i , g i ∈ S i := O S i ,s i and ord C i g i = ord s i g i = d.
Furthermore, the initial form in C i g i ∈ (gr

I C i S i ) d is the strict transform of in C f ∈ (gr I C S) d ≃ S/(u 1 , . . . , u n-1 )[U 1 , . . . , U n-1 ] d .
Proof. We have

S i = S i-1 [u (i) 1 , . . . , u (i) n-1 ] (u (i) 1 ,...,u (i)
n ) , where u (i)

j := u (i-1) j /u (i-1) n , 1 ≤ j ≤ n -1, u (i) n := u (i-1) n
for every i ≥ 1, with u (0)

j := u j , 1 ≤ j ≤ n. Then C i = V (u (i) 1 , . . . , u (i)
n-1 ) with these notations. There is an expansion

f = (u (i-1) n ) m i-1 g i-1 , g i-1 := x∈S γ(x) (i-1) (u (i-1) 1 ) x 1 • • • (u (i-1) n ) xn ∈ S i-1 , where γ(x) (i-1) ∈ S i-1 is a unit for each x ∈ S, S ⊂ N n a finite set, m i-1 ∈ N, g i-1 ∈ (u (i-1) n ). Since ord C f = d, it can be assumed without loss of generality that d = min x∈S {x 1 + • • • + x n-1 }. Therefore d = ord C i-1 g i-1 ≤ d i-1 := ord s i-1 g i-1 = min x∈S {| x |}.
Note that the initial form in C i-1 f is given by

in C i-1 f = x 1 +•••+x n-1 =d γ(x) (i-1) (u (i-1) n ) xn (U (i-1) 1 ) x 1 • • • (U (i-1) n-1 ) x n-1 ,
where γ(x

) (i-1) , u (i-1) n ∈ S i-1 /(u (i-1) 1 , . . . , u (i-1)
n-1 ) denote the classes of the corresponding elements in S i-1 . After blowing up, we get an expansion

f = (u (i) n ) m i-1 +d i-1 g i , g i := x∈S γ(x) (i-1) (u (i) 1 ) x 1 • • • (u (i) n-1 ) x n-1 (u (i) n ) |x|-d i-1 ∈ S i . Let A i-1 := {x ∈ S : x 1 + • • • + x n-1 < d i-1 }. For each x ∈ A i-1 , we have | x | -d i-1 < x n . We deduce: 0 ≤ min x∈A i {x n } < min x∈A i-1 {x n }.
This proves that there exists i 0 ≥ 0 such that A i = ∅ for every i ≥ i 0 . Then d i = d for i ≥ i 0 . This proves the first statement in the lemma, taking m := m i 0di 0 ≥ 0. Finally, this construction preserves the initial form in C f , i.e.

in C i f = u -(m+di) n (in C f ) u i n U (i) 1 , . . . , u i n U (i) n ,
and this concludes the proof. Proof. Our function (m, ω, κ) refines the multiplicity function m on X , and our notion of permissible blowing up refines the Hironaka-permissibility. We may thus apply the well known constructibility of multiplicity and Hironakapermissibility. It is therefore sufficient to prove the first statement when m(y) = p. For the second statement, we take a nonempty Zariski open set U 1 ⊆ Y such that Z is Hironaka permissible at every y ′ ∈ U 1 .

Let W := η(Y), s := η(y), W Z := η(Z) for the second statement. We pick an adapted r.s.p. (u 1 , . . . , u ns ) of S s , where E s = div(u 1 • • • u es ). For every y ′ ∈ U 1 there exists an adapted r.s.p. (u 1 , . . . , u

n y ′ ) of S η(y ′ ) (i.e. E η(y ′ ) = div(u 1 • • • u e y ′ ), e y ′ ≥ e s ) such that S s is the localization of S η(y ′ ) at some prime I(W y ′ ) = ({u j } j∈J y ′ ), J y ′ ⊆ {1, . . . , n y ′ }.
After possibly shrinking U 1 ⊆ Y, it can be assumed without loss of generality that e y ′ = e s for every y ′ ∈ U 1 .

We now choose any point y 0 ∈ U 1 . Let (u 1 , . . . , u n 0 ; Z) be well adapted coordinates at y 0 , s 0 := η(y 0 ), S 0 := S s 0 . There is a corresponding expansion

h = Z p + f 1,Z Z p-1 + • • • + f p,Z ∈ S 0 [Z], f 1,Z , . . . , f p,Z ∈ S 0 .
After possibly restricting again U 1 , we may assume that the rational functions u 1 , . . . , u n 0 , f 1,Z , . . . , f p,Z are regular at η(y ′ ) for every y ′ ∈ U 1 . Moreover, we have in S η(y ′ ) I(W ) = ({u j } j∈J ) (and I(W Z ) = ({u j } j∈J Z ) for the second statement) with J Z ⊆ J = {1, . . . , n}, n y ′ ≥ n, subsets which do not depend on y ′ . We fix an associated expansion at s 0 :

f i,Z = x∈S i γ(i, x)u ix 1 1 • • • u ixn 0 n 0 ∈ S 0 , 1 ≤ i ≤ p,
with S i ⊂ ( 1 i N) n 0 finite and γ(i, x) ∈ S 0 a unit for each x ∈ S i . After possibly restricting again U 1 , it may also be assumed that each γ(i, x) appearing in some f i,Z , 1 ≤ i ≤ p, is a regular function at η(y ′ ). By Proposition 2.12, the polyhedra ∆ S 0 (h; {u j } j∈J ; Z) (and ∆ S 0 (h; {u j } j∈J Z ; Z)) are minimal.

(3.76)

We define A i ⊂ ( 1 i N) J (and A i,Z ⊂ ( 1 i N) J Z for the second statement) to be the respective images of S i by the projections pr J : R n 0 → R J and pr J Z : R n 0 → R J Z . Given a ∈ A i , we let:

γ(i, a) := pr J (x)=a γ(i, x) j ∈J u ix j j ∈ S 0 .
By definition of ǫ(y), we have:

ǫ(y) = p min 1≤i≤p min a∈A i {| a |: γ(i, a) = 0} - es j=1 H j .
(3.77)

Let B ⊂ Q n be the set of (i, a) achieving equality on the right hand side of (3.77). The initial form polynomial in m Ss h is thus of the form Case 1. Suppose that B 0 = ∅. We define:

in m Ss h = Z p + (i,a)∈B γ(i, a) j∈J U ia j j Z p-i ∈ G(m Ss )[Z], ( 3
U := {y ′ ∈ U 1 : ∀(i, a) ∈ B 0 , γ(i, a) is a unit in S η(y ′ ) }.
Since γ(i, a) is nonzero for (i, a) ∈ B by (3.77), U is a nonempty Zariski open subset of Y. To y ′ ∈ U, we associate x ∈ ∆ S η(y ′ ) (h; u 1 , . . . , u n y ′ ; Z) (depending on (i, a)) by

x j = a j if j ∈ J x j = 0 if j ∈ J
Computing initial forms from Definition 2.3 with

α y ′ := (1, . . . , 1) ∈ R n y ′ , δ α y ′ (h; u 1 , . . . , u n y ′ ; Z) = δ(y), the corresponding initial form polynomial in α y ′ h = Z p + p i=1 F i,Z,α y ′ Z p-i ∈ G(m S η(y ′ ) )[Z] (3.79) is such that F i,Z,α y ′ = 0 for some i = p or F p,Z,α y ′ ∈ k(y ′ )[U p 1 , . . . , U p n y ′ ]
. Therefore δ(y ′ ) = δ(y) and we deduce that ǫ(y ′ ) = ǫ(y) for every y ′ ∈ U.

(3.80)

To prove the first statement, note that we are already done by (3.80) if ǫ(y) = 0. Assume now that ǫ(y) > 0. If i 0 (y) = p -1, there exists some (p -1, a 0 ) ∈ B 0 for some a 0 ∈ N J . Let y ′ ∈ U and pick well adapted coordinates (u 1 , . . . , u n y ′ ; Z y ′ ) at y ′ . The corresponding initial form polynomial

in m S η(y ′ ) h = Z p y ′ -G p-1 y ′ Z y ′ + F p,Z y ′ ∈ G(m S η(y ′ ) )[Z y ′ ] is such that < G y ′ >=< U a 0 > (resp. G y ′ = 0) if i 0 (y) = p -1 (resp. if i 0 (y) = p). We have F p,Z y ′ = (p,a)∈B 0 λ y ′ (p, a)U a + Ψ y ′ ⊆ G(m S η(y ′ ) ) ǫ(y) , where λ y ′ (i, a) ∈ k(y ′ ), λ y ′ (i, a) = 0, Ψ y ′ ∈ k(y ′ )[{U p j } j∈J ]
for every (p, a) ∈ B 0 and every y ′ ∈ U. Comparing with Definition 2.68, we have ω(y ′ ) = ω(y), κ(y ′ ) = 1 if κ(y) = 1 for y ′ ∈ U. This proves the first statement in case 1.

For the second statement, we are also done if ǫ(z) = ǫ(y), i.e. if Z is of the first kind at y. Suppose that Z is permissible of the second kind at y. In particular, we have ǫ(y) > 0. There exist j 1 (y) ∈ J\J Z and j ′ (y) ∈ J\J Z , j ′ (y) ≥ e s + 1, satisfying the conclusion of Proposition 3.7. Let y ′ ∈ U and pick well adapted coordinates (u 1 , . . . , u n y ′ ; Z y ′ ) at y ′ . The corresponding initial form polynomial (3.80) again satisfies

H -1 y ′ G p y ′ ⊆ U j 1 (y) k(y ′ )[U 1 , . . . , U n y ′ ] ǫ(y)
and there is an expansion

H -1 y ′ F p,Z y ′ =< j ′ ∈J ′ U j ′ Φ j ′ ({U j } j∈J ) + Ψ({U j } j∈J ) >⊆ G(m S η(y ′ ) ) ǫ(y)
with Φ j ′ (y 0 ) = 0, hence Y is permissible of the second kind at y ′ and the conclusion follows. Since ({u j } j∈J ; Z) are well adapted coordinates at y, there exists a vertex a 0 ∈ ∆ Ss (h; {u j } j∈J ; Z), (p, a 0 ) ∈ B which is not solvable, i.e. γ(p, a 0 ) ∈ k(y) p . Let B 1 ⊆ B 0 be the nonempty subset defined by

B 1 := {(p, a) ∈ B : γ(p, a) ∈ k(y) p }.
Given (p, a) ∈ B 1 , we define a morphism:

η (p,a) : Y (p,a) := Spec O U 1 [T ] (T p -γ(p, a)) -→ U 1 .
Note that Y (p,a) is integral and η (p,a) is finite and purely inseparable. We define: We now prove the first statement. Let y ′ ∈ U and pick well adapted coordinates (u 1 , . . . , u n y ′ ; Z y ′ ) at y ′ . Let

U := {y ′ ∈ U 1 : ∀(p, a) ∈ B 1 , η -1 (p,a) (y ′ ) red is a regular point of Y (p,a) }.
B(y ′ ) := {(p, a) ∈ B 1 : (a) is satisfied}. Suppose that B(y ′ ) = ∅. We get δ(y ′ ) = δ(y), i 0 (y ′ ) = p and the initial form polynomial in m S η(y ′ ) h ∈ G(m S η(y ′ ) )[Z y ′ ] is in m S η(y ′ ) h = Z p y ′ + (p,a)∈B(y ′ ) λ y ′ (p, a)U a + Ψ p y ′
where λ y ′ (p, a) ∈ k(y ′ ) p and Ψ y ′ ∈ k(y ′ )[{U p j } j∈J ]. This shows that ω(y ′ ) = ǫ(y ′ ) = ǫ(y) = ω(y), the right hand side equality by (3.82). Moreover κ(y ′ ) ≥ 2, so y ′ ∈ Ω(Y).

Suppose on the contrary that B(y ′ ) = ∅. We get

δ(y ′ ) = δ(y) + 1 p , i 0 (y ′ ) = p (since δ(y ′ ) ∈ N)
and the initial form polynomial in m S η(y

′ ) h ∈ G(m S η(y ′ ) )[Z y ′ ] is in m S η(y ′ ) h = Z p y ′ + (p,a)∈B 1 V y ′ (p, a)U a + Ψ y ′ , where V y ′ (p, a) ∈< U 1 , . . . , U n y ′ > \ < {U j } j∈J >, Ψ y ′ ∈ k(y ′ )[{U j } j∈J ] pδ(y)+1 .
This shows that ω(y ′ ) = ǫ(y ′ )-1 = ǫ(y) = ω(y), applying again (3.82). Moreover κ(y ′ ) ≥ 2, so y ′ ∈ Ω(Y). This concludes the proof of the first statement.

For the second statement, note that Z is necessarily of the first kind at y in case 2, since (3.81) is not compatible with Proposition 3.7. With notations as above, Z is then permissible of the first kind (resp. of the second kind) at We do not know if the sets Perm(Y, Z) as in the theorem are constructible subsets of Y. An important issue about permissibility is addressed below in Question 3.27.

y ′ if B(y ′ ) = ∅ (resp. if B(y ′ ) = ∅).
About a possible extension of our methods to a global Resolution of Singularities statement, we remark the following: let S be an excellent regular domain, η : X → S be a finite morphism, x ∈ X be such that (X , x) → S η(x) satisfies the assumption of Theorem 3.22. It is easily seen that its conclusion extends to some affine neighbourhood U of x on X .

Example 3.25.

Let S = k[[u 1 , u 2 , u 3
]], k a (nonperfect) field of characteristic p > 0 and λ, µ ∈ k be p-independent. We take:

h = Z p -(u 1 u 2 ) p-1 Z + λu p 3 + u 3 u p-1 1 + µu p 1 ∈ S[Z], E = div(u 1 u 2 ).
The coordinates (u 1 , u 2 , u 3 ; Z) are well adapted to (S, h, E). Let

x := (Z, u 1 , u 2 , u 3 ), y := (Z, u 1 , u 3 ).

We have H(x) = (1), m(x) = m(y) = p, and compute:

in m S h = Z p + λU p 3 + U 3 U p-1 1 + µU p 1 , i 0 (x) = p, ω(x) = ǫ(x) -1 = p -1.
137

On the other hand, we have:

in m S η(y) h = Z p -(U 1 u 2 ) p-1 Z + λU p 3 + U 3 U p-1 1 + µU p 1 , i 0 (y) = p -1, ǫ(y) = p.
In order to compute ω(y), we must introduce a truncation operator

T y : k(y)[U 1 , U 3 ] p → k(y)[U 1 , U 3 ] p
as in Definition 2.68 and get T y F p,Z,y = λU p 3 , so ω(y) = p > ω(x). This proves that the set X (p,p) := {z ∈ X : (m(z), ω(z)) ≥ (p, p)} is not Zariski closed.

Proposition 3.26. Let (X , x) be as in the theorem. The set

Ω + (X ) := {y ∈ X : (m(y), ω(y)) > (p, 0)} ⊆ X is Zariski closed and of dimension at most n -2.
Proof. Let ξ ∈ X be the generic point of an irreducible component of η -1 (E). Then (m(ξ), ǫ(ξ)) ≤ (p, 0), so ξ ∈ Ω + (X ). Therefore it is sufficient to prove that Ω + (X ) is Zariski closed.

We will use the Nagata Criterion to prove openness of X \Ω + (X ). By Theorem 3.22, it is sufficient to prove that Ω + (X ) is stable by specialization. Let y 0 y 1 be a specialization in X and assume that y 1 ∈ Ω + (X ). Since the multiplicity does not decrease by specialization [START_REF]Étude locale des singularités, Cours de 3 ème cycle[END_REF] Theorem 3.9 p.II-30, we may assume that m(y 1 ) = p. We are done unless m(y 0 ) = p which we assume from now on. Let Y 0 := {y 0 }.

By localizing η at η(y 1 ), it can be furthermore assumed that y 1 = x. Arguing by induction on the dimension of Y 0 , it can be furthermore assumed that Y 0 is a curve. Let

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ) ← • • •
be a sequence of local blowing ups at closed points belonging to the strict transform of Y 0 . We have m(x r ) ≥ m(y 0 ) = p ibid., so m(x r ) = p for every r ≥ 0. Since S is excellent, the strict transform of Y 0 in X r is Hironaka permissible for r >> 0. By construction, these maps induce local isomorphisms at y 0 .

We then have (m(x r ), ω(x r )) ≤ (p, 0) by Proposition 2.79, hence ω(x r ) = 0 since m(x r ) = p for every r ≥ 0. In other words, after possibly replacing (X , x) by (X r , x r ) for some r ≥ 0, it can be assumed that Y 0 is Hironaka permissible. Then there exist well adapted coordinates (u 1 , . . . , u n ; Z) at x such that

I(W 0 ) = ({u j } j∈J 0 ), W 0 := η(Y 0 )
with J 0 = {1, . . . , n}\{j ′ } for some j ′ (since Y 0 is a curve). We let s 0 := η(y 0 ), S 0 := S s 0 . By Proposition 2.12, the polyhedron ∆ S (h; {u j } j∈J ; Z) is minimal, so we deduce that ǫ(y 0 ) ≤ ǫ(x).

Since ω(x) = 0 by assumption, we have ω(y 0 ) = 0 except possibly if

ǫ(y 0 ) = ǫ(x) = 1. Since ω(x) = 0, the initial form polynomial in W 0 h ∈ G(m S )[Z] then satisfies H -1 W 0 F p,Z,W 0 =< j∈J 0 γ j U j >⊆ G(W 0 ) 1 = S/I(W 0 )[{U j } j∈J 0 ],
and there exists j 0 ∈ J 0 , e + 1 ≤ j 0 ≤ n such that γ j 0 is a unit in S/I(W 0 ). This gives ω(y 0 ) = 0 if i 0 (y) = p. If i 0 (y) = p -1, we must introduce a truncation operator

T 0 : G(m S 0 ) pδ(y 0 ) → G(m S 0 ) pδ(y 0 ) ,
as in Definition 2.68 in order to compute ω(y 0 ). However, T 0 proceeds from Definition 2.59 in the special case pδ(y 0 ) = 1 + j∈J 0 H j . Lemma 2.61 then implies that

H -1 W 0 KerT 0 ⊆< {U j } j∈J 0 ,j≤e >⊂ G(m S 0 ) pδ(y 0 ) .
Since j 0 ≥ e+1, we thus have H W 0 U j 0 KerT 0 and this proves that ω(y 0 ) = 0 as required.

A very special case of the following question (for µ a discrete valuation with some extra assumption) has been answered in the affirmative in Proposition 3.17 above. See also Theorem 6.3 for a related result. Question 3.27. Let Y = Y 0 be an integral closed subscheme with generic point y, m(y) = p, ω(y) > 0, and let µ be a valuation centered at m S . Does there exist a finite sequence of permissible local blowing ups along µ:

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ) with centers Z i ⊂ (Y i , x i ), Y i denoting the strict transform of Y in (X i , x i ), 0 ≤ i ≤ r, such that Y r is permissible at x r ?
Then R ′ is an iterated quadratic transform of R. Proposition 4.3. (Cossart-Jannsen-Saito) Let S be a regular Noetherian irreducible scheme of dimension three which is excellent and X ֒→ S be a reduced subscheme.

There exists a composition of blowing ups along integral regular subschemes σ : S ′ → S such that the strict transform X ′ ֒→ S ′ of X has strict normal crossings with the reduced exceptional divisor E of σ. Moreover σ restricts to an isomorphism π : X ′ \σ -1 (SingX) ≃ X\SingX. Proposition 4.4. (Cossart-Piltant) Let S be a regular Noetherian irreducible scheme of dimension three which is excellent and I ⊆ O S be a nonzero ideal sheaf. There exists a finite sequence

S =: S(0) ← S(1) ← • • • ← S(r)
with the following properties:

(i) for each j, 0 ≤ j ≤ r -1, S(j + 1) is the blowing up along a regular integral subscheme Y(j) ⊂ S(j) with Y(j) ⊆ {s j ∈ S(j) : IO S(j),s j is not locally principal}.

(ii) IO S(r) is locally principal.

Proof. The assumption "X/k is quasi-projective" is not used in the proof of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Proposition 4.2. The equicharacteristic assumption is used only via the power series expansions used for defining E and the characteristic polygon "∆(E; u 1 , u 2 ; y) prepared" on pp.1061-1062 of ibid.. But this is also characteristic free by [START_REF] Cossart | Characteristic polyhedra of singularities without completion[END_REF] Theorem II.3.

Reduction to local uniformization and proof of the corollaries.

We now reduce Theorem 1.1 to its local uniformization form (LU) below. Let (A, m, k) be a quasi-excellent local domain with quotient field K. Recall that quasi-excellent rings are Noetherian by Definition [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] (7.8.2) and Remark (7.8.4)(i). We consider the following Local Uniformization problem:

(LU) for every valuation v of K, with valuation ring

(O v , m v , k v ) such that A ⊂ O v ⊂ K, m v ∩ A = m, k v |k algebraic,
there exists a finitely generated A-algebra T , A ⊆ T ⊆ O v , such that T P is regular, where P := m v ∩ T . Zariski's proof of the Fundamental Theorem (quoted from [START_REF] Zariski | Reduction of the singularities of algebraic three dimensional varieties[END_REF] on p.539) only requires two results: [START_REF] Zariski | Reduction of the singularities of algebraic three dimensional varieties[END_REF] Theorem 7 of section 19 and the Lemma on p. 538. In our characteristic free context, these are respectively Lemma 4.5 below and Proposition 4.2.

Y =: Y 0 ← Y 1 ← • • • ← Y r+1 = Y ′ (4.1)
of blowing ups along regular centers

Z i ⊆ Y i such that (i) Z i is fundamental for ρ i : Y i • • • -→ X , 0 ≤ i ≤ r; (ii) ρ • π is a morphism on π -1 (F ∩ RegY), where π : Y ′ → Y is the composed map.
Proof. This lemma rephrases [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Proposition 4.7, using the characteristic free Proposition 4.4. We denote by

F • := F ∩ RegY, dimF • ≤ 1.
Let F ⊆ F be the Zariski closure of F • in Y and G ⊆ F be its onedimensional component (possibly G = ∅). We construct π as a composition of blowing ups along regular subschemes mapping to F.

Step 1: let

π 1 : Y i 1 → Y (4.2)
be the minimal composition of blowing ups at closed points such that the strict transform G ′ of G is a disjoint union of regular curves, followed by the blowing up along G ′ . Let

ρ 1 : Y i 1 • • • -→ X
denote the composed map ρ • π 1 , F 1 its fundamental locus. We now denote

F • 1 := F 1 ∩ π -1 1 (RegY)
and

F 1 ⊆ F 1 its Zariski closure in Y i 1 .
Let furthermore G 1 ⊆ F 1 be the union of its one-dimensional irreducible components whose image in Y has dimension one. We now iterate this construction. Applying a classical result on quadratic sequences in regular local rings of dimension two (e.g. [START_REF] Zariski | Commutative Algebra II[END_REF] appendix 5, Theorem 3 and (E) on p.391), we construct

π n : Y in → Y such that ρ • π n is a morphism away from π -1 n ((F ∩ RegY)\{x 1 , . . . , x k }),
where x 1 , . . . , x k are finitely many closed points.

Step 2: let Z be the closure of the graph of ρ • π n . Since X is projective, Z is isomorphic to the blowing up of Y n along a certain ideal sheaf

I n ⊆ O Y in . Since π -1 n (RegY) ⊆ RegY n , there exists I ⊆ O Y in with V (I) ⊆ π -1 n (x 1 ) ∪ . . . ∪ π -1 n (x k ), dimV (I) ≤ 1, (4.3)
such that Z is isomorphic to the blowing up of Y in along I above π -1 n (RegY). Applying Proposition 4.4 to I ⊆ O Y in concludes the proof. Proposition 4.6. Let X be a reduced and separated Noetherian scheme which is quasi-excellent and of dimension at most three. Let X 1 , . . . , X c be the irreducible components of X . Assume that (LU) holds for every local ring of the form A = O X i ,x i which is of dimension three, 1 ≤ i ≤ c. Then Theorem 1.1 holds for X . Proof. Suppose that (i) and (ii) in Theorem 1.1 have been proved. Apply Proposition 4.3 to

X := π -1 (SingX ) red ⊆ X ′ ,
then blow up along X ′ : we get (iii). It remains to prove (i) and (ii).

Step 1: it can be assumed that X is irreducible of dimension three.

There is a finite birational morphism

f : c i=1 X i → X ,
isomorphic above RegX . The theorem holds for X if it holds for each X i . Resolution of singularities is known if dimX ≤ 2 [START_REF] Lipman | Desingularization of two-dimensional schemes[END_REF], so we may assume that dimX = 3.

Step 2: it can be assumed that X = SpecA is affine. This is based on Lemma 4.5. Consider open sets U ⊆ X satisfying (i) and (ii) in Theorem 1.1, i.e. there exists π U : U ′ → U proper and birational, such that RegU ′ = U ′ and π -1 U (RegU) ≃ RegU. (4.4)

We assume furthermore that a finite affine covering

U = U 1 ∪ • • •∪ U n is given such that π -1 U (U i ) → U i is projective. (4.5)
Claim: if two open sets U 1 and U 2 satisfy (4.4) and (4.5), so does U 1 ∪ U 2 w.r.t. the union of their respective coverings. Since X is Noetherian, this claim completes reduction step 2. We now prove the claim. Let V := U 1 ∩ U 2 . Denote by π i : U ′ i -→ U i the given resolutions of singularities satisfying (4.4) and (4.5). Let

F 1 ⊆ U ′ 1 ∩ π -1 1 (V)
be the fundamental locus of the birational map

ρ : U ′ 1 ∩ π -1 1 (V) • • • -→ U ′ 2 ∩ π -1 2 (V),
and

F 1 ⊆ U ′ 1 be its Zariski closure in U ′ 1 .
By (4.4), we have:

π 1 (F 1 ) ⊆ SingU 1 .
In particular, we may replace U ′ 1 by any blow up along a regular center contained in F 1 . We apply Lemma 4.5 to π -1 i (U j 1 j 2 ) → U j 1 j 2 , i = 1, 2 for each U j 1 j 2 := U j 1 ∩ U j 2 with obvious notations.

When some Z i in Lemma 4.5 is a curve, it can be assumed that Z i is regular away from (the inverse image of) V by blowing up closed points beforehand. Furthermore the sequences (4.1) for distinct U j 1 j 2 's glue together, which follows from the Definitions (4.2)-(4.3). We may thus assume that ρ is a morphism.

(4.6)

Let F 2 ⊆ U ′ 2 ∩ π -1
2 (V) be the fundamental locus of ρ -1 and consider the associated sequence (4.1). We will only perform step 1 in the proof of Lemma 4.5.

When Z i is a closed point mapping to V, we apply Proposition 4.4 beforehand to

I(Z i )O U ′
1 in order to preserve (4.6). When Z i is an irreducible curve with generic point ξ i , whose image in V has dimension one, the ideal

I(Z i )O U ′ 1 is invertible above ξ i by Proposition 4.2. Applying Proposition 4.4 beforehand to I(Z i )O U ′
1 , we also preserve (4.6) while U ′ 1 is unchanged away from the inverse image of finitely many closed points of V. It can be assumed that Z i is regular away from the inverse image of V by blowing up closed points beforehand as above.

Summing up, it can be assumed that (4.6) holds and that ρ -1 is a morphism (hence an isomorphism by (4.6)) away from π -1 2 (x 1 ), . . . , π -1 2 (x k ), x 1 , . . . , x k ∈ V finitely many closed points. (4.7)

We may then glue U ′ 1 and U ′ 2 \{π -1 2 (x 1 ), . . . , π -1 2 (x k )} along

π -1 1 (V\{x 1 , . . . , x k }) = π -1 2 (V\{x 1 , . . . , x k })
to some proper morphism π W : W ′ → W := U 1 ∪ U 2 . By construction, π W satisfies (4.4) and (4.5) for each (4.7). This concludes the proof of the claim, hence of step 2. is a nonempty open property for any reduced Y which is of finite type over X because A is excellent. This applies in particular to any projective closure of SpecT , T as in (LU). Hence Theorem 1.1(i) is reduced to the following patching problem: let

U j 1 ⊆ U 1 . Let U j 2 ⊆ U 2 be fixed, so π -1 2 (U j 2 ) → U j 2 is projective. Now π -1 1 (U j 1 j 2 ) → U j 1 j 2 is projective for each U j 1 ⊆ U 1 , so π W (U j 2 ) → U j 2 projective follows from
X 1 -→ SpecA, X 2 -→ SpecA
be projective birational morphisms. There exists Y -→ SpecA projective birational and morphisms [START_REF] Zariski | Reduction of the singularities of algebraic three dimensional varieties[END_REF] on p.539, Zariski's Patching Theorem only requires Proposition 4.2 and Lemma 4.5 (here in our characteristic free context) in order to deduce step 3 from (LU).

π i : Y -→ X i , i = 1, 2, such that π -1 1 (RegX 1 ) ∪ π -1 2 (RegX 2 ) ⊆ RegY. As indicated in
Step 4: achieving (ii). Let π : X ′ → X be as achieved in step 3, i.e. projective birational with RegX ′ = X ′ . Let F ⊆ X be the fundamental locus of π -1 . We define

F 1 := Zariski closure in X of F ∩ RegX .
Note that F 1 has dimension at most one. We only sketch the argument and refer to [START_REF] Cossart | Modèle projectif régulier et désingularisation[END_REF] (see also [START_REF] Piltant | An axiomatic version of Zariski's Patching Theorem[END_REF] section 6) for the details. There exists a commutative diagram

X ′ e ′ ←-Y ′ ↓ ↓ X e ←-Y (4.8)
such that e (resp. e ′ ) is a composition of blowing ups with regular centers mapping to SingX (resp. to π -1 (SingX )). Let π ′ : Y ′ → Y be the resulting morphism. This diagram has the following property: let G ⊂ Y be the fundamental locus of π ′ -1 , and

F ′ 1 ⊆ G be the strict transform of F 1 . Then any connected component of G containing points of SingY is disjoint from F ′ 1 (in particular F ′ 1 ⊂ RegY)
. This is achieved as follows: (a) by iterating finitely many blowing ups of X at intersection points of F 1 and SingX , then applying Proposition 4.4, we first obtain e, e ′ such that F ′ 1 ⊂ RegY. (b) by applying the techniques of step 2 above those irreducible curves C ⊆ G only such that

C F ′ 1 , C ∩ F ′ 1 = ∅,
then applying Proposition 4.4 to get e ′ , we disconnect F ′ 1 from components of G containing points of SingY.

By (4.8), there exists U ⊆ RegY such that the fundamental locus of π ′ -1 (U) → U is a projective subscheme (of dimension at most one) containing

F ′ 1 . We define Z ⊂ Y ′ × X Y by composing the diagonal embedding ∆ Y ′ : Y ′ → Y ′ × X Y ′ with the second projection 1 × π ′ above Y ′ × X U. Then Z → X

Reduction to cyclic coverings.

In this section, we reduce the local uniformization form (LU) of the previous section to Theorem 1.5. This reduction is performed in two steps: first to complete local domains, then to cyclic coverings of degree p in residue characteristic p > 0. The first step is adapted from the descent methods of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Proposition 9.1 for (LU) inside the Henselization of finitely generated algebras of dimension three. Descent from complete local rings to Henselian local rings, i.e. algebraization of (LU), is proved in any dimension in [START_REF] Illusie | Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasiexcellents[END_REF] Proposition 6.2, but this does not imply Proposition 4.8 below.

Proposition 4.8. Assume that (LU) holds for every complete local domain of dimension three. Then Theorem 1.1 holds.

Proof. By Proposition 4.6, it is sufficient to prove that (LU) holds for every quasi-excellent local domain (A, m A , k) of dimension three. As an indication, the general strategy of the proof is deducing (LU) for A from Theorem 1.1 for Spec Â. We will choose an extension v of v to Spec  and a suitable resolution of singulaties Ŷ → Spec  (Lemma 4.9) such that Ŷ algebraizes at the center ŷ of v. By general facts about excellent rings and Zariski's Main Theorem, Ŷ algebraizes to a regular local ring T P at ŷ (see (4.12) below).

Let v be a valuation of K as in (LU). Denote by

Γ v := K × /O × v , r := dim Q (Γ v ⊗ Z Q
) the value group and rational rank of v. To begin with, we may assume that dimO v = 1, i.e. Γ v ⊂ (R, ≥), applying [START_REF] Novacoski | Reduction of Local Uniformization to the rank one case[END_REF] Theorem 1.1 (valid in all dimensions) or using the dimension three techniques in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Proposition 5.1. We may also assume that the residue extension

k v |k is algebraic: if x ∈ O v has transcendental residue, replacing A by B := A[x] mv ∩A[x]
gives a reduction on dimension, since dim B < dim A by the dimension formula.

Since A is local quasi-excellent, its formal completion  w.r.t. m A is reduced [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Let X := SpecA, X := Spec  and f : X → X be the completion morphism. By assumption in this proposition and Proposition 4.6, Theorem 1.1 holds for X . Let π : Ŷ → X be the corresponding resolution of singularities. Let ŷ ∈ Ŷ be the center of v. Since k v|k is algebraic and Â/ P1 is universally catenary, we have

d = dimO Ŷ,ŷ .
By [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF](7.8.3)(v), we have Sing X = f -1 (SingX ). Therefore there exists g ∈ A, g = 0 such that π is an isomorphism above Xg = Spec Âg by Theorem 1.1(ii). Let also

f 1 , . . . , f r ∈ A such that v(f 1 ), . . . , v(f r ) are Q-linearly independent in Γ v and set h := gf 1 • • • f r ∈ A.
We have:

Lemma 4.9. With notations as above, it can be assumed that

hO Ŷ,ŷ = m ÂO Ŷ,ŷ = (û 1 • • • ûr ), (4.9)
where (û 1 , . . . , ûd ) is a r.s.p. of O Ŷ,ŷ . In particular

v(û 1 ), . . . , v(û r ) ∈ Γ v ⊗ Z Q
and these values are Q-linearly independent.

Proof. This is [29] Proposition 6.2, taking into account Proposition 4.3. Note that it is not necessary to assume here that dimO v = 1 because h ∈ A.

We now conclude the proof which is easily adapted from [29] Proposition 9.1. By elementary linear algebra, there exists an r × r matrix M ∈ M(r, Z), a = detM > 0 such that

g j := r i=1 f m ij i = δj ûa j ∈ O Ŷ,ŷ ∩ K, (4.10)
where δj

∈ O Ŷ,ŷ is a unit, 1 ≤ j ≤ r. Let Qj := (û j ) ∩ Â, r + 1 ≤ j ≤ d.
By construction (4.9), we have O Ŷ,û j =  Qj , so (û j ) is the strict transform of Qj at ŷ. Since A is dense in  for the m A -adic topology, the right-hand side equality in (4.9) implies: there exists g ′ r+1 , . . . , g ′ d ∈ A and positive integers m ij , 1 ≤ i ≤ r, r + 1 ≤ j ≤ d, such that:

u ′ j := g ′ j r i=1 û-m ij i ∈ O Ŷ,ŷ
and (û 1 , . . . , ûr , u ′ r+1 , . . . , u ′ d ) is a r.s.p. of O Ŷ,ŷ . Let now

g j := g ′ j a r i=1 g -m ij i = u ′ j a r i=1 δ-m ij j ∈ O Ŷ,ŷ ∩ K (4.11)
and T be the integral closure of A[g 1 , . . . , g d ] in K. By [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Corollary 7.7.3, T is a finitely generated A-algebra. Furthermore, we have

A ⊆ T ⊆ O Ŷ,ŷ ∩ K ⊂ O v ∩ K = O v (4.12)
by (4.10)-(4.11). To complete the proof, it must be proved that T P is regular, where P := m v ∩ T . By [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Lemma 7.9.3.1, it is sufficient to prove that T ′ := T ⊗ A Â is regular at the center P ′ := m ŷ ∩ T ′ of v. Since T P is normal, T ′ P ′ is also normal ibid. and (7. Proof. By Proposition 4.8, it is sufficient to prove that (LU) holds for every complete local domain (A, m, k) of dimension three. Let (O v , m v , k v ) be the given valuation ring as in (LU). We may assume here that chark v = p > 0, the equicharacteristic zero version of Theorem 1.1 being known. As in Proposition 4.8, it is sufficient to deal with the case dimO v = 1.

By Noether normalization [START_REF] Matsumura | Commutative ring theory[END_REF] Theorem 29.4(iii), there exists a complete regular local domain S ⊆ A such that A is a finite S-module, dimS = 3. We will prove that the equal characteristic techniques of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] extend to our situation. Let F be the quotient field of S, so the field extension K|F is finite algebraic. By [START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF] Corollary 7.7.3, the integral closure of A in any finite extension of F is a finite A-module.

Let K sep ⊆ K be the separable closure of F . We first reduce to the case K sep = K. If charK = 0, we already have K sep = K and there is nothing to prove. Assume p := charK > 0. The extension K|K sep is a tower of purely inseparable extensions of degree p = charK:

K sep =: K 0 ⊂ K 1 ⊂ • • • ⊂ K n := K, n ≥ 0.
Let i ≥ 1 and assume that (LU) holds for the integral closure S i-1 of S in K i-1 . We have:

K i = K i-1 (x 1/p i ), x i ∈ K p i-1
. By Proposition 4.4 (applied to the ideal (f, g) where x i = f g ), we may take x i ∈ S i-1 , where S i-1 is given by (LU) for S i-1 . So

h := X p -x ∈ S i-1 [X]
satisfies the assumption of Theorem 1.5(i). We conclude that (LU) holds for S i which completes the induction step. From now on, we assume that K|F is separable.

Let K|K be a Galois closure and v be an extension of v to K. Ramification theory of valuations [START_REF] Zariski | Commutative Algebra II[END_REF] section 12 provides a diagram of fields

QF (A) = K ⊆ K i ⊆ K r ⊆ K ↑ ↑ ↑ QF (S) = F ⊆ F i ⊆ F r (4.13)
as in the proof of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Theorem 8.1. More precisely, F i (resp. K i ) is the inertia field with respect to v of the field extension K|F (resp. K|K); F r (resp. K r ) is the ramification field with respect to v of the field extension K|F (resp. K|K).

The left-hand side (resp. middle) inclusions in this diagram are unramified (resp. totally ramified Abelian of order prime to p). The extension K r |F r is a tower of totally ramified Galois extensions of degree p. Remark 4.11. Theorem 1.5 is actually required only to deal with those ramified extensions of degree p which are immediate (same value group and same residue field) w.r.t. the corresponding restrictions of v. For extensions of degree p which are not immediate, a much simpler proof is available, vid.

[29] Proposition 6.3 in the equicharacteristic case.

In order to connect ramification theory of valuations and ramification theory of S-algebras essentially of finite type, we restate [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Theorem 7.2 in our context as Proposition 4.13 below. Definition 4.12. Let (R, m, k) be an excellent regular local ring, L|QF (R) be a finite field extension and w be a valuation of L, with valuation ring

(O w , m w , k w ) such that R ⊂ O w ⊂ L, m w ∩ R = m, k w |k algebraic. A normal local model of O w |R is the localization B P of a finitely generated R-algebra B, R ⊆ B ⊆ O w , QF (B) = L such that B is normal, where P := m w ∩ B.
Let L ′ |L be a finite field extension and w ′ be an extension of w to L ′ . Given a normal local model

B P of O w |R, we define a normal local model B ′ of O w ′ |R by localizing the integral closure B of B in L ′ at P ′ := m w ′ ∩ B. Note that B ′ is actually a normal local model because R, hence B, is excellent. Assume that L ′ |L is Galois. Note that if B ′ is a normal local model of O w ′ |R then B ′ ∩ L = B ′ Gal(L ′ |L) is a normal local model of O w |R. By G s (B ′ |B), (resp. G i (B ′ |B))
, we mean the splitting group (resp. inertia group) of B ′ |B. For ramification theory of local rings, we refer to [START_REF] Abhyankar | Ramification theoretic methods in algebraic geometry[END_REF] (see also [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] section 2 equations (2)(3) for a quick summary of the required notions and notations).

Finally, we denote by G s (w ′ |w) (resp. G i (w ′ |w), G r (w ′ |w)) the splitting group (resp. the inertia group, ramification group) of O w ′ |O w from classical valuation theory: we refer to [START_REF] Zariski | Commutative Algebra II[END_REF] chapter VI, section 12, see also [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] section 2 pp. 1056-7.

Proposition 4.13. (Galois Approximation). Let L ′ |L be a finite Galois extension and w ′ be an extension of w to L ′ . There exists a normal local model B 0 of O w |R such that for any normal local model B of O w |R with B 0 ⊆ B, the following holds:

(1) G s (w ′ |w) = G s (B ′ |B) and G i (w ′ |w) = G i (B ′ |B); (2) the normal model B r := B ′ G r (w ′ |w) of O w r |R satisfies B r /m B r = B i /m B i ,
where B i is the inertia ring of B ′ over B, i.e. B i = B ′ G i (B ′ |B)
, and w r is the restriction of w ′ to L r := L ′ G r (w ′ |w) . Moreover the representation

ρ : G i (w ′ |w)/G r (w ′ |w) → GL(m B r /m 2 B r ), g → (x → g.x) is faithful and diagonalizable.
Proof. Since O w is the direct union of all its normal local models B, its integral closure O w in L ′ is the direct union of all corresponding integral closures B in L ′ . Since the extensions of O w to L ′ are the localizations of O w at its maximal ideals m 1 , . . . , m s , any B 0 such that for 1 ≤ i ≤ s, the m i ∩ B 0 's are pairwise distinct satisfies the statement about splitting groups in (1) of the proposition.

Let now B be any normal local model of

O w /R such that B 0 ⊂ B. There is an inclusion G i (w ′ /w) ⊆ G i (B ′ /B). Let t 1 , .
. . , t f be elements of O w ′ whose residues t 1 , . . . , t f generate k w ′ as a k w vector space. Enlarging B 0 , it can be assumed that k w /(B 0 /m B 0 ) is algebraic and that t 1 , . . . , t f ∈ B ′ 0 . Then any g ∈ G i (B ′ /B) acts trivially on k w ′ , hence g ∈ G i (w ′ /w). This concludes the proof of [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF].

We now turn to the proof of ( 2). On the one hand, the residue extension

B ′ m B ′ | B m B
is generated by purely inseparable elements (Theorem 1.48 [START_REF] Abhyankar | Ramification theoretic methods in algebraic geometry[END_REF]). On the other hand the field extension QF (B r )|QF (B i ) is Galois of degree prime to p since

H := G i (w ′ /w)/G r (w ′ /w) ≃ Hom(Γ w ′ /Γ w , k w ′ ) × . (4.14)
by [START_REF] Zariski | Commutative Algebra II[END_REF] Theorems 24 and 25. Hence B ′ m B ′ = B i m B i . From now on, for x ∈ m B r , we write x for its initial form in m B r /m 2 B r . Consider the following representation of

H ρ : H → GL m B r /m 2 B r , h → x → h.x . (4.15)
By [START_REF] Zariski | Commutative Algebra II[END_REF] middle of page 78, Hom(Γ w ′ /Γ w , k w ′ ) × is the entire character group, so k w ′ contains the group µ ǫ of ǫ th -roots of unity, where ǫ is the exponent of the Abelian group H, and ǫ is prime to p. Since k w ′ |k w r is purely inseparable (Theorem 1.48 [START_REF] Abhyankar | Ramification theoretic methods in algebraic geometry[END_REF]), we also have µ ǫ ⊆ k w r . Enlarging B 0 , we may assume that µ ǫ ⊆ B i m B i . Now, any irreducible representation of H over B i m B i has degree one, since H is Abelian and µ ǫ ⊆ B i m B i . Therefore, ρ is diagonal up to choosing a basis (x 1 , . . . , x n ) of m B r /m 2 B r . We write ρ(h).x j =: χ j (h)x j , for 1 ≤ j ≤ n and h ∈ H, where χ j ∈ Hom(H, B i m B i ) × .

Let Lr := QF ( Br ) and Li := QF ( Bi ). Since w ′ /w i is totally ramified, we also have Gal( Lr / Li ) = H with the natural extension of the H-action to formal completions. By Hensel's Lemma, the embedding µ ǫ ⊆ B i m B i lifts to an embedding µ ǫ ⊆ Bi . Let

y j := 1 | H | h∈H χ j (h -1 )(h.x j ) ∈ Br . (4.16)
It is immediately checked that y j = x j and that h.y j = χ j (h)y j for each h ∈ H. After replacing x j with y j , it can therefore be assumed that h.x j = χ j (h)x j (4.17)

for each h ∈ H and 1 ≤ j ≤ n, i.e. the action is faithful and diagonal on Br .

We now complete the proof of Proposition 4.10. To emphasize the dependence on v, we say that (LUv) holds if (LU) holds for a particular v. With notations as in (4.13), we denote by v 0 , v i 0 , v r 0 , v i , v r the respective restrictions of v to F , F i , F r , K i and K r . The strategy is to prove successively the implications

(LUv 0 ) =⇒ (LUv i 0 ) =⇒ (LUv r 0 ) =⇒ (LUv r ) =⇒ (LUv i ) =⇒ (LUv).
Note that (LUv 0 ) holds by construction since S is regular. Firstly, we apply Proposition 4.13 (1) with

R = S, L = F, L ′ = F i , w = v i 0 .
By Proposition 4.4, we may assume that B 0 ⊂ S. By Proposition 4.13 (1), the corresponding ring S ′ from Definition 4.12, is local-étale over S, hence S ′ is regular. So (LUv i 0 ) holds (this follows the argument in [29] Corollary 7.3).

Then (LUv r 0 ) holds because F r |F i is a tower of ramified Galois extensions of prime degrees l = p: the proof relies on the Perron algorithm as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Proposition 6.3 and this is characteristic free.

To prove that (LUv r ) holds, we may assume that K r |F r is a single Galois extension of degree p. Let x ∈ O v r be a primitive element with minimal polynomial

h := X p + f 1 X p-1 + • • • + f p ∈ O v r 0 [X]
. By Proposition 4.4, we may take f 1 , . . . , f p ∈ T r , where T r is a local uniformization, since (LUv r 0 ) holds: we have the assumptions of Theorem 1.5(ii) which states that (LUv r ) holds.

To prove that (LUv i ) holds, we may assume that K r |K i is a single Galois extension of prime degree l = p. By Proposition 4.13 (2), the representation ρ is faithful and diagonal. Using elementary linear algebra [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] p. 1080, we may assume that ρ has the form:

(x 1 , x 2 , x 3 ) → (ζx 1 , x 2 , x 3 ),
where (x 1 , x 2 , x 3 ) is a suitable r.s.p. of B r and ζ ∈ µ l . In this situation, we have

B r = B i ⊕ B i x 1 ⊕ • • • ⊕ B i x l-1 1 . Let u i := h l (x i ), 1 ≤ i ≤ 3 where h is the generator of H. This means that B r (u 1 , u 2 , u 3 )B r ∼ -→ B r m B i B r ∼ = B r m B r [X 1 ] (X l 1 )
. By flatness, m B i = (u 1 , u 2 , u 3 ): B i is regular.

To prove that (LUv) holds, let K s be the splitting field with respect to v of the field extension K|K. Firstly, we apply Proposition 4.13 (1) with

R = A, L = K, L ′ = K, w = v.
Let T i be a given regular local model of O v i |A. By Proposition 4.4, we may assume that B 0 ⊂ T i . Let T be the localization of the integral closure of T i in K at the center of v; let T := T i ∩ K s = (T i ) Gal(K i |K s ) be the fixed ring. By Proposition 4.13 (1), we have

Gal(K|K s ) = G s (v|v s ) = G s (T |T ) and G i (v|v s ) = G i (T |T ).
This shows that:

G i (T i |T ) = G s (T |T ) G i (T |T ) = Gal(K i |K s ).
By [START_REF] Raynaud | Anneaux locaux henséliens[END_REF] Theorem 2 p. 110, T i is local-étale over T . Since T i is regular, so is T . Therefore (LUv s ) holds. Let B 0 be as above, let B s 0 be the localization of the integral closure of B 0 in K s at the center of v s . By Proposition 4.4, we may assume that B s 0 ⊂ T . Note that B0 = Bs 0 .

We claim that there exist

g 1 , g 2 , g 3 ∈ T ∩ K such that (g 1 , g 2 , g 3 )T = m T .
The construction is the same as in the proof of Proposition 4.8, using (4.18), see Lemma 4.9 up to the end of the proof of Proposition 4.8. By Zariski's Main Theorem [START_REF] Raynaud | Anneaux locaux henséliens[END_REF] Theorem 1 p. 41, T is the localization of the integral closure of B 0 [g 1 , g 2 , g 3 ] in K s at the center of v s . Let T 0 be the localization of the normalization of B 0 [g 1 , g 2 , g 3 ] at the center of v, T = T0 : T 0 is regular and (LUv) is proved.

Normal crossings divisors conditions.

In this section, we consider a pair (S, h) satisfying the assumptions of Theorem 1.5, i.e. such that (G) holds. We construct a sequence π : X ′ → X of blowing ups along Hironaka-permissible centers in such a way that every x ′ ∈ π -1 (x) has either m(x ′ ) < p, or (m(x ′ ) = p and x ′ satisfies condition (E)). This is proved in Corollary 4.19 below. Assumption (G) is not required here and we prove a more general version for arbitrary multiplicity in Proposition 4.15.

Lemma 4.14. Let S, h ∈ S[X] (2.1) and η : X → SpecS be given. Assume that dimS = 3 and that h is reduced. There exists a composition of Hironakapermissible blowing ups (2.17) w.r.t. E = ∅:

X π ←-X ′ ↓ ↓ SpecS σ ←-S ′ such that π(Sing m X ′ ) ⊆ η -1 (m S ).
Proof. This statement means that there exists a diagram

X =: X 0 π 0 ←-X 1 π 1 ←-• • • π n-1 ←-X n =: X ′ ↓ ↓ ↓ SpecS =: S 0 σ 0 ←-S 1 σ 1 ←-• • • σ n-1 ←-S n =: S ′ (4.19)
where each morphism π i , 0 ≤ i ≤ n -1, is the blowing up along a Hironakapermissible center Y i ⊂ X i w.r.t. the reduced exceptional divisor E i of π (i) : X i → X . It can be assumed that dim(Sing m X ) ≥ 1.

Let y i ∈ X i denote the generic point of such a Hironaka-permissible center Y i ⊂ X i w.r.t. E i . We define:

∆ i := {y ∈ Sing m X i : dim O X i ,y = dim O X ,π (i) (y) = 1}, δ i := max{δ(y), y ∈ ∆ i }, N i := ♯{y ∈ ∆ i : δ(y) = δ i }. Let i ≥ 0. We claim that    (δ i+1 , N i+1 ) = (δ i , N i ) if dim O X ,π (i) (y i ) ≥ 2 (δ i+1 , N i+1 ) < (δ i , N i ) if dim O X ,π (i) (y i ) = 1 . (4.20)
Namely, this is an obvious consequence of the definition

if dim O X ,π (i) (y i ) ≥ 2. If dim O X ,π (i) (y i ) = 1, let y ∈ X i+1 with π i (y) = y i . We have (m(y), δ(y)) ≤ (m(y i ), δ(y i ) -1)
by Proposition 2.18 applied for n = 1 and the claim follows Pick y ∈ ∆ i with δ(y) = δ i and denote Y := {y} ⊂ X i . By Proposition 4.3, there exists a composition of blowing ups X i ′ → X i with regular centers contained in the successive strict transforms of Y such that

η i ′ (Y ′ ) has normal crossings with E i ′ , where Y ′ denotes the strict transform of Y in X i ′ . Then Y ′ itself and each blowing up center in X i ′ → X i are Hironaka-permissible w.r.t. E i ′ because m(y) = m.
We have (4.20). Since ∆ i is a finite set and δ i ∈ 1 m N, there exists an index i 1 > i such that ∆ i 1 = ∅ and this is preserved by further Hironaka-permissible blowing ups w.r.t. E = ∅.

(δ i ′ , N i ′ ) = (δ i , N i ) by (4.

20). Taking as blowing up center

Y i ′ := Y ′ also gives (δ i ′ +1 , N i ′ +1 ) < (δ i , N i ) by
Since ∆ i 1 = ∅, we are done unless π (i 1 ) (Sing m X i 1 ) = C, where C has pure dimension one. Let C ⊂ SpecS be an irreducible component of η(C) and s be its generic point. Note that the stalk (X i ) s at s of the S-scheme X i is embedded in the regular scheme of dimension three SpecS s [X] for i = 0 and in an iterated blowing up along regular centers of the former for i ≥ 1. By Proposition 4.3, there exists a composition of Hironaka-permissible blowing ups X ′ s → (X i 1 ) s w.r.t. (E i 1 ) s such that Sing m X ′ s = ∅. Let Y s ⊆ (X i 1 ) s be a Hironaka-permissible center and Y ⊆ X i 1 be its Zariski closure, so in particular we have Y ⊆ Sing m X i 1 . Since ∆ i 1 = ∅, Y is either (1) a curve mapping onto C, or (2) a surface mapping to some irreducible component of E i 1 .

In situation [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF], there exists a composition of blowing ups along closed points

X i ′ 1 → X i 1 such that η i ′ 1 (Y ′ ) has normal crossings with E i ′ 1 , where Y ′ denotes the strict transform of Y in X i ′ 1 . In situation (2), Y itself is Hironaka-permissible w.r.t. E i 1 and we let i ′ 1 := i 1 .
In both situations, we may blow up X i ′ 1 along Y ′ and iterate: this produces an index i 2 ≥ i 1 and a composition of Hironaka-permissible blowing ups

X i 2 → X i 1 w.r.t. E i 1 such that η -1 (s) ∩ π (i 2 ) (Sing m X i 2 ) = ∅.
Applying this construction to the finitely many irreducible components of η(C) proves the lemma.

Proposition 4.15. Let X ′ satisfy the conclusion of Lemma 4.14 and E ′ ⊂ S ′ be the reduced exceptional divisor of σ. Let D ⊂ S ′ be a reduced divisor.

There exists a composition of Hironaka-permissible blowing ups (2.17) w.r.t. E ′ :

X ′ π ′ ←-X ′′ ↓ ↓ S ′ σ ′ ←-S ′′ such that the strict transform D ′′ of D is disjoint from η ′′ (Sing m X ′′ ), where η ′′ : (X ′′ , x ′′ ) → S ′′ is the local projection at x ′′ ∈ Sing m X ′′ .
Proof. We take S ′ = SpecS. The problem is to find a sequence (4.19) which monomializes P := I(D) ⊂ S, i.e. such that P n := P O Sn is a monomial with components at normal crossings with E n . Let us write P i := H i Q i where H i is a monomial whose components are components of E i . At the beginning, H = H 0 = 1. The strategy is to get

P n = H n , Q n = 1 at the end.
We consider the idealistic exponents (see [START_REF] Hironaka | Idealistic exponents of singularity[END_REF] p. 54) (h, m) and (Q, b) living in SpecS [Z], where b =ord m S (Q). We make a descending induction on b: the case b = 0 means that we get the conclusion of 4.15. Each pair of blowing ups π i , σ i is locally centered at some Y i and η(Y i ) respectively, and is Hironaka-permissible for h (resp. Q i ) w.r.t. E i .

Let

P i+1 =: H i+1 Q i+1 where Q i+1 is the strict transform of Q i . This means that (Q i+1 , b) is the transform of (Q i , b). When ord x i+1 (Q i+1 ) < b,
we have strictly improved and we go on with the new idealistic exponent (Q i+1 , b ′ ), with b ′ :=ord x i+1 (Q i+1 ). To define a sequence of σ i is a consequence of [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF] Theorem 0.3 (Canonical embedded resolution with boundary), the problem is the sequence of π i , i.e. to define the pair (σ i , π i ).

Notation 4.16. To avoid cumbersome notations, from now on, x i , S i , X i ,etc. i are denoted by x, S, X ,etc. and x i+1 , S i+1 , X i+1 ,etc. i+1 by x ′ , S ′ , X ′ ,etc. ′ . Let us define Vdir(x, D) as Vdir(h)+Vdir(Q). This is a vector space of codimension τ (x, D) in the Zariski's tangent space Spec(gr

(m S ,Z) (S[Z])) of Spec(S[Z]) at x. Of course, τ (x, D) ≥ 2. We denote by IDir(x, D) ⊂gr (m S ,Z) (S[Z]) = k(x)[Z, U 1 , U 2 , U 3 ] the ideal of Vdir(x, D).
Lemma 4.17. Let π be the blowing up along Y which is permissible for both (h, m) and

(Q, b). Let x ′ ∈ π -1 (x) be such that m(x ′ ) = m(x) = m and ord x ′ Q ′ = b. Then x ′ is on Proj(k(x)[Z, U 1 , U 2 , U 3 ]/IDir(x, D)).
In particular, x ′ is on the strict transform of div(Z).

Proof. By Proposition 2.55 and Remark 2.56, we have Dir(F ) = Max(F ) except if p = 2 and

F = λ(Z 2 + λ 2 U 2 1 + λ 1 U 2 2 + λ 1 λ 2 U 2 3 ) α , [k 2 (λ 1 , λ 2 ) : k 2 ] = 4 (4.21)
up to a linear change of variables, λ = 0, α ≥ 1. Then π is the blowing up centered at x. Since m(x ′ ) = m(x), we have

x ′ := V (U 2 1 + λ 1 U 2 3 , U 2 2 + λ 2 U 2 3 , Z 2 + λ 1 λ 2 U 2 3 ) on π ′ -1 (x) = Proj(k[Z, U 1 , U 2 , U 3 ]/(F )).
Since ord x ′ Q ′ = b, the initial of Q cannot satisfy (4.21) (only the last three variables occur). Therefore

x ′ ∈ Proj(k(x)[Z, U 1 , U 2 , U 3 ]/IDir(h)) ∩ Proj(k(x)[Z, U 1 , U 2 , U 3 ]/IDir(Q)) = Proj(k(x)[Z, U 1 , U 2 , U 3 ]/IDir(x, D)). (4.22)
Let us come back to the proof of Proposition 4.15. We discuss according to the value of τ (x, D).

When τ (x, D) = 4, the blowing-up centered at x makes b strictly drop. When τ (x, D) = 2 or 3, then, if we blow up along x, then τ (x ′ , D ′ ) ≥ τ (x, D). In case τ (x, D) = 3, we make only blowing ups at closed points. Either for some n, (m(x n ),ord xn (Q n )) < lex (m, b), then we stop at this n; or we have equality for n ≥ 0. Then, τ (x n , D n ) = 3, n ≥ 0, by an usual argument, the x n are all on the strict transform of a curve C n which, for n >> 0 is permissible for both (h, m) and (Q, b) and η(C n ) is transverse to E n . Then at step n in (4. [START_REF] Cossart | Desingularization of embedded excellent surfaces[END_REF], we blow up along C n . By Lemma 4.17, (m(x n+1 ),ord

x n+1 (Q n+1 )) < lex (m, b). When τ (x, D) = 2, we can choose Z, u 3 such that Vdir(Q) =< U 3 >, Vdir(h) ≡< Z > mod(U 3 ). Remark 4.18. If there is a component Y of dimension 2 in Sing(h, m) ∩ Sing(Q, b),
then we can choose the parameters so that I(Y ) = (Z, u 3 ). Then Q ∈ (z, u 3 ) b , i.e. Q = u b 3 , up to multiplication by an invertible. Then, if Y has normal crossing with E, we blow up along Y : π is the blowing up along Y and σ is the identity. In fact in S, we just add η(Y ) = div(u 3 ) to E and we get b = 0.

We also note that (h, m) ∩ (Q, b) = (hQ, m + b). In other words, we have

Sing(h, m) ∩ Sing(Q, b) = Sing(hQ, m + b)
and permissible centers are the same for (hQ, m + b) and for (h, m) ∩ (Q, b).

Then we apply those techniques from [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF] 10, 11, 12. More precisely, if for some n 0 the number b just strictly drops, we call "old components" the components of E n 0 at x n 0 which are components of H and, for n ≥ n 0 , at x n , n ≥ n 0 with b(x n ) = b(x n 0 ), the strict transforms of this old components. The first step is to reach the case where x n is not on the strict transform of this old components: the invariant is (m, b, o(x)) where o(x) is the number of these old components. In the language of idealistic exponents, we desingularize (hQQ O , mbo(x)) where Q O is the equation of the reduced divisor whose components are the old ones. Then we look at the directrix of hQQ O . When its codimension denoted by τ (hQQ O ) is 3 or 4, we play the same game that above with τ (x, D) = 3 or 4. We reach the case where τ (hQQ O ) = 2. This means that either Q O = 1 (no old component) or there is one old component which is tangent to Q.

Then we look at the characteristic polyhedron ∆(hQQ 0 , z, u 3 , u 1 , u 2 ) as in [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF] Section 7.

• Case ∆(hQQ 0 , z, u 3 , u 1 , u 2 ) = ∅. This is equivalent to hQQ 0 ∈ (z, u 3 ) mbo(x) , i.e. this is equivalent to dim(Sing(hQQ

O , mbo(x)) = 2. So QQ O = u mbo(x) 3 , call Y := V (z, u 3 ), in fact, at step n 0 , as b(x 0 ) = b(x), Q was a b(x 0 ) power and, if at x there is one old component, it is a factor of Q: this is impossible, therefore o(x) = 0.
So, at x, E is a union of components which are exceptional divisors of the blowing ups σ n , n ≥ n 0 . By [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF] Theorem 8.3, they are transverse to u 3 : Y is permissible for (hQQ O , mbo(x)) and transverse to E. We apply the first statement of Remark 4.18.

• Case where dim(Sing(hQQ O , mbo(x)) ≤ 1. Then, we apply [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF] Theorem 5.28 which gives the result if chark(x) ≥ 3. This hypothesis p = 2 is used just to get Dir(F ) = Max(F ) at each step, but we showed above in Lemma 4.17, that the only case where Dir(F ) = Max(F ) stops after blowing up the closed point x.

Corollary 4.19. Assume that charS/m S = p > 0 and (S, h) satisfies condition (G). There exists a composition of Hironaka-permissible blowing ups (2.17) w.r.t. E = ∅: X

π ′′ ←-X ′′ ↓ ↓ SpecS σ ′′ ←-S ′′
such that η ′′ (Sing p X ′′ ) ⊆ σ ′′ -1 (m S ) and condition (E) holds at every s ′ ∈ η ′′ (Sing p X ′′ ), where η ′′ : X ′′ → S ′′ is the projection.

Proof. This is a direct application of Lemma 4.14 in the purely inseparable case ((iii) of condition (G)). If η is separable and charS = p, we apply Proposition 4.15 to the strict transform in S ′ of D := div(Disc X (h)) and the conclusion follows.

Assume that charS = 0. Let D ′ 1 be the strict transform of div(pDisc X (h)) in S ′ and D ′ 2 be the union of those components of E ′ of characteristic zero. We apply Proposition 4.15 to D := D ′ 1 ∪D ′ 2 . Let E ′′ be the exceptional divisor of σ ′′ and s ′ ∈ η ′′ (Sing p X ′′ ). Since all blowing up centers of σ ′ are Hironakapermissible w.r.t. E ′ , they map to η(x) and are thus of characteristic p = charS/m S . We deduce from Proposition 4.15 that any irreducible component of E ′′ passing through s ′ has characteristic p and that (ii) of Definition 2.32 holds.

5 Projection number κ(x) ∈ {1, 2, 3, 4}, Projection Theorem.

Let (S, h, E) satisfy assumptions (G) and (E). In this section, we perform induction on the dimension dimS[Z] = 4 of the ambient space of X , vid. introduction. This step is for now far out of reach in higher dimensions and little more than definitions could be stated. We reduce Theorem 1.5 to Theorem 5.5 below (Corollary 5.6) which is proved in the next sections.

Projection number κ(x).

For y ∈ X , s := η(y) ∈ SpecS, the assignment κ(y) ≥ 2 has so far been used to express κ(y) = 1; we now distinguish κ(y) = 2, 3, 4 when (ω(y) > 0, κ(y) ≥ 2). This completes our definition of the complexity function (2.76):

ι : X → {1, . . . , p} × N × {1, . . . , 4}, y → (m(y), ω(y), κ(y)).

The projection number κ(y) expresses the transverseness of Vdir(y) w.r.t. E s . We claim no further invariance property w.r.t. regular local base change than that of Theorem 2.74 when κ(y) ≥ 2.

Since our assumptions (G) and (E) are stable when changing (S, h, E) to (S s , h s , E s ) (Notation 2.23), we may assume that s = m S . The following definition is for codimension three, the remark afterwards for codimension two. One has ω(y) = ǫ(y) = 0 in codimension one. We denote

E = div(u 1 • • • u e ) as before. Definition 5.1. (Projection Number). Assume that m(x) = p, ω(x) > 0 and κ(x) ≥ 2, where η -1 (m S ) = {x}. We let κ(x) := 4 if Vdir(x) ⊆< U 1 , . . . , U e > .
(5.1)

Assume now that κ(x) = 4. We let κ(x) := 3 if (ω(x) = ǫ(x) -1 and one of the following conditions is satisfied):

(1) E = div(u 1 ) and there exist well adapted coordinates (u 1 , u 2 , u 3 ; Z) at

x such that

Vdir(x) ⊆< U 1 , U 3 > and H -1 ∂F p,Z ∂U 2 ⊆< U ω(x) 1
>;

(2) E = div(u 1 u 2 ).

Finally, we let κ(x) := 2 if κ(x) = 3, 4.

Remark 5.2. When dimO X ,y = 2, m(y) = p, ω(y) > 0 and κ(y) ≥ 2, we define: if

E s = div(u 1 u 2 ), let κ(y) := 4; if E s = div(u 1 ), let: κ(y) :=    2 if ω(y) = ǫ(y) and Vdir(y) < U 1 > 3 if ω(y) = ǫ(y) -1 4 if ω(y) = ǫ(
in m S h =              Z p + λU pd 1 1 U ω(x) 3 , E = div(u 1 ), ω(x) ≡ 0 mod p Z p + λU pd 1 1 U pd 2 2 U ω(x) 3 , E = div(u 1 u 2 ), ω(x) ≡ 0 mod p Z p + λU pd 1 1 U 2 U ω(x) 3
, E = div(u 1 ), ω(x) ≡ 0 mod p (5.2) which are three kinds of the special case κ(x) = 2(*) (Definition 7.10).

Projection Theorem.

We now turn to the statement of the Projection Theorem. We assume that ω(x) > 0, so (X , x) is (analytically) irreducible by Theorem 2.36. Let µ be a valuation of L = k(X ) centered at x. We will consider finite sequences of local blowing ups along µ:

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ) (5.3)
with Hironaka-permissible centers Y i ⊂ (X i , x i ), where x i , 0 ≤ i ≤ r, denotes the center of µ. We require that our assumptions (G) and (E) be preserved by such blowing ups and that

(m(x i ), ω(x i )) ≤ (m(x i-1 ), ω(x i-1 )), 1 ≤ i ≤ r.
This certainly holds when the blowing up centers are permissible of the first or second kind by Propositions 2.28, 2.34 and Theorem 3.13. Another example is blowing up along codimension one centers of the form V (Z, u j ) with

d j ≤ 1, 1 ≤ j ≤ e.
In chapter 8, we will use another kind of Hironakapermissible blowing up with the same property. We recall that all permissibility conditions (Definitions 2.20, 3.1 and 3.5) always refer to the reduced total transform E i of E in S i , where there are projections

η i : (X i , x i ) -→ SpecS i , 0 ≤ i ≤ r.
Similarly, ω(x i ), ǫ(x i ), κ(x i ) are always computed w.r.t. E i .

We emphasize that we do not require any particular behavior about the numbers κ(x i ) along the process (5.3). Our goal is to eventually achieve κ(x r ) < κ(x) and we may have κ(x i ) > κ(x) for some i, 1 ≤ i < r.

Our strategy consists in looking for expansions of in m S (h) in each case κ(x) = 2, 3, 4 which are stable by permissible blowing ups. These stable expansions are denoted respectively (*) (Definition 7.10), (**) and (T**) (Definition 8.1). Our first goal is to reach these conditions. Achieving this first step involves sequences of blowing ups (5.3) where we may have κ(x i ) > κ(x). See for example Proposition 7.11 which relies on Lemma 7.1 and Propositions 7.6 and 7.7. Definition 5.4. Assume that m(x) = p and ω(x) > 0. Given any finite sequence (5.3), we say that x r is very near x if ι(x r ) ≥ ι(x).

Let a ∈ {1, . . . , 4}. We say that x is resolved for (p, ω(x), a) (resp. resolved for m(x) = p) if for every valuation µ of L = k(X ) centered at x, there exists a finite and independent sequence (5.3) (cf. Definition 2.77) such that ι(x r ) < (p, ω(x), a) (resp. m(x r ) < p). We simply say that x is good if x is resolved for ι(x).

The following Projection Theorem is proved in the next sections: Corollary 6.4, Theorem 7.26, Theorem 9.6, ibid., for κ(x) = 1, 2, 3, 4 respectively. Theorem 5.5. (Projection Theorem). Assume that (S, h, E) satisfies assumption (G) and (E), with m(x) = p and ω(x) > 0.

For every valuation µ of L = k(X ) centered at x, there exists a finite and independent composition of local Hironaka-permissible blowing ups (5.3) such that ι(x r ) < ι(x), i.e. x is good.

6 Maximal contact, resolution of κ(x) = 1.

We assume in the whole section that (S, h, E) satisfies conditions (G) and (E). We consider here any refinement C of the function x → (m(x), ω(x)) on X .

Fix an irreducible component div(u 1 ) ⊆ E. Let µ be a valuation of L = k(X ) centered at x. We consider in this chapter finite sequences (5.3) of local blowing ups along µ:

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ), (6.1)
with permissible centers of the first kind Y i ⊂ (X i , x i ), where x i , 0 ≤ i ≤ r, denotes the center of µ. It is furthermore assumed that (1) η i (Y i ) belongs to the strict transform of div(u 1 ) in SpecS i , where

η i : (X i , x i ) -→ SpecS i is the projection, vid. Proposition 2.22, and (2)
C is not increasing along (6.1), i.e. C(x i ) ≤ C(x i-1 ), 1 ≤ i ≤ r. Definition 6.1. We say that div(u 1 ) ⊆ E ⊂ X has "maximal contact" (resp. "weak maximal contact") for some refinement C if for every µ, any sequence (6.1) (resp. the quadratic sequence (6.1) with Y i := {x i }) satisfies the following:

C(x r ) = C(x) =⇒ x r maps to the strict transform of div(u 1 ). (

Remark 6.2. Take C = ι, where κ(x) = 1. Then div(u 1 ) ⊆ E has maximal contact for C if U 1 divides H -1 G p , with notations as in Definition 2.68. This follows from Theorem 3.13.

The purpose of this section is to prove Theorem 6.3 below: the value C(x) of any such refinement can be lowered by permissible blowing ups of the first kind. A direct application proves Theorem 5.5 for κ(x) = 1. Further applications are given in chapter 8. The proof of this theorem uses a secondary invariant γ(x) ∈ N which is defined and studied afterwards, viz. (6.7) and (6.9). Theorem 6.3. Assume that div(u 1 ) has maximal contact for C. Let µ be a valuation of L = k(X ) centered at x, where m(x) = p and ω(x) > 0. There exists a finite and independent composition of local permissible blowing ups of the first kind:

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ), (6.3
)

where x i ∈ X i is the center of µ, such that C(x r ) < C(x) or x r is resolved for m(x) = p.
Proof. By Proposition 3.26, the set

Ω + (X ) := {y ∈ X : (m(y), ω(y)) > (p, 0)} ⊆ X
is Zariski closed and of dimension at most one. By performing the quadratic sequence (6.1), it can be assumed that there exist well adapted coordinates (u 1 , u 2 , u 3 ; Z) at x such that any one dimensional irreducible component Y of Ω + (X ), with η(Y) contained in div(u 1 ) either: (a) maps to an intersection of components of E, i.e.

η(Y) = V (Z, u 1 , u j ), div(u j ) ⊆ E, j ≥ 2, or (b) η(Y) = V (Z, u 1 , u 3 ), E ⊆ div(u 1 u 2 ).
Furthermore, there exists at most one Y satisfying (b) and such Y is permissible of the first kind by Proposition 3.17 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF]. Let X ′ → (X , x) be the blowing up along such Y. Replacing (X , x) by (X ′ , x ′ ), where x ′ is the center of µ, we may therefore assume that any one-dimensional irreducible component Y of Ω + (X ), with η(Y) contained in div(u 1 ), satisfies (a) above.

Consider now the quadratic sequence (6.1) and apply Proposition 6.16 below. If alternative (ii) of that proposition holds, the theorem follows from Proposition 3.17 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF], since the conclusion of Proposition 3.17 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] does not hold by the above preparation of Ω + (X ). Assume then that alternative (i) of Proposition 6.16 holds. Then the conclusion follows from Proposition 6.17 below. Corollary 6.4. Projection Theorem 5.5 holds when κ(x) = 1.

The arguments are quite similar to [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 4 pages 1957 and following and we sketch the argument below. This section may serve as an introduction to the more involved material in the next chapter. Notation 6.5. We assume that div(u 1 ) has maximal contact or weak maximal contact. Then, we may also assume that div(u 1 u 2 ) ⊆ E. Indeed, after the first blowing up, E ′ will contain at least two components: the strict transform of div(u 1 ) and the new exceptional component. Cases 1 and 2: ǫ(x) = ω(x) and (E = div(u 1 u 2 ) or E = div(u 1 u 2 u 3 ) respectively). Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates. Consider the characteristic polyhedron

∆ S (h; u 1 , u 2 , u 3 ; Z) ⊂ R 3 ≥0
in the affine space with origin v 0 := (d 1 + ω(x)/p, d 2 , d 3 ) with the convention d 3 = 0 when div(u 3 ) ⊆ E. Perform the stereographic projection p ′ 2 from v 0 on the plane x 1 = 0, followed by the homothety of center (0, 0) and ratio p ω(x) . Let p 2 be the resulting map. Analytically, we have:

p 2 : (x 1 , x 2 , x 3 ) → (y 2 , y 3 ) := 1 ω(x) p -(x 1 -d 1 ) (x 2 -d 2 , x 3 -d 3 ). (6.4)
We denote for simplicity

∆ 2 (x) := p 2 (∆(h; u 1 , u 2 , u 3 ; Z) ∩ {0 ≤ x 1 -d 1 < ω(x)/p}). (6.5)
There are associated invariants:

               A j (x) := inf {y j | (y 2 , y 3 ) ∈ ∆ 2 (x)} B(x) := inf {y 2 + y 3 | (y 2 , y 3 ) ∈ ∆ 2 (x)} C(x) := B(x) -A 2 (x) -A 3 (x) ≥ 0 in case (2) C(x) := B(x) -A 2 (x) ≥ 0 in case (1) β(x) := inf {y 3 | (A 2 (x), y 3 ) ∈ ∆ 2 (x)} β 2 (x) := sup {y 3 | (y 2 , y 3 ) ∈ ∆ 2 (x), y 2 + y 3 = B(x)} . (6.6)
The main secondary invariant is:

γ(x) :=    max{1, ⌈β(x)⌉} if E = div(u 1 u 2 ) 1 + ⌊C(x)⌋ if E = div(u 1 u 2 u 3 ) . (6.7)
Note that ∆ 2 (x) = ∅: this follows from (6.4) and the definition of d 1 . Therefore A 2 (x), A 3 (x), B(x) < +∞.

It is easily seen that ∆ 2 (x) ⊆ R 2 ≥0 is a polygon. Since all vertices of ∆ S (h; u 1 , u 2 , u 3 ; Z)-(d 1 , d 2 , d 3 ) have module at least ǫ(x) p , we have B(x) ≥ 1.

Case 3: ǫ(x) = 1 + ω(x), E = div(u 1 u 2 ). The definition is the same as in cases 1 and 2 except that v 0 is replaced by v ′ 0 := (d 1 + ω(x)/p, d 2 , 1/p). Analytically, we have:

p 2 : (x 1 , x 2 , x 3 ) → (y 2 , y 3 ) := 1 ω(x) p -(x 1 -d 1 ) (x 2 -d 2 , x 3 -1/p). (6.8)
Note that the image set ∆ 2 (x) defined by (6.5) may contain points with negative third coordinate. The invariants A 2 (x), B(x), C(x) := B(x) -A 2 (x) and β(x) are defined as in cases 1 and 2. We let:

γ(x) := max{1 + ⌊β(x)⌋, 1}. (6.9)

These definitions depend in principle on (u 1 , u 2 , u 3 ), but certainly not on Z such that (u 1 , u 2 , u 3 ; Z) are well adapted coordinates. Indeed, the above definitions are given in terms of ∆(h; u 1 , u 2 , u 3 ; Z). It can be proved that the numbers A j (x), B(x), C(x), β(x) and γ(x) are actually independent of (u 1 , u 2 , u 3 ; Z) once the numbering of the components of E is fixed. We skip this fact here and refer to the next chapter (Theorem 7.18 and Definition 7.19 in particular) for similar issues. Remark 6.6. The numbers B(x), A j (x) can be computed directly from the equation h.

In cases 1-2, let (a, b) be positive real numbers such that

a(d 1 + ω(x) p ) + b(d 2 + d 3 ) = 1
with the convention d 3 = 0 when div(u 3 ) ⊆ E. Define a monomial valuation v (a,b,b) on S[Z] by setting weights:

v (a,b,b) (u 1 ) = a, v (a,b,b) (u 2 ) = v (a,b,b) (u 3 ) = b, v (a,b,b) (Z) = 1. Then B(x) = sup{ a b |v (a,b,b) (h) = p}.
The pair (a, b) giving the sup above is said to "define B(x)" (viz. We denote:

H B := in v (a,b,b) (h) = Z p + 1≤i≤p Φ i Z p-i , Φ i ∈ k(x)[U 1 , U 2 , U 3 ], (6.11)
where (a, b) "defines B(x)". By Theorem 2.36, we have

Φ i = 0, 1 ≤ i ≤ p -2 and -Φ p-1 = G p-1
where G is a constant times a monomial in U 1 , . . . , U e . We expand the corresponding initial form as in (6.11) and let

U -pd 1 1 U -pd 2 2 U -pd 3 3 Φ p = λU ω(x) 1 + ω(x) i=1 U ω(x)-i 1 F i (U 2 , U 3 ), λ ∈ k(x), (6.12)
where Indeed, b = 1, a > 1 gives the same contradiction as above. More generally, let σ 2 be a compact face of ∆ 2 (x). The topological closure of the set σ := ∆ S (h; u 1 , u 2 , u 3 ; Z) ∩ p -1 2 (σ 2 ) is a compact face of ∆ S (h; u 1 , u 2 , u 3 ; Z) defined by a weight vector α := α σ 2 . The corresponding initial form polynomial is written

F i ∈ k(x)[U 2 , U 3 ]
H α :=Z p + 1≤i≤p Φ i,α Z p-i , Φ i,α ∈ gr α (S), (6.14)
In case 3, there exists a unique compact face σ ⊂ ∆ S (h; u 1 , u 2 , u 3 ; Z) whose image by p 2 is the face y 2 + y 3 = B(x), maximal for this property. For B(x) = 1,

σ in := {x ∈ R 3 ≥0 : x 1 + x 2 + x 3 = δ(x)}
obviously has this property. For B(x) > 1, we expand the corresponding initial form as in (6.11) and let

U -pd 1 1 U -pd 2 2 Φ p = U ω(x) 1 (λ 3 U 3 + λ 2 U 2 ) + ω(x) i=1 U ω(x)-i 1 F i (U 2 , U 3 ), (6.15) with λ 2 , λ 3 ∈ k(x), F i ∈ k(x)[U 2 , U 3 ] homogeneous of degree 1 + iB(x).
In cases 1-2-3, let (a, b) be positive real numbers such that

a(d 1 + ω(x) p ) + bd 2 = 1.
We have similarly:

A 2 (x) = sup{ a b |v (a,b,0) (h) = p},
this suitable pair (a, b) is also said to "define A 2 (x)". We denote:

H 2 :=in v (a,b,0) (h) = Z p + 1≤i≤p φ i Z p-i , φ i ∈ S (u 1 , u 2 ) [U 1 , U 2 ], (6.16)
where (a, b) "defines A 2 (x)"( [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Theorem I.4, valuation µ 1 on page 1962). We expand the φ i , 1 ≤ i ≤ p:

φ i = ω(x) j=0 U j 1 U b(i,j) 2 φ i,j , b(i, j) = i b -jA 2 (x), φ i,j ∈ S (u 1 , u 2 )
,

where 1 b = A 2 (x)(d 1 + ω(x) p ) + d 2 .
All proofs are based on the following elementary lemma: Lemma 6.7. Let (R, m, k) be a regular local ring of dimension two, m = (v 2 , v 3 ), chark = p > 0. Let f ∈ R with initial form

in m f = V a 2 2 V a 3 3 F (V 2 , V 3 ) ∈ G(m), in m f ∈ G(m) p .
Let furthermore P (t) ∈ R[t] be monic of degree d ≥ 1 with irreducible residue

P (t) ∈ k[t], R ′ := R v 3 v 2 (v 2 ,v) , v := P v 3 v 2
and for every α ∈ R ′ , α := α mod(v 2 ) ∈ R ′ v 2 R ′ . We define:

a ′ := max g ′ ∈R ′ {ord v 1 (f -g ′ p )}, e ′ := max g ′ ∈R ′ {ord ṽ ( v -a ′ 2 (f -g ′ p )) : ord v 2 (f -g ′ p )) = a ′ }.
The following hold:

and Φ p-1 = 0 implies

Φ p-1 = λU (p-1)(d 1 + ω(x) p ) 1 U (p-1)d 2 2 U (p-1)d 3 3 , λ ∈ k(x) * or β(x ′ ) = 0 if x ′ is in case 1, β(x ′ ) < 0 if x ′ is in case 3. (6.19)
If moreover x is in case 1 and β(x) > 0, we have

   β(x ′ ) ≤ β(x) if x ′ is in case 1 β(x ′ ) < β(x) if x ′ is in case 3 . ( 6.20)
(a, b) defined in (6.11). Clearly v (a ′ ,b ′ ,0) does not depend on the choice of v.

An easy computation gives with the notations of (6.11):

H ′ 2 = Z ′ p + U 2 -p+1 Φ p-1 (U ′ 1 U 2 , U 2 , ū3 U 2 )Z ′ + U 2 -p Φ p (U ′ 1 U 2 , U 2 , ū3 U 2 ) when a > b H ′ 2 = Z ′ p + U 2 -p+1 Φ p-1 (ū 1 U 2 , U 2 , ū3 U 2 )Z ′ + U 2 -p Φ p (ū 1 U 2 , U 2 , ū3 U 2 ) when a = b (6.22) Φ i (U ′ 1 U 2 , U 2 , ū3 U 2 ) ∈ k(x)[U ′ 1 , U 2 , ū3 ] = S ′ (u ′ 1 ,u 2 ) [U ′ 1 , U 2 ] when a > b, resp. Φ i (ū 1 U 2 , U 2 , ū3 U 2 ) ∈ k(x)[ū 1 , U 2 , ū3 ] = S ′ (u 2 ) [U 2 ] when a = b. As H ′ 2 = Z ′ p , a ′ x 1 + b ′ x 2 + 0x 3 = 1 is the equation of a face of ∆(h ′ ; u ′ 1 , u 2 , v; Z ′
). This face cannot be solved by translation on Z ′ as H B of (6.11) is not a p-th power: an eventual translation minimizing the polyhedron Z ′ ← Z ′ + θ, θ ∈ S ′ , will verify v (a ′ ,b ′ ,0) (θ) ≥ 1. Furthermore, the initial form polynomial H 2 in (6.16) at x ′ is H ′ 2 and has A 2 (x ′ ) = B(x) -1: this gives the equality in (6.17). When Φ p-1 = 0, it is a monomial in U 1 , U 2 (case 1) or in U 1 , U 2 , U 3 (case 2) by Theorem 2.36. Let:

Φ p-1 = λU (p-1)d 1 +d 1 U (p-1)d 2 +e 2 U (p-1)d 3 +f 3 , λ ∈ k(x), d + (p -1)d 1 , e + (p -1)d 2 , f + (p -1)d 3 ∈ N when λ = 0, d, e, f ∈ Q ≥0 , d ≤ (p -1) ω(x) p . (6.23)
When Φ p-1 = 0 and d < (p -1) ω(x) p , then e + f = i 0 B(x) with i 0 = (p -1) ω(x) pd. In this case, the coefficient of Z ′ in H ′ 2 is:

λU ′ 1 (p-1)d 1 U i 0 (B(x)-1) 2 U ′ 1 d × invertible when a > b, λ ū1 (p-1)d 1 ū1 d × invertible when a = b. (6.24)
As this coefficient is invariant by an eventual translation

Z ′ ← Z ′ + θ with v (a ′ ,b ′ ,0) (θ) ≥ 1.
we get β(x ′ ) = 0 when x ′ is in case 1, and β(x ′ ) = -1 i 0 < 0 when x ′ is in case 3. This gives in this case all the equalities and inequalities in (6.17) (6.18)(6.20) and [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF].

From now on, we suppose

Φ p-1 = 0 or Φ p-1 = λU (p-1)(d 1 + ω(x) p ) 1 U (p-1)d 2 2 U (p-1)d 3 3 . (6.25) Let Φ p = U pd 1 1 U pd 2 2 U pd 3 3 ω(x) i=0 U ω(x)-i 1 F i (U 2 , U 3 ), deg(F i ) = iB(x) when F i = 0. Let i 0 :=sup{i|F i = 0} > 0 and w := (d 1 + ω(x)-i 0 p , d 1 + d 2 + d 3 -1 + i 0 B(x)
p , w 3 ) be the vertex of smallest abscissa of the face of equation a

′ x 1 + b ′ x 2 + 0x 3 = 1 of ∆(h ′ ; u ′ 1 , u 2 , v; Z ′ ). This vertex w is defined by the monomial U p(d 1 +d 2 +d 3 -1)+i 0 B(x) 2 U ′ 1 pd 1 +ω(x)-i 0 F i 0 (1, ū3 ): we have w 3 = 1 p ord v (F i 0 (1, ū3
)). In the case where (6.26) this vertex w is not solvable and we get

pd 1 + ω(x) -i 0 = 0 mod p or p(d 1 + d 2 + d 3 -1) + i 0 B(x) = 0 mod p,
β(x ′ ) = 1 i 0 ord v (F i 0 (1, ū3 )) when x ′ is in case 1 , β(x ′ ) = 1 i 0 (ord v (F i 0 (1, ū3 )) -1) when x ′ is in case 3 , (6.27)
which gives (2) in this case. When (6.26) is not true, a translation

Z ′ ← Z ′ + θ, with θ = γu ′ 1 d 1 + ω(x)-i 0 p u d 1 +d 2 +d 3 -1+ i 0 B(x) p 2 v x 3 , γ ∈ S ′ , γ invertible,
may solve w. By (6.25), the eventual contribution of the coefficient of Z ′ to the term of degree 0 in Z ′ of h ′ will be divisible by u ′ (p-1)(d 1 + ω(x) p ) 1

u ′ d 1 + ω(x)-i 0 p 1 , as (p-1)(d 1 + ω(x) p )+d 1 + ω(x) -i 0 p = pd 1 +ω(x)- i 0 p > pd 1 +ω(x)-i 0 = pw 1 ,
the eventual translation translation may only spoil vertices of this face with a bigger abscissa, when a > b: it will just add a p-th power to U

p(d 1 +d 2 +d 3 -1)+i 0 B(x) 2 U ′ 1 ω(x)-i 0 F i 0 (1, ū3 ), resp. U p(d 1 +d 2 +d 3 -1)+i 0 B(x) 2 ū1 ω(x)-i 0 F i 0 (1, ū3 ) when a = b. Let F i 0 (U 2 , U 3 ) =: U a 2 2 U a 3 3 F (U 2 , U 3 ) with a 2 maximal, a 3 = 0 if x is in case 1, a 3 maximal if x is in case 2. We have the inequality: β(x ′ ) ≤ord v (F i 0 (1, ū3 )+ θ p )/i 0 , with strict inequality when x ′ is in case 3. When x is in case 1 or 2, β(x) ≥ deg(F ) i 0 , C(x) ≥ deg(F ) i 0 .
The inequalities (6.18) and (6.20) follow from the fact that

β(x ′ ) ≤ 1 i 0 ord v(F i 0 (1, ū3 ) + θ p )
and Lemma 6.7 (2) which give ord v(F i 0 (1, ū3 ) + θ p ) ≤ degF d + 1 and in case of equality, ord v (F i 0 (1, ū3 ) + θ p ) = 0 mod p. Finally, assume that x is in case 1. Let

in m S (h) =: Z p + λU (p-1)d 1 1 U (p-1)d 2 2 U a 1 U b 2 Z + U pd 1 1 U pd 2 2 ω(x) i=0 U b i 1 G i (U 2 , U 3 ), λ ∈ k(x), a + (p -1)d 1 , b + (p -1)d 2 ∈ N when λ = 0, b i ∈ N, G i ∈ k(x)[U 2 , U 3 ]. (6.
28) It is clear that, when λ = 0, x ′ is in case 1. Suppose λ = 0. Let i 0 :=sup{i|G i = 0} > 0. When i 0 > 0, the proof runs along the same lines as above: x ′ is in case 1. When i 0 = 0, then

in m S (h) =: Z p + λ ′ U pd 1 1 U pd 2 2 U ω(x) 1 , d 1 + d 2 + ω(x) p = δ(x), λ ′ ∈ k(x) * . (6.29)
The first face of ∆(h; u 1 , u 2 , u 3 ; Z) has only one vertex: w = (d 1 + ω(x) p , d 2 , 0) which will give the vertex of smallest ordinate

w ′ = (d 1 + ω(x) p , δ(x) -1, 0) of ∆(h ′ ; u ′ 1 , u 2 , v; Z ′ ). When    d 1 + ω(x) p ∈ N, or δ(x) -1 ∈ N, or λ ′ ∈ k(x ′ ) p , (6.30)
w ′ is not solvable and ω(x ′ ) = ǫ(x ′ ), x ′ is in case 1. When none of the conditions above are satisfied, then the coordinates of w = (d 1 + ω(x) p , d 2 , 0) are in N and, as w is not solvable, λ ′ ∈ k(x) p : so λ ′ ∈ k(x ′ ) p \ k(x) p , k(x ′ ) is inseparable over k(x). Corollary 6.10. With hypotheses and notation of 6.8, assume that x is in case 1 with β(x) = 2. Then

β(x 1 ) ≤ 2 (β(x 1 ) < 2 if k(x 1 ) = k(x)) if x 1 is again in case 1.
If x 1 is in case 3, and k(x 1 ) = k(x), we get β(x 1 ) < 1.

Proof. The only case to consider is k(x 1 ) = k(x) and (6.25). As β(x ′ ) ≤

1 i 0 ord v (F i 0 (1, ū3 ) + θ p ), the result is clear except if: • i 0 = 1, pd 1 + ω(x) -1 = 0 mod p, • d = 2 =deg(F ),
With the the notations of Lemma 6.7, β(x ′ ) = e ′ . By Lemma 6.7 ( 2), e ′ <deg(F ) = 2.

Following now [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Theorem I.5 on page 1964: Proposition 6.11. With hypotheses and notations as above, assume that x is in case 3. Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x and assume furthermore that

x ′ ∈ Spec(S[ u 1 u 2 , u 3 u 2 ][Z ′ ]/(h ′ )), h ′ := u -p 2 h, Z ′ := Z u 2 .
If C(x ′ ) = C(x), we have

A 2 (x ′ ) = B(x) -1, γ(x ′ ) ≤ γ(x),
and there exist well adapted coordinates (u ′ 1 := u 1 /u 2 , u 2 , v; Z ′ ) at x ′ such that the following holds:

(1) if x ′ is in case 1, then β(x ′ ) ≤ γ(x) d + 1;
(2) if x ′ is in case 3, then

β(x ′ ) ≤ max{β(x), 0}
and β(x ′ ) < β(x) if (k(x ′ ) = k(x) and β(x) > 0);

Proof. As in the preceeding proof, we look at the initial form polynomial H B (6.11) corresponding to the valuation v a,b,b with B(x) = a b . By (6.15), the term of degree 0 in Z is:

Φ p = U pd 1 1 U pd 2 2 [(λ 3 U 3 + λ 2 U 2 )U ω(x) 1 + ω(x) i=1 U ω(x)-i 1 F i (U 2 , U 3 )]. (6.31)
The initial form polynomial (6.11) is in fact the form H 2 at x ′ and has A 2 (x ′ ) = B(x) -1, the term of degree 0 in Z is:

H ′ 2 := in v (a ′ ,b ′ ,0) (h ′ ) with a ′ := a-b 1-b and b ′ := b 1-b with (a, b) defined in
U ′ 1 pd 1 U ′ 2 ǫ(x)-p [(λ 3 u ′ 3 + λ 2 )U ′ 1 ω(x) + ω(x) i=1 U ′ 1 ω(x)-i U ′ 2 i(B(x)-1) F i (1, u ′ 3 )], when a > b, u ′ 1 pd 1 U ′ 2 ǫ(x)-p [(λ 3 u ′ 3 + λ 2 )u ′ 1 ω(x) + ω(x) i=1 u ′ 1 ω(x)-i U ′ 2 i(B(x)-1) F i (1, u ′ 3 )], when a = b, (6.32)
and when Φ p-1 = 0, the upper bounds for β(x ′ ) follow from Lemma 6.7. By (6.9), note that

degF i (U 2 , U 3 ) -iA 2 (x) ≤ iγ(x)
in (6.15) whenever F i (U 2 , U 3 ) = 0. When Φ p-1 = 0, we apply Lemma 6.7 to U pd 2 2 F i (U 2 , U 3 ) with a 2 > pd 2 + iA 2 (x) and a 3 = 0. This gives

iβ(x ′ ) ≤ iγ(x) d + 1.
One deduces the upper bounds of ( 1) or (2) and γ(x ′ ) ≤ γ(x).

If Φ p-1 = 0, it is a monomial in U 1 , U 2 by Theorem 2.36 and, we are in the case (6.23) with d 3 = f = 0 and

a(d 1 + d p -1 ) + b(d 2 + e p -1 ) = a(d 1 + ω(x) p ) + b(d 2 + 1 p ),
which leads to

d = (p -1) ω(x) p ,
when d < (p-1) ω(x) p , using the same arguments as in the proof of Proposition 6.8, we get β(x ′ ) = 0 (resp. β(x ′ ) < 0) if x ′ is in case 1 (resp. in case 3). (6.33) It may be possible that d > (p -1) ω(x) p , then e = 0 and ω(x) = (p-1)b a ≤ p -1, in this extreme case, we conclude as above, using the index i 0 :=sup{i|F i = 0} > 0. Corollary 6.12. With hypotheses and notation of 6.11, assume that β(x) = 1 and x ′ is in case 1. Then, one of the following is true:

(1) β(x ′ ) < 2 (2) x is in case 1 with: β(x) = 2 and Φ p,α = ω(x) i=0 U ′ 1 ω(x)-i Φ p,α,i (U 2 , V )
has Φ p,α,1 = 0 with notations as in (6.14), where σ 2 := {(A 2 (x), 2)}.

Proof. Indeed, in the proof above, we get iβ(x ′ ) ≤ iγ(x) d + 1. So (1) is true when i = 1, when i = 1, we get [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF].

Following [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma I.5.3 on page 1966: Proposition 6.13. With hypotheses and notations as above, let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x and assume furthermore that

x ′ = (Z ′ := Z/u 3 , u ′ 1 := u 1 /u 3 , u ′ 2 := u 2 /u 3 , u 3 ). If C(x ′ ) = C(x), then x ′ is in case 2, (u ′ 1 , u ′ 2 , u 3 ; Z ′ ) are well adapted coordi- nates at x ′ , A 3 (x ′ ) = B(x) -1, β(x ′ ) = A 2 (x) + β(x) -1, γ(x ′ ) ≤ γ(x),
and the following holds:

(1) if x is in case 1, then C(x ′ ) ≤ min{β(x) -C(x) -A 3 (x), C(x)}; (2) if x is in case 2, we have C(x ′ ) ≤ min{β(x) -C(x), C(x)}. (3) if x is in case 3, we have C(x ′ ) ≤ min{β(x) -C(x), C(x) -β 2 (x)}.
Proof. The argument is the same as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma I.5.3 on page 1966. This relies on the characteristic free Proposition 2.18 which asserts that no changes in Z ′ need to be performed in order to get well adapted data. It is easy to see that ∆ 2 (x ′ ) is obtained from ∆ 2 (x) by applying the affine transformation: (v 2 , v 3 ) → (v 2 , v 2 + v 3 -1) and adding quadrants. In fact we focus on two vertices (maybe equal) of ∆ 2 (x): (A 2 (x), β(x)) and (B(x)β 2 (x), β 2 (x)). They become two vertices of ∆ 2 (x ′ ): (A 2 (x), β(x) + A 2 (x) -1) and (B(x)β 2 (x), B(x) -1) which are respectively the vertex of smallest abscissa and the vertex of smallest ordinate of ∆ 2 (x ′ ). So

C(x ′ ) ≤ β(x) + A 2 (x) -1 -(B(x) -1) = β(x) + A 2 (x) -B(x) in case 2 = β(x) -C(x) -A 3 (x) in case 1,3 = β(x) -C(x) C(x ′ ) ≤ B(x) -β 2 (x) -A 2 (x) in case 2 ≤ C(x) in cases 1,3 = C(x) -β 2 (x) (6.
34) This gives all statements except "γ(x ′ ) ≤ γ(x)".

(2) C(x) = 0, β 2 (x) = -1, C(x 1 ) = 1, A 3 (x 1 ) = A 2 (x) -1, β(x 1 ) = A 2 (x).
The vertices of smallest abscissa of smallest ordinate of ∆ 2 (x ′ ) are: (1) (A 2 (x), 1 + A 2 (x)) and (1 + A 2 (x), A 2 (x)), ( 2) (A 2 (x), A 2 (x)) and (A 2 (x), A 2 (x) -1).

These facts imply that in both cases:

∆ 2 (x 1 ) = (A 2 (x 1 ), A 3 (x 1 )) + {(y 2 , y 3 ) ∈ R 2 ≥0 : y 2 + y 3 ≥ 1}. When x 2
is again in case 2, we get C(x 2 ) = 0 by (6.34). Otherwise, we may assume that C(x 2 ) = C(x) and apply Proposition 6.8 to estimate γ(x 2 ). We get γ(x 2 ) = 1 if k(x 2 ) = k(x) by ( 2) of this proposition.

Assume that k(x 2 ) = k(x). We claim that, when x 1 is not in case 2*, the following sharper bound holds, which concludes the proof:

β(x 2 ) ≤ 1 (resp. β(x 2 ) ≤ 0) (6.35) if x 2 is in case 1 (resp. in case 3). There are associated d ′ 1 , d ′ 2 , d ′ 3 ∈ 1/pN at x 1 with d ′ 1 = d 1 , d ′ 2 = d 2 and d ′ 3 = d 1 + d 2 -1 + ω(x) p (resp. d ′ 3 = d 1 + d 2 -1 + 1 + ω(x) p )
if x is in case 1 (resp. in case 3).

The initial form H B (6.12) at x 1

H B = Z ′ p -Z ′ Φ p-1 + U ′ 1 pd ′ 1 U ′ 2 pd ′ 2 U pd ′ 3 3 ω(x) i=0 U ′ 1 ω(x)-i F i (U ′ 2 , U 3 ) (6.36) has F 1 (U ′ 2 , U 3 ) = 0 and is of the form F 1 (U ′ 2 , U 3 ) = U ′ 2 a 2 U a 3 3 F (U ′ 2 , U 3 ), a 2 + a 3 + deg(F ) = 1 + A 2 (x 1 ) + A 3 (x 1
) where a 2 ≥ A 2 (x), a 3 ≥ A 3 (x), and either F ∈ k(x) or

a 2 = A 2 (x 1 ), a 3 = A 3 (x 1 ) ∈ N and F = λ 2 U ′ 2 + λ 3 U 3 , λ 3 = 0. (6.37)
The point x 2 has for parameters (Y,

v 1 , v 2 , v) := (Z ′ /u ′ 2 , u ′ 1 /u ′ 2 , u ′ 2 , u ′ 3 /u ′ 2 + λ), λ ∈ S ′ , λ invertible. The initial form H 2 at x 2 H 2 = Y p -Y G ′′ p-1 +V pd ′ 1 1 V p(d ′ 1 +d ′ 2 +d ′ 3 )+ω(x)-p 2 ω(x) i=0 V 1 ω(x)-i V i(A 2 (x 1 )+A 3 (x 1 )) 2 F " i (1, v-λ) • when x i is in case 1: ⌈β(x i )⌉ = 2, thus 1 < β(x i ) ≤ 2, • when x i is in case 2: ⌊C(x i )⌋ = 1, thus 1 ≤ C(x i ) < 2,
• when x i is in case 3:

⌊β(x i )⌋ = 2, thus 1 ≤ β(x i ) < 2.
We now derive a contradiction by studying different cases. (a) if x is in case 1 with β(x) < 2, we are done. Indeed take the smallest i ′ as above, by the last statement of Proposition 6.8, x i 0 is in case 1 for i 0 ≤ i ′ , by Proposition 6.8(1), β(

x i ′ -1 ) ≤ β(x) < 2. Either k(x i ′ ) = k(x i ), Proposition 6.8 gives β(x i ′ ) ≤ 1 when x i ′ is in case 1, β(x i ′ ) < 1 when x i ′ is in case 3. Or x i ′ is
in case 2 (point at infinity), then C(x i ′ ) < 1 by Proposition 6.13 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF].

Assume that x is in case 1 with β(x) = 2. By Proposition 6.8 and Corollary 6.10, we obtain

β(x 1 ) ≤ 2 (β(x 1 ) < 2 if k(x 1 ) = k(x)) if x 1 is again in case 1. If x 1 is in case 3, we get β(x 1 ) < 1.
Assume that x is in case 3. If Proposition 6.11 applies, we obtain β(x 1 ) ≤ β(x) (with strict equality if k(x 1 ) = k(x)) if x 1 is again in case 3. If x 1 is in case 1, we get β(x 1 ) ≤ 2; if furthermore β(x) = 1, by corollary6.12 the inequality is strict unless x 1 satisfies the assumptions of Lemma 6.15 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF]. We deduce: (b) if x is in case 3 with β(x) = 1, we are done: this follows from Lemma 6.15 and the previous comments. (c) if x is in case 1 with β(x) = 2, we are done: we may assume that Proposition 6.13 applies by the previous comments; we reach (a)(b) or the assumptions of Lemma 6.15(1) at x 2 since it is assumed that γ(x 2 ) = 2. (d) the remaining cases: x is in case 2 or x is in case 3 with β(x) > 1. (d-1) x is in case 2. The result is trivial if x i is in case 2 for every i >> 0. Indeed, let (β 3 (x), A 3 (x)) the vertex of smallest ordinate of ∆ 2 (x), then at x ′ the point at infinity, the vertices of smallest abscissa (resp. smallest ordinate)

(A 2 (x), β + A 2 (x) -1) resp. (B(x) -β 2 (x), B(x) -1): when C(x) > 0, we get (β(x) -A 3 (x)) + (β 3 (x) -A 2 (x)) > (β(x ′ ) -A 3 (x ′ )) + (β 3 (x ′ ) -A 2 (x ′ ))
, by symmetry there is the same inequality when x ′ is the point of parameters (Z/u 2 , u 1 /u 2 , u 2 , u 3 /u 2 ). Otherwise, note that: by Proposition 6.8,

C(x 1 ) ≤ C(x) if x 1 is in case 2; β(x 1 ) ≤ C(x) if x 1 is in case 3. (d-2)
x is in case 3 with β(x) > 1. Note that C(x 1 ) < β(x) if x 1 is in case 2; by Proposition 6.11(2), β(x 1 ) ≤ β(x) if x 1 is in case 3 with strict inequality when k(x 1 ) = k(x). In the case where x i is in case 3 for i >> 0, we reach β(x i ) < 1 or k(x i ) = k(x i+1 ) for i >> 0, in the latter, we are in case (ii) of the proposition.

for every i ≥ 0. Applying Proposition 3.17, we are done if alternative (2) of this proposition holds; if alternative (1) holds, it can be assumed that there exists a permissible curve of the first kind Y = V (Z, u 1 , u 3 ) ⊆ (X , x). Then x is resolved by blowing up Y: in view of Definition 6.1, we need only to consider the point x ′ := (Z/u 3 , u 1 /u 3 , u 2 , u 3 ) and get ω(x ′ ) < ω(x) from Proposition 2.18. This proves the proposition under the extra assumption (6.39).

We now consider several cases which are proved consecutively: (a) x is in case 1. We have A 2 (x) ≥ 1 if the extra assumption (6.39) does not hold. Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x and note that Y := V (Z, u 1 , u 2 ) is a permissible curve of first kind. Blowing up along Y, as div(u 1 ) has maximal contact for C, the only point with C(x 1 ) = C(x) is x 1 = (Z/u 2 , u 1 /u 2 , u 2 , u 3 ), in which case x 1 is again in case 1 with

(A 2 (x 1 ), β(x 1 )) = (A 2 (x) -1, β(x)).
The proof concludes by induction on A 2 (x). Before going along with the proof in cases 2 and 3, we make the following remark: Remark 6.18. Assume that x is in case 2 with A 2 (x) ≥ 1. Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x and denote Y := V (Z, u 1 , u 2 ) with generic point y. Since ǫ(y) = ǫ(x), Y is permissible of the first kind if and only if it is Hironaka-permissible w.r.t. E, i.e. if m(y) = m(x) = p. Thus:

Y is permissible of the first kind ⇔ d 1 + d 2 + ω(x) p ≥ 1. (6.40)
Suppose that Y is Hironaka-permissible. Blowing up along Y and arguing as in (a), we achieve:

x 1 in case 2, A 2 (x 1 ) = A 2 (x) -1, A 3 (x 1 ) = A 3 (x). (6.41)
This proves that it can be assumed to begin with that

A j (x) < 1 or d 1 + d j + ω(x) p < 1 (6.42)
for each of j = 2, 3.

(m(x i ), ω(x i )) = (m(x i-1 ), ω(x i-1 )) and κ(x i ) > κ(x i-1 ) = 2. In fact such x i will fit the hypotheses of one of the technical lemmas. As a consequence of subsection 7.1, such x i is always resolved for (p, ω(x), 2). Let us note that the hypotheses of the lemmas are not so restrictive and some other useful cases of points x resolved for (p, ω(x), 2) occur. In subsection 7.2, we reduce the problem to a special case (*) (Definition 7.10). By Proposition 7.11(ii), apart the cases of subsection 7.1, this case κ(x) = 2 and (*) is stable when

Y i = x i .
The end of the chapter is completely devoted to the resolution of the case κ(x) = 2 and (*).

Up to the end of this chapter, "resolved" stands for "resolved for (p, ω(x), 2)".

Preliminaries.

In this section, we study points x ′ obtained by performing a permissible blowing up and such that (m(x ′ ), ω(x ′ )) = (m(x), ω(x)) and κ(x ′ ) > κ(x) = 2. Lemma 7.1. Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x. Assume that ǫ(x) = ω(x) ≥ 2, κ(x) ≥ 2 and div(u 1 ) ⊆ E.

Assume furthermore

(d 1 , d 2 + 1/p, d 3 + ω(x)/p) is the only vertex v = (v 1 , v 2 , v 3 ) of ∆ S (h; u 1 , u 2 , u 3 ; Z) in the region v 1 = d 1 .
Then x is resolved.

Proof. Since κ(x) ≥ 2, there is an expansion

in m S h = Z p + F p,Z , H -1 F p,Z ⊆ k(x)[U 1 , U 2 , U 3 ] ω(x) .
Any vertex of ∆ S (h; u 1 , u 2 , u 3 ; Z) ∩ {x :

x 1 + x 2 + x 3 = δ(x)} lies in the region v 1 > d 1
by assumption and we deduce that

U 1 ∈ Vdir(x). Let in v h = Z p + p i=1 F i,v Z p-i ∈ k(x)[U 1 , U 2 , U 3 ][Z]
be the initial form polynomial with respect to v. By Theorem 2.36, we have

F i,v = 0, 1 ≤ i ≤ p -2, and F p-1,v = -G p-1 v since ǫ(x) > 0. Moreover G p-1 v = 0 implies that v ∈ N 3 , E = div(u 1 u 2 u 3 ) and (Disc Z (h)) = (u pd 1 1 u pd 2 +1 2 u pd 3 +ǫ(x) 3 ) p-1 . (7.2)
with the usual convention: a(3) := 0 when div(u 1 u 2 ) = E. We define F := HU ω-i 3 F 0 ; assume that F ∈ G(m S ) p and that < U 3 , U j > Vdir(J(F, E, m S )) for j = 1 and j = 2.

Then Vdir(J(F, E, m S )) =< U 3 , U 1 + λU 2 >, λ = 0, (7.6) and the following holds:

(i) if i ≡ 0 modp, there exists 0 = c ∈ k(x) such that

F -cHU ω-i 3 (U 1 + λU 2 ) i ∈ G(m S ) p ;
(ii) if i ≡ 0 modp, let a j := a(j), 1 ≤ j ≤ 3, and a := i = 0. Then:

a 3 + ω -a ≡ 0 modp, a 1 a 2 = 0 and a 1 + a 2 + a = p. (7.7)
In particular p ≥ 3. There exists 0 = c ∈ k(x) p such that

F -cU a(3)+ω-i 3 Φ i (U 1 , λU 2 ) ∈ G(m S ) p ,
where

Φ i (U 1 , U 2 ) := (-1) a 2 U a(1) 1 U a(2) 2 a k=0 a 2 + k -1 k U a-k 1 (U 1 + U 2 ) i-a+k . (7.8)
Proof.

[30] II.5 p.1896 for (i) and (7.7). It remains to prove that there exists 0 = c ∈ k(x) p such that

H 0 F 0 -cΦ i (U 1 , λU 2 ) ∈ (k(x)[U 1 , U 2 ]) p , H 0 := U a(1) 1 U a(2) 2
.

It is easily checked that (7.14) holds when

F = U a(3)+ω(x)-i 3 Φ i (U 1 , λU 2 ). Note that H -1 0 Φ i (U 1 , λU 2 ) = (-1) a 2 λ a(2) a 2 + a a U i 1 + • • • . (7.9)
Let (λ l ) l∈Λ be an absolute p-basis of k(x) and let

D l := ∂ ∂λ l D j := U j ∂ ∂U j , j = 1, 2.
We expand

F 0 := αU i 1 + α 1 U i-1 1 U 2 + • • • , α, α 1 ∈ k(x). (7.10) Since H -1 0 D l • (H 0 F 0 ) ∈< (U 1 + λU 2 ) i > by (7.14), l ∈ Λ ∪ {1, 2}, it is easily seen that α = 0. Suppose that α ∈ k(x) p . Since H -1 0 D l • (H 0 F 0 ) ∈< (U 1 + λU 2 ) i >, l
∈ Λ, and this polynomial is divisible by U 2 , we have D l • H 0 F 0 = 0 for l ∈ Λ by (7.14). We deduce that H 0 F 0 ∈ k(x) p [U 1 , U 2 ] and in particular that λ ∈ k(x) p . Let

F ′ := H 0 F 0 -cΦ i (U 1 , λU 2 ), c := α(-1) a 2 λ -a(2) a 2 + a a -1 ∈ k(x) p .
By construction, we have H -1 D l • F ′ = 0, l ∈ Λ ∪ {1, 2}, and (ii) is proved.

Suppose that α ∈ k(x) p . Without loss of generality, it can be assumed that α = λ l for some l ∈ Λ. For (7.14). This proves that

l ′ = l, U 2 divides H -1 0 D l ′ • (H 0 F 0 ), so D l ′ • (H 0 F 0 ) = 0 by
F 0 ∈ k(x) p (α)[U 1 , U 2 ]. We have    H -1 0 D l • (H 0 F 0 ) = U i 1 + (D l • α 1 )U i-1 1 U 2 + • • • H -1 0 D 1 • (H 0 F 0 ) = (a 1 + a)αU i 1 + (a 1 + a -1)α 1 U i-1 1 U 2 + • • • from which we deduce the identity    aλ = D l • α 1 a(a 1 + a)αλ = (a 1 + a -1)α 1 . (7.11)
Therefore (a 1 + a -1)α 1 = (a 1 + a)(D l • α 1 ). Expanding α 1 =: p-1 j=0 c p j α j , we then deduce that α 1 = c p j α j , where (a 1 + a)j ≡ a 1 + a -1 modp. (7.12)

Since a 1 + a + a 2 = p in this case (ii), we get a 2 (j -1) ≡ 1 modp from (7.12). One deduces from (7.11)-(7.12) that α = dλ a (2) for some d ∈ k(x) p , d = 0.

The proof now concludes as in the above case α ∈ k(x) p .

Lemma 7.4.

Let F 0 ∈ k(x)[U 1 , U 2 , U 3 ] ω , ω ∈ N \ {0}, F 0 = 0. Take E := div(u 1 u 2 u 3 ) and let (a(1), a(2), a(3)) ∈ N 3 , H := U a(1) 1 U a(2) 2 U a(3) 3 ∈ G(m S ) = k(x)[U 1 , U 2 , U 3 ].
Let F := HF 0 . Assume that:

Vdir(J(F, E, m S )) =< U 1 + λ 2 U 2 + λ 3 U 3 >, λ 2 λ 3 ∈ k(x) * . (7.13)
Then ω = 0 mod p and there exists 0 = c ∈ k(x) such that

F -cH(U 1 + λ 2 U 2 + λ 3 U 3 ) ω ∈ G(m S ) p .
Proof. Instead of quoting [START_REF] Cossart | Polyèdre caractéristique d'une singularité[END_REF] Proposition E.5.1 page 33, we give a short argument. Let

F 0 := 0≤a,b,c≤ω λ a,b,c U a 1 U b 2 U c 3 , λ a,b,c ∈ k(x).
It is clear that λ ω,0,0 λ 0,ω,0 λ 0,0,ω ∈ k(x) * . As U This leads to ω = 0 mod p.

Let G := F 0 -λ ω,0,0 (U 1 + λ 2 U 2 + λ 3 U 3 ) ω , we have Vdir(J(HG, E, m S )) =< U 1 + λ 2 U 2 + λ 3 U 3 > or HG ∈ G(m S ) p . (7.14)
As deg U 1 (G) < ω, the first is impossible, the second is true. This gives the result.

Lemma 7.5. Assume that E = div(u 1 ). If (u 1 , u 2 , u 3 ; Z) are well adapted coordinates at x, then

in E h = Z p + U pd 1 1 F ∈ S/(u 1 )[U 1 ][Z], F = 0.
Proof. This is obvious if charS = p > 0 and h is purely inseparable (case (c) of assumption (G)). Otherwise, (E) implies that Disc Z (h) = γu D 1 for some D ≥ p(p -1)d 1 and γ ∈ S a unit. Let

in E h = Z p + p i=1 U id 1 1 F i Z p-i , F i ∈ S/(u 1 )[U 1 ] id 1 ,
where

F i = 0 if id 1 ∈ N.
Since charS/(u 1 ) = p > 0, condition (G) implies that in E h has p distinct roots over an algebraic closure of k(E) if F i = 0 for some i = p. But then D = p(p -1)d 1 : a contradiction since ǫ(x) > 0. We deduce that F i = 0, 1 ≤ i ≤ p -1. We have F p = 0 by Proposition 2.12.

Proposition 7.6. Assume that ǫ(x) = ω(x), κ(x) ≥ 2 and E = div(u 1 ). Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x. Assume furthermore that S/(u 1 ) ≃ k(x)[u 2 , u 3 ] (u 2 ,u 3 ) and the following two conditions are satisfied:

(i) the initial form polynomial in E h of Lemma 7.5 is of the form

in E h = Z p + U pd 1 1 F , F ∈ k(x)[u 2 , u 3 ] 1+ω(x) ;
(ii) we have

Vdir(x) + Vdir ∂F ∂u 2 , ∂F ∂u 3 =< U 2 , U 3 >,
where Vdir(x) denotes the image of Vdir(x) in < U 2 , U 3 >.

Then x is resolved.

Proof. The proof is the same as that of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] II.3 p.1890 and we only indicate the necessary changes. Since κ(x) ≥ 2, we have

in m S h = Z p + F p,Z , H -1 F p,Z ⊆ k(x)[U 1 , U 2 , U 3 ] ω(x) (7.15)
and U 1 ∈ Vdir(x) as in the beginning of the proof of Lemma 7.1. We discuss according to the value of τ ′ (x).

• Assume that τ ′ (x) = 3. The proposition follows from Theorem 3.13.

• Assume that τ ′ (x) = 2. Note that ω(x) ≥ 2. Since E = div(u 1 ) and U 1 ∈ Vdir(x), we have Vdir(x) =< U 1 , λ 2 U 2 + λ 3 U 3 >, (λ 2 , λ 3 ) = (0, 0). By symmetry, it can be assumed that λ 2 = 1. If λ 3 = 0, we let v 2 := u 2 + γ 3 u 3 , where γ 3 ∈ S is a preimage of λ 3 ∈ S/(u 1 ) ≃ k(x)[u 2 , u 3 ] (u 2 ,u 3 ) . Let (u 1 , v 2 , u 3 ; Z 1 ) be well adapted coordinates at x, Z 1 = Zφ, φ ∈ S. By Lemma 7.5, we have ord

u 1 φ > d 1 . Therefore in E h = Z p 1 + U pd 1 1 (F + φ p ),
where φ = 0 (resp.

φ = cl 0 (u -d 1 1 φ) ∈ S/(u 1 )) if d 1 ∈ N (resp d 1 ∈ N). Note that (1 + ω(x) ≡ 0 modp and φ ∈ k(x)[u 2 , u 3 ] (1+ω(x))/p ) if φ = 0.
Assumptions (i) and (ii) are then unchanged, so it can be assumed w.l.o.g. that Vdir(x) =< U 1 , U 2 >. Assumption (ii) now implies

F (u 2 , u 3 ) ∈< u 1+ω(x) 2 > (resp. F (u 2 , u 3 ) ∈< u 1+ω(x) 2 , u 3 u ω(x) 2 >) if ω(x) ≡ 0 modp (resp. if ω(x) ≡ 0 modp).
Let X ′ -→ (X , x) be the blowing along x and x ′ ∈ X ′ be the center of µ. By Theorem 3.13, we have ι(x ′ ) ≤ (p, ω(x), 1) except possibly if

x ′ = (Z ′ := Z/u 1 , u ′ 1 := u 1 /u 3 , u ′ 2 := u 2 /u 3 , u 3 ), since Vdir(x) =< U 1 , U 2 >. If ι(x ′ ) ≥ (p, ω(x), 2)
, there are two cases to be considered as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] end of p.1891:

Case 1:

F (u 2 , u 3 ) = λ 0 u 1+ω(x) 2 + λ 1 u 3 u ω(x) 2
, λ 1 = 0. Then (X ′ , x ′ ) satisfies the assumption of Lemma 7.1 (instead of ibid. II.1 on p.1885) whose conclusion proves the proposition.

Case 2:

F (u 2 , u 3 ) = λ 0 u 1+ω(x) 2 + λ 1 u 3 u ω(x) 2 + λ 2 u 2 3 u ω(x)-1 2
, λ 2 = 0 and ω(x) -1 ≡ 0 modp. Then τ (x ′ ) = 3 by the characteristic free ibid. Lemma II.3.3 on p.1892. Blowing up again x ′ then gives ι(x ′′ ) ≤ (p, ω(x), 1) by Theorem 3.13, where x ′′ is the center of µ.

• Assume that τ ′ (x) = 1. We have Vdir(x) = k(x)U 1 and F p,Z = λU

pd 1 +ω(x) 1 in (7.15). Assumption (ii) now reads Vdir ∂F ∂u 2 , ∂F ∂u 3 =< U 2 , U 3 > . (7.16)
Let X ′ -→ (X , x) be the blowing along x and x ′ ∈ X ′ be the center of µ. By Theorem 3.13, we have ι(x ′ ) ≤ (p, ω(x), 1) except possibly if η ′ (x ′ ) lies on the strict transform of div(u 1 ). By symmetry between u 2 , u 3 , it can be assumed that x

′ = (Z ′ := Z/u 3 , u ′ 1 := u 1 /u 3 , u ′ 2 = P (u 2 /u 3 ), u 3 ), where
Then x is resolved.

Proof. This is a simpler variation of Proposition 7.6 and we build upon its proof. To begin with, let (u 1 , u 2 , u ′ 3 ; Z ′ ) be another set of well adapted coordinates at x. There is an equality

U ′ 3 = λ 3 U 3 + λ 2 U 2 + λ 1 U 1 ∈ G(m S ) 1 =< U 1 , U 2 , U 3 >, λ 3 = 0.

The corresponding initial form polynomial in

E h = Z ′ p + U pd 1 1 F ′ satisfies Φ ′ := cl ω(x)+1 F ′ = F (U 2 , λ -1 3 (U ′ 3 -λ 2 U 2 )) + Θ p ∈ k(x)[U 2 , U ′ 3 ] ω(x)+1 , where Θ ∈ k(x)[U 2 , U ′ 3 ] (ω(x)+1)/p , Θ = 0 if d 1 ∈ N or if ω(x) + 1 ≡ 0 modp. We deduce that ∂Φ ′ ∂U ′ 3 = 0 and Vdir( ∂Φ ∂U 2 ) =< U 2 , U ′ 3 > . (7.18)
In other terms, (i) and (ii) remains valid for the well adapted coordinates (u 1 , u 2 , u ′ 3 ; Z ′ ). Also note that no Φ satisfies (ii) when ω(x) + 1 ≡ 0 modp, since then

∂Φ ∂U 3 = 0 =⇒ Φ ∈ k(x)[U p 2 , U p 3 ] =⇒ ∂Φ ∂U 2 = 0. (7.19)
Let X ′ -→ (X , x) be the blowing along x, x ′ ∈ X ′ be the center of µ and suppose that ι(x ′ ) ≥ (p, ω(x), 2). We discuss according to the values of τ ′ (x) as in the proof of Proposition 7.6.

• Assume that τ ′ (x) = 3. The proposition follows from Theorem 3.13.

• Assume that τ ′ (x) = 2. By (7.18), it can be assumed that Vdir(x) =< U 1 , U 2 > or Vdir(x) =< U 1 , U 3 >. The polynomial assumption of Proposition 7.6 (i) on F is used only in cases 1 and 2 of the corresponding proof. Therefore under the assumptions of this proposition, it is sufficient to prove that

     Φ ∈< U 1+ω(x) 2 , U 3 U ω(x) 2 > if Vdir(x) =< U 1 , U 2 > Φ ∈< U 1+ω(x) 3 , U 2 U ω(x) 3 > if Vdir(x) =< U 1 , U 3 > (7.20) and that      Φ ∈< U 1+ω(x) 2 , U 3 U ω(x) 2 , U 2 3 U ω(x)-1 2 > if Vdir(x) =< U 1 , U 2 > Φ ∈< U 1+ω(x) 3 , U 2 U ω(x) 3 , U 2 2 U ω(x)-1 3 > if Vdir(x) =< U 1 , U 3 > (7.21) if furthermore ω(x) -1 ≡ 0 modp. By (ii), we have Φ ∈ k(x)[U 2 , U p 3 ]\k(x)[U 2 ] and ∂Φ ∂U 2 ∈< U ω(x) 3
> and (7.20) follows easily. Furthermore, (7.21) reduces to (7.20) except possibly if p = 2; but assumption (ii) then implies that ω(x) ≡ 0 mod2 by (7.19).

• Assume that τ ′ (x) = 1. We have Vdir(x) =< U 1 >. The polynomial assumption Proposition 7.6 (i) on F is also used only in cases 1 and 2 of the corresponding proof.

If k(x ′ ) = k(x), one is then reduced to proving (7.20)-(7.21) and the proof is identical as in (b).

If [k(x ′ ) : k(x)] ≥ 2, the argument in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] proof of II.3 (cases 1 and 2 on p.1894) shows that p = 2, ω(x) = 3 and [k(x ′ ) : k(x)] = 2; but assumption (ii) then implies ω(x) ≡ 0 mod2 by (7.19) and the conclusion follows.

Proposition 7.8. Assume that E = div(u 1 ), ǫ(x) = ω(x), κ(x) = 2 and

Vdir(x) + k(x)U 1 =< U 1 , U 2 , U 3 > . Then x is good. Proof. This follows from Theorem 3.13 if Vdir(x) =< U 1 , U 2 , U 3 >, i.e. τ ′ (x) = 3.
Assume that τ ′ (x) = 2. Since Vdir(x) and ι(x) do not depend on the choice of well adapted coordinates, it can be assumed w.l.o.g. that Vdir(x) =< U 2 , U 3 >. Since ǫ(x) = ω(x), there is an expansion

in m S h = Z p + F p,Z , H -1 F p,Z ⊆ k(x)[U 2 , U 3 ] ω(x) .
Let µ be a valuation of L = k(X ) centered at x, X 1 -→ X be the blowing up along x and x 1 ∈ X 1 be the center of µ. By Theorem 3.13, ι(x 1 ) < ι(x) If (ǫ(x 1 ) = ω(x) and Ψ = 0), then x 1 satisfies the assumptions of Lemma 7.1, so x 1 is resolved.

If (ǫ(x 1 ) = ω(x) and 0 = Ψ ∈< V ′ 2 ω(x) >), we have J(F p,Z ′ 1 , E 1 , m S 1 ) ≡< V ′ 2 ω(x) > mod(U 1 ) ∩ G(m S 1 ) ω(x) .
Therefore ι(x 1 ) = ι(x) and x 1 satisfies condition (*1).

If (ǫ(x 1 ) = ω(x) and Ψ ∈< V ′ 2 ω(x) >), we have κ(x ′ ) = 2 and Vdir(x ′ ) + k(x ′ )U 1 =< U 1 , V ′ 2 , U ′ 3 >, so x 1 is good by Lemma 7.8. If ǫ(x 1 ) = 1 + ω(x), we have H ′ -1 ∂F p,Z ′ 1 ∂V ′ 2 ≡< U ′ 3 ω(x) > mod(U 1 , V ′ 2 ) ∩ G(m S 1 ) ω(x 1 ) .
Then there exist well adapted coordinates of the form (u 1 , v ′ 2 , v 3 ; Z ′ ) at x 1 satisfying Definition 7.10, so ι(x 1 ) = ι(x) and x 1 satisfies condition (*3). This concludes the proof of (ii) when τ ′ (x) = 1.

• Assume that τ ′ (x) = 2. Up to a change of well adapted coordinates, it is easily seen that x belongs to one of the following types:

(T0) ω(x) = ǫ(x), E = div(u 1 ) and Vdir(x) =< U 3 , U 2 >; (T1) ω(x) = ǫ(x) -1 and Vdir(x) =< U 3 , U 2 >; (T2) ω(x) = ǫ(x), E = div(u 1 u 2 ) and Vdir(x) =< U 3 , U 1 +λU 2 > with λ = 0; (T3) ω(x) = ǫ(x) and Vdir(x) =< U 3 , U 1 >; (T4) ω(x) = ǫ(x) -1 and Vdir(x) =< U 3 , U 1 >.
Claim: suppose x is of type (Tk), 0 ≤ k ≤ 4. Then x 1 is resolved or one of the following properties hold: (a) ι(x 1 ) = ι(x) and x 1 satisfies condition (*);

(b) ι(x 1 ) = ι(x), τ ′ (x 1 ) = 2 and x 1 is of type (Tl) with l ≤ k.

If moreover x satisfies condition (*), then x 1 is resolved or (a) holds.

To prove the claim, we do a case by case analysis. If k = 0, then x is good by Proposition 7.8.

Assume that k = 1. There is an expansion (7.22) with

G = 0 and U -pd 1 1 F p,Z = F 1+ω(x) (U 2 , U 3 ) + 1+ω(x) i=1 F 1+ω(x)-i (U 2 , U 3 )U i 1 . Since Vdir(x) =< U 2 , U 3 >, we have      Vdir ∂F 1+ω(x) ∂U 2 , ∂F 1+ω(x) ∂U 3 =< U 2 , U 3 > F 1+ω(x)-i (U 2 , U 3 ) ∈ k(x)[U p 2 , U p 3 ], 1 ≤ i ≤ 1 + ω(x) . (7.29)
Assume that ι(x ′ ) ≥ ι(x). By Theorem 3.13, x 1 = x ′ , where

x ′ := (Z ′ := Z/u 1 , u 1 , u ′ 2 := u 2 /u 1 , u ′ 3 := u 3 /u 1 ).
We have

E ′ = div(u 1 ), S ′ /(u 1 ) ≃ k(x)[u ′ 2 , u ′ 3 ] (u ′ 2 ,u ′ 3 ) and H(x ′ ) = (u p(d 1 -1)+1+ω(x) 1
).

Assume that ι(x ′ ) ≥ ι(x). By Proposition 2.18, ∆ S ′ (h

′ ; u 1 , u ′ 2 , u ′ 3 ; Z ′ ) is min- imal. The initial form in E ′ h ′ of Lemma 7.5 is of the form: in E ′ h ′ = Z ′ p + U p(d 1 -1)+1+ω(x) 1   F 1+ω(x) (u ′ 2 , u ′ 3 ) + 1+ω(x) i=1 F 1+ω(x)-i (u ′ 2 , u ′ 3 )   .
This proves that F i (U 2 , U 3 ) = 0, 2 ≤ i ≤ 1 + ω(x). We consider two cases:

Case 1: F ω(x) (U 2 , U 3 ) = 0. If ǫ(x ′ ) = ω(x)
, then x ′ satisfies all assumptions of Proposition 7.6 by (7.29), so x is good.

If ǫ(x ′ ) = ǫ(x), then ι(x ′ ) = ι(x) and H ′ -1 ∂F p,Z ′ ∂U ′ j ≡< ∂F 1+ω(x) ∂U j (U ′ 2 , U ′ 3 ) > mod(U 1 ) ∩ G(m S ′ ) ω(x) , 209
for j = 2, 3 again by (7.29). We conclude that τ ′ (x ′ ) = 3 (so x is good) or x ′ is again of type (T1) as required. If x satisfies condition (*), so does x ′ . Case 2:

F ω(x) (U 2 , U 3 ) = 0. We have ǫ(x ′ ) = ω(x) and in m S ′ h ′ = Z ′ p +U p(d 1 -1)+1+ω(x) 1 (F ω(x) (U ′ 2 , U ′ 3 )+U 1 Φ ′ ), Φ ′ ∈ k(x ′ )[U 1 , U ′ 2 p , U ′ 3 p ]. Therefore ι(x ′ ) = ι(x). If F ω(x) (U 2 , U 3 ) is monic in U 2 or in U 3 , then x ′ satis- fies condition (*1)
. Otherwise x ′ is of type (T0) and the conclusion follows.

Note that if ω(x) = 1, x is of type (T1) and satisfies condition (*3). So we may assume from this point on that ω(x) ≥ 2.

Assume that k = 2. There is an expansion (7.22) with G = 0 and

F p,Z = U pd 1 1 U pd 2 2 ω(x) i=0 F i (U 1 , U 2 )U ω(x)-i 3
.

Note that F i (U 1 , U 2 ) = 0 whenever ω(x)i ≡ 0 modp, since ω(x) = ǫ(x); we have F i = 0 for some i, 0 ≤ i ≤ ω(x) -1 since κ(x) = 2; moreover F 0 = 0 iff x satisfies condition (*). Assume that ι(x ′ ) ≥ ι(x). By Theorem 3.13, we have

x 1 = x ′ := (X ′ := Z/u 1 , u 1 , u ′ 2 := 1 + γu 2 /u 1 , u ′ 3 := u 3 /u 1 ),
γ ∈ S being a preimage of λ. We have

E ′ = div(u 1 ), k(x ′ ) = k(x) and H(x ′ ) = (u p(d 1 +d 2 -1)+ω(x) 1
).

Assume that ι(x ′ ) ≥ ι(x). Since Vdir(x) =< U 3 , U 1 + λU 2 >, we consider two cases deduced from Lemma 7.3: Case 1: ω(x) ≡ 0 modp. By Lemma 7.3(i), it can be assumed w.l.o.g that

F pi (U 1 , U 2 ) = c pi (U 1 + λU 2 ) pi , c pi ∈ k(x), 1 ≤ i ≤ ω(x) p . (7.30)
After blowing up, there is an expansion in m S ′ h ′ = X ′ p + F p,X ′ , where

U -pd ′ 1 1 F p,X ′ = (-λ) -pd 2 ω(x)/p i=0 c pi U ′ 2 pi U ′ 3 ω(x)-pi + U 1 Φ ′ , (7.31)
for some Φ

′ ∈ k(x)[U 1 , U ′ 2 p , U ′ 3 p ], d ′ 1 := d 1 + d 2 -1 + ω(x)/p. If d ′ 1 ∈ N, then ǫ(x ′ ) = ω(x) and ι(x ′ ) = ι(x). Moreover k(x ′ )U 1 + Vdir(x ′ ) =< U 1 , U ′ 2 , U ′ 3 >, so τ ′ (x ′ ) = 3 or x ′ is of type (T0). In both cases, x is good. If (d 1 , d 2 ) ∈ N 2
, it can be assumed furthermore that c pi = 0 or c pi ∈ k(x) p in (7.30). We have d ′ 1 ∈ N and we also get ǫ(x ′ ) = ω(x) and ι(

x ′ ) = ι(x). Since J(F p,Z , E, x) = H -1 < ∂F p,Z ∂λ l l∈Λ 0 >
with notations as in (2.49), we get in any case since k(

x ′ ) = k(x): k(x ′ )U 1 + Vdir(x ′ ) =< U 1 , U ′ 2 , U ′ 3 > . Therefore τ ′ (x ′ ) = 3 or x ′ is of type (T0), so x is good. If d ′ 1 ∈ N, d 2 ∈ N, we define I := {i : (-λ) -pd 2 c pi ∈ k(x) p }.
If I = ∅, we also get ǫ(x ′ ) = ω(x) and ι(x ′ ) = ι(x). If ω(x) ∈ I, x ′ satisfies condition (*1); otherwise x ′ is good.

If I = ∅, let (u 1 , u ′ 2 , u ′ 3 ; Z ′ ) be well adapted coordinates at x ′ . We denote by a ∈ F p the residue of pd 2 . Since d 2 ∈ N, we have a = 0. The initial form in E ′ h ′ of Lemma 7.5 is of the form:

in E ′ h ′ = Z ′ p + U pd ′ 1 1 F ′ (u ′ 2 , u ′ 3 ) ∈ S ′ /(u 1 )[U 1 , Z ′ ],
where

S ′ /(u 1 ) ≃ k(x)[u ′ 2 , u ′ 3 ] (u ′ 2 ,u ′ 3 ) . The form Φ ′ := cl ω(x)+1 F ′ is given by Φ ′ = -a(-λ) -pd 2 ω(x)/p i=0 c pi U ′ 2 pi+1 U ′ 3 ω(x)-pi ∈ k(x)[U ′ 2 , U ′ 3 ] ω(x)+1 .
If ǫ(x ′ ) = ω(x), x ′ thus satisfies all assumptions of Proposition 7.7, so x is good. Otherwise, we have ǫ(x ′ ) = 1 + ω(x) and

k(x ′ )U 1 + Vdir(x ′ ) =< U 1 , U ′ 2 , U ′ 3 > .
Therefore ι(x ′ ) = ι(x) and x is good (if τ (x ′ ) = 3) or x ′ is of type (T1). If x satisfies condition (*2), i.e. c 0 = 0, then x ′ satisfies condition (*3).

Case 2: ω(x) ≡ 0 modp. Recall that F i (U 1 , U 2 ) = 0 whenever ω(x)i ≡ 0 modp. Therefore a := ω(x) = i whenever F i = 0. Let a j := pd j , j = 1, 2. By Lemma 7.3(ii), we have a 1 a 2 = 0, a 1 + a 2 + a = p. Moreover, it can be assumed w.l.o.g. that

U a(1) 1 U a(2) 2 F i (U 1 , U 2 ) = c i Φ i (U 1 , λU 2 ), c i ∈ k(x) p , 1 ≤ i ≤ ω(x), (7.32)
with notations as in (7.8). After blowing up, the initial form in E ′ h ′ of Lemma 7.5 is of the form:

in E ′ h ′ = Z ′ p + U pd ′ 1 1 F ′ (u ′ 2 , u ′ 3 ) ∈ S ′ /(u 1 )[U 1 , Z ′ ],
where

S ′ /(u 1 ) ≃ k(x)[u ′ 2 , u ′ 3 ] (u ′ 2 ,u ′ 3 ) . The form Φ ′ := cl ω(x)+1 F ′ is given explic- itly by Φ ′ = a 2 + a a + 1 ⌊ω(x)/p⌋ i=0 c pi+a U ′ 2 a+pi+1 U ′ 3 ω(x)-a-pi ∈ k(x)[U ′ 2 , U ′ 3 ] ω(x)+1 .
If ǫ(x ′ ) = ω(x), x ′ thus satisfies all assumptions of Proposition 7.7, so x is good. Otherwise, we have ǫ(x ′ ) = 1 + ω(x) and

k(x ′ )U 1 + Vdir(x ′ ) =< U 1 , U ′ 2 , U ′ 3 > .
Therefore ι(x ′ ) = ι(x) and x is good (if τ (x ′ ) = 3) or is of type (T1). Note that x did not satisfy condition (*2): since J(F p,Z , E, m S ) ⊂ k[U 1 , U 2 , U p 3 ] ω(x) and ω(x) ≡ 0 modp, J(F p,Z , E, m S ) contains no monic polynomial in U 3 .

Assume that k = 3. There is an expansion (7.22) with G = 0 and

U -pd 1 1 U -pd 2 2 F p,Z = ω(x) i=0 λ i U ω(x)-i 3 U i 1 .
Assume that ι(x ′ ) ≥ ι(x). By Theorem 3.13, we have x 1 = x ′ , where

x ′ := (Z ′ := Z/u 2 , u ′ 1 := u 1 /u 2 , u 2 , u ′ 3 := u 3 /u 2 ). By Proposition 2.18, ∆ S ′ (h ′ ; u ′ 1 , u 2 , u ′ 3 ; Z ′ ) is minimal and we have in m S ′ h ′ = Z ′ p + U ′ 1 pd 1 U pd ′ 1 2 ( ω(x) i=0 λ i U ′ 3 ω(x)-i U ′ 1 i + U 2 Φ ′ ), where d ′ 1 := d 1 + d 2 -1 + ω(x)/p, Φ ′ ∈ k(x ′ )[U ′ 1 , U 2 , U ′ 3 p ]. since it is assumed that ι(x ′ ) ≥ ι(x). Then ι(x ′ ) = ι(x) and k(x ′ )U 2 + Vdir(x ′ ) =< U ′ 1 , U 2 , U ′ 3 > .
We conclude that τ ′ (x ′ ) = 3 (so x is good) or x ′ is of either type (T2) or (T3).

If moreover x satisfies condition (*), i.e. λ 0 = 0, then x ′ satisfies condition (*2).

Assume that k = 4. We have

H -1 G p ⊆ k(x)U ω(x)+1 1
and there is an expansion (7.22) with

U -pd 1 1 F p,Z = F 1+ω(x) (U 1 , U 3 ) + 1+ω(x) i=1 F 1+ω(x)-i (U 1 , U 3 )U i 2 . (7.33)
Assume that ι(x ′ ) ≥ ι(x). By Theorem 3.13, we have x 1 = x ′ , where

x ′ := (Z ′ := Z/u 2 , u ′ 1 := u 1 /u 2 , u 2 , u ′ 3 := u 3 /u 2 ), E ′ = div(u ′ 1 u 2 ). By Proposition 2.18, ∆ S ′ (h ′ ; u 1 , u ′ 2 , u ′ 3 ; Z ′ ) is minimal. We deduce from (7.33) that F 1+ω(x)-i (U 1 , U 3 ) = 0, 2 ≤ i ≤ 1 + ω(x),
since it is assumed that ι(x ′ ) ≥ ι(x). Since κ(x) = 2, we deduce from Definition 5.1 that

F ω(x) (U 1 , U 3 ) ∈< U ω(x) 1
> .

(7.34)

In particular, we get from (7.33):

ǫ(x ′ ) = ω(x) and Vdir(x ′ ) < U ′ 1 , U 2 > . The initial form polynomial in m S ′ h ′ is therefore given by in m S ′ h ′ = Z ′ p + U ′ 1 pd 1 U pd ′ 2 2 (F ω(x) (U ′ 1 , U ′ 3 ) + U 2 Φ ′ ) (7.35) where d ′ 2 := d 1 + d 2 -1 + (1 + ω(x))/p, Φ ′ ∈ k(x ′ )[U ′ 1 , U 2 , U ′ 3 ]. This proves that ι(x ′ ) = ι(x).
Suppose that x satisfies condition (*3), i.e. F ω(x) (U 1 , U 3 ) is unitary in U 3 . We deduce from (7.35) that x ′ satisfies condition (*2). Otherwise, U 1 divides F ω(x) (U 1 , U 3 ) and we deduce from (7.34) that

k(x ′ )U 2 + Vdir(x ′ ) =< U ′ 1 , U 2 , U ′ 3 > .
for every i ≥ 0. The above claim shows that x r is of type (T1) for every r >> 0. We get x r resolved for some r ≥ 0 arguing as in the above proof of (i), so x is good.

A direct consequence of Proposition 7.11(iii) and Remark 7.9 is: Corollary 7.12. Projection Theorem 5.5 holds when κ(x) = 2 and ω(x) ≡ 0 modp. One may take all local blowing ups in (5.3) permissible of the first kind if p = 2 or if ω(x) ≥ 3.

Remark 7.13. Assume that κ(x) = 2, ω(x) ≡ 0 modp and use notations as in Proposition 7.11. Suppose that x satisfies condition (*1) or (*2) and x 1 satisfies condition (*3). It follows from the above proof that x 1 is resolved or there exist well adapted coordinates (u 1 , u 2 , u 3 ; Z) at x 1 such that

H -1 ∂F p,Z ∂U 2 ≡< Φ(U 2 , U 3 ) > mod(U 1 ) ∩ G(m S ) ω(x) , (7.36)
where

Φ(U 2 , U 3 ) ∈ k(x 1 )[U 2 , U 3 p ]
. This is precisely the definition used by the authors for κ(x) = 2 when ǫ(x) = 1 + ω(x) in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] I.1(ii) on p.1899.

Suppose now that κ(x) = 2, x satisfies condition (*3) and (u 1 , u 2 , u 3 ; Z) are well adapted coordinates satisfying the requirements in Definition 7.10. It also follows from the above proof that x is good or

H -1 ∂F p,Z ∂U 2 =    < U ω(x) 3 > if τ ′ (x) = 1 < F ω(x) (U 1 , U 3 ) > if Vdir(x) =< U 1 , U 3 > .
In particular, (7.36) holds in both cases with < Φ >=< U ω(x) 3

>. We deduce the following: there exists r ≥ 0 such that x r is resolved or for every r >> 0, we have (ι(x r ) = ι(x), x r satisfies condition (*)) and x r satisfies condition ( * 3) =⇒ (7.36) holds at x r .

Namely, otherwise we would have (ι(x r ) = ι(x), τ ′ (x r ) = 2 and x r is of type (T1)) for every r >> 0 by the above. But this implies that x r is resolved for some r ≥ 0 (viz. proof of Proposition 7.11(i) for τ ′ (x) = 2).

This matches the present definition of κ(x) = 2 with that used in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF], and reduces the proof to the same situation (7.36).

Monic expansions: secondary invariants.

Proposition 7.11(i) has reduced the proof of the Projection Theorem to those points with κ(x) = 2 satisfying condition (*). Moreover, we may assume that ω(x) ≡ 0 modp by Corollary 7.12. For such points, we introduce a new invariant γ(x) ∈ N in Definition 7. [START_REF] Cossart | Desingularization of embedded excellent surfaces[END_REF].

We assume in this section and in the following one that ω(x) ≡ 0 modp and x satisfies condition (*).

Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates satisfying the condition in Definition 7.10. If x satisfies condition (*1) or (*2) (resp. condition (*3)), then

v 0 := (b 0 , ω(x) p ), b 0 := (d 1 , d 2 ) (resp. b 0 := (d 1 , 1 p )) (7.37)
is a vertex of ∆ S (h; u 1 , u 2 , u 3 ; Z). Consider the projection from the point v 0 :

p ′ 2 : R 3 \{x 3 = ω(x)/p} -→ A := b 0 + {(x 1 , x 2 , 0), (x 1 , x 2 ) ∈ R 2 }.
We view here A as an affine plane with origin b 0 and coordinates (x 1 , x 2 ). Of course A as a set is independent of our choice of b 0 . Let

p 2 := τ • p ′ 2 , where τ : A -→ A, b 0 + (y 1 , y 2 ) → b 0 + 1 ω(x) p (y 1 , y 2 ).
Analytically, we have:

p 2 : (x 1 , x 2 , x 3 ) → b 0 + (x 1 , x 2 ) -b 0 ω(x) p -x 3 . (7.38)
From now on, we will use affine coordinates in A, i.e. (y 1 , y 2 ) ∈ R 2 represents the point b 0 + (y 1 , y 2 ) ∈ A.

In explicit terms, when a monomial, say

u pd 1 1 u pd 2 2 u ω(x)-i 3 u a 1 1 u a 2 2 , i > 0, defines the vertex x = (d 1 + a 1 p , d 2 + a 2 p , ω(x) -i p ) ∈ ∆ S (h; u 1 , u 2 , u 3 ; Z), we have p 2 (x) = ( a 1 i , a 2 i ) (resp. p 2 (x) = ( a 1 i , a 2 -1 i ))
in cases (*1)(*2) (resp. in case (*3)).

In order to prove the lemma, we must understand the limit points p 2 (x) ∈ ∆ 2 (h; u 1 , u 2 ; u 3 ; Z) when x ∈ ∆ S (h; u 1 , u 2 , u 3 ; Z) tends to the hyperplane {x 3 = ω(x)/p}. By convexity, we have

x ∈ Conv v∈V {v + R 3 ≥0 } .
• Assume that x satisfies condition (*1) or (*2). Let v ∈ V\V -. Since v j ≥ d j , j = 1, 2, and v 3 ≥ ω(x)/p, we have v = v 0 . One deduces immediately that

∆ 2 (h; u 1 , u 2 ; u 3 ; Z) = Conv {p 2 (v) + R 2 ≥0 , v ∈ V -} .
All statements in the lemma follow easily.

• Assume that x satisfies condition (*3). Let a = (a 1 , a 2 , a 3 ) ∈ V -be chosen in such a way that

(α 2 + β 2 , -β 2 ) := a 1 + a 2 -d 1 -1 p ω(x) p -a 3 , -a 2 + 1 p ω(x) p -a 3 (7.42)
is minimal for the lexicographical ordering, viz. (7.38). We now prove [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF]. Let v ∈ V\V -. Since v 3 > 0, Theorem 2.36 implies that

in v h = Z p + λU pv , λ = 0. If v = v 0 , we therefore have v 3 ≥ 1 + ω(x) p or v = v k := (d 1 + k p , 0, ω(x) p ) for some k ≥ 1. (7.43) Let α := ( ω(x) p , ω(x) p , α 2 + β 2 ) ∈ R 3 >0 , L α (x 1 , x 2 , x 3 ) := x 1 + x 2 + (α 2 + β 2 )x 3 .
By (7.42)-(7.43), we have .44) This shows that v 0 , a ∈ σ α , where σ α is the compact face of the polyhedron ∆ S (h; u 1 , u 2 , u 3 ; Z) defined by α. In particular we have proved that

L α (v 0 ) = L α (b 0 ) + ω(x) p (α 2 + β 2 ) = L α (a) L α (v) ≥ L α (v 0 ) if v ∈ {v 0 , a} . ( 7
α 2 + β 2 = B(h; u 1 , u 2 ; u 3 ; Z).
Similarly, let

α ′ := ( ω(x) p α ′ 1 , ω(x) p , α ′ 1 α 2 + β 2 ) ∈ R 3 >0 , where α ′ 1 > 1 is chosen in such a way that L α ′ (v) > L α ′ (a) for every v ∈ V -. Such α ′
1 > 1 exists thanks to the minimal property in (7.42). We now have

L α ′ (v 0 ) = L α ′ (b 0 ) + ω(x) p (α ′ 1 α 2 + β 2 ) = L α ′ (a) L α ′ (v) > L α ′ (v 0 ) if v ∈ {v 0 , a}
and this proves that the line (v 0 a) meets ∆ S (h; u 1 , u 2 , u 3 ; Z) along an edge. This completes the proof of (1), and of (4) when σ 2 = {(α 2 , β 2 )}.

Statement ( 4) is proved along the same lines for arbitrary

σ 2 ⊆ ∆ 2 (h; u 1 , u 2 ; u 3 ; Z) ∩ {y 2 ≥ β 2 }
and we omit the proof. Then (3) is a consequence of (4) because V -is a finite set.

To prove (5) when B(h; u 1 , u 2 ; u 3 ; Z) > 1, note that equality possibly holds in (7.44) only if v = v 1 and the conclusion follows.

If B(h; u 1 , u 2 ; u 3 ; Z) = 1, we have α = (1, 1, 1) with notations as above and σ in is the compact face of ∆ S (h; u 1 , u 2 , u 3 ; Z) generated by σ • . Corollary 7.16. With notations as above, let:

∆ + 2 (h; u 1 , u 2 ; u 3 ; Z) := ∆ 2 (h; u 1 , u 2 ; u 3 ; Z) ∩ {y 2 ≥ β 2 (h; u 1 , u 2 ; u 3 ; Z)}. Then ∆ + 2 (h; u 1 , u 2 ; u 3 ; Z) = Conv {p 2 (x) + R 2 ≥0 , x ∈ S} , where S is the set of vertices x ∈ ∆ S (h; u 1 , u 2 , u 3 ; Z) with 0 ≤ x 3 < ω(x) p and y 2 := (p 2 (x)) 2 ≥ β 2 (h; u 1 , u 2 ; u 3 ; Z).
Taking σ = σ α as in Lemma 7.15(4) or (5), we deduce from Theorem 2.36 that:

in

α h = Z p + F p-1,Z,α Z + F p,Z,α ∈ gr α S[Z].
is a vertex of both ∆ + 2 (h; u 1 , u 2 ; u 3 ; Z) and ∆ + 2 (h; u 1 , u 2 ; û3 ; Ẑ). Let x be the closed point of X and assume that

A 1 (x) > A 1 := min y∈∆ + 2 (h;u 1 ,u 2 ;u 3 ;Z) {y 1 }. (7.47)
Let J := {1, 3} and consider the weight vector α := ( ω(x) p , A 1 ) ∈ R J >0 . We consider the initial form polynomial

in α h = Z p + p i=1 F i,Z,α Z p-i ∈ (gr α S)[Z], where    gr α S = S/(u 1 )[U 1 ] ⊆ gr α Ŝ = Ŝ/(u 1 )[U 1 ] if A 1 = 0 gr α S = S/(u 1 , u 3 )[U 1 , U 3 ] ⊆ gr α Ŝ = Ŝ/(u 1 , u 3 )[U 1 , U 3 ] if A 1 > 0 .
Case 1: A 1 = 0. One deduces from the above algorithm and (7.46) that there exists some ĉ ∈ (u 2 ) Ŝ/(u 1 ) such that

F i, Ẑ,α = ĝi U id 1 1 u d 2,i 2 (u 3 -ĉ) i ω(x) p , 1 ≤ i ≤ p -1 (7.48)
for some ĝi ∈ Ŝ/(u 1 ) (ĝ i = 0 if d 1 ∈ N), d 2,i ≥ id 2 , and

   F p, Ẑ,α = lU pd 1 1 u pd 2 2 (u 3 -ĉ) ω(x) in cases ( * 1) or ( * 2) F p, Ẑ,α = lU pd 1 1 u 2 (u 3 -ĉ) ω(x) in case ( * 3) (7.49)
for some l ∈ Ŝ/(u 1 ) a unit. The regular local ring T := (gr α S) (U 1 ,u 2 ,u 3 ) is excellent and the polynomial in α h ∈ T [Z] satisfies the assumptions of Proposition 2.12. Let

Ξ := Spec(T [Z]/(in α h)), Ξ := Spec( T [Z]/(in α h)). Since v 0 is a nonsolvable vertex of ∆ T (in α h; U 1 , u 2 , u 3 ; Z), we deduce from (7.48)-(7.49) that V := V ( Ẑ, u 3 -ĉ) ⊆ Sing p Ξ ⊆ V ( Ẑ, U 1 u pd 2 2 (u 3 -ĉ)). (7.50)
Since T is excellent, one deduces that the Zariski closure V of V in Ξ is contained in Sing p Ξ. Let P : Ξ -→ SpecT be the projection. By (7.50), P (V ) is an irreducible component of P (Sing p Ξ) contained in div(U 1 u pd 2 2 (u 3ĉ)). Since each of div(U 1 ), div(u 2 ) is Zariski closed, there exist δ′ ∈ Ŝ/(u 1 ) a unit such that u ′ 3 := δ′ (u 3ĉ) ∈ S/(u 1 ). Let u ′ 3 ∈ S be a preimage of u ′ 3 . Applying again Proposition 2.12, there exist well adapted coordinates (u 1 , u 2 , u ′ 3 ; Z ′ ) at x satisfying Definition 7.10 and such that min

y∈∆ + 2 (h;u 1 ,u 2 ;u ′ 3 ;Z ′ ) {y 1 } > A 1 . (7.51)
Case 2: A 1 > 0. The argument runs along the same lines: we now have some ĉ ∈ (u 2 ) Ŝ/(u 1 , u 3 ), (7.50) is replaced by

V ( Ẑ, U 3 -ĉU A 1 1 ) ⊆ Sing p Ξ ⊆ V ( Ẑ, U 1 u pd 2 2 (U 3 -ĉU A 1 1
)), with Ξ as above and (7.51) holds. Applying this procedure and (7.51) finitely many times, it can be assumed w.l.o.g. that A 1 = A 1 (x). When x satisfies condition (*1) or (*2), one introduces similarly

A 2 := min y∈∆ 2 (h;u 1 ,u 2 ;u 3 ;Z) {y 2 } ≤ min y∈∆ + 2 (h;u 1 ,u 2 ;u 3 ;Z) {y 2 } = β 2 (h; u 1 , u 2 ; u 3 ; Z).
The same argument shows that there exist well adapted coordinates (u 1 , u 2 , u 3 ; Z) at x satisfying Definition 7.10 and well 2-adapted coordinates (u 1 , u 2 ; û3 ; Ẑ) of X = Spec( Ŝ[X]/(h)), Ẑ = Z -φ, such that

A j := min y∈∆ 2 (h;u 1 ,u 2 ;u 3 ;Z) {y j } = min y∈∆ + 2 (h;u 1 ,u 2 ;û 3 ; Ẑ) {y j }, j = 1, 2. (7.52)
Finally, if x satisfies condition (*1) or (*2) (resp. (*3)), (7.52) (resp. (7.51)) proves that the region ∆ 2 (h; u 1 , u 2 ; u 3 ; Z)\∆ 2 (h; u 1 , u 2 ; û3 ; Ẑ) ⊆ R 2 ≥0 (resp. ∆ + 2 (h; u 1 , u 2 ; u 3 ; Z)\∆ + 2 (h; u 1 , u 2 ; û3 ; Ẑ)) is bounded. Therefore the above algorithm and (7.46) can repeat only finitely many times. This proves the existence of well 2-adapted coordinates for arbitrary S.

Let then (u 1 , u 2 ; u 3 ; Z) be well 2-adapted coordinates and define the curve Y := V (Z, u 1 , u 3 ) ⊂ X . By Proposition 2.12, the polyhedron

∆ Ŝ (h; u 1 , u 3 ; Z) = pr {1,3} ∆ Ŝ (h; u 1 , u 2 , u 3 ; Z)
is minimal and we have

ǫ(y) = ω(x) × min{1, A 1 (x)}. ( 7
(y) ≥ ǫ(y) = ω(x) ≥ p. By (7.53), we have ǫ(y) = ǫ(x) -1. Suppose that i 0 (y) = p -1. Let W := η(Y), so we have in W h = Z p -G p-1 W Z + F p,W,Z ∈ G(W )[Z] with δ(y) ∈ N, G W = g W U δ(y) 1 and 0 = cl p(p-1)δ(y) Disc Z h =< g p(p-1) W U p(p-1)δ(y) 1 >∈ G(W ) p(p-1)δ(y)
by Theorem 2.36. Since E = div(u 1 ), g W ∈ S/(u 1 , u 3 ) is a unit by assumption (E). We then get

ǫ(x) ≤ ord m S (H(x) -(p-1) f p p-1,Z ) p -1 = ǫ(y) = ǫ(x) -1,
a contradiction. Therefore Definition 3.5(ii) is satisfied because i 0 (y) = p. Finally it follows from Definition 7.10(ii) that Definition 3.5(iii) is satisfied.

The previous theorem shows that the following invariants are actually independent of the choice of well 2-adapted coordinates.

It follows from Definition 7.17 that y ′ is not 2-solvable, since y is not. The lemma follows easily. Proposition 7.21. Assume that κ(x) = 2 and x satisfies condition (*). If γ(x) = 0, then x is good.

Proof. By Theorem 7.18, there exist well 2-adapted coordinates (u 1 , u 2 ; u 3 ; Z) at x. The assumption γ(x) = 0 means that (x is in case (*1) and β(x) = 0) or (x is in case (*3) and β(x) < 0).

Assume that x is in case (*1). We have

∆ 2 (h; u 1 , u 2 ; u 3 ; Z) = (A 1 (x), 0) + R 2 ≥0 .
Since B(x) ≥ 1 (viz. (7.39)), we have A 1 (x) ≥ 1.

Assume that x is in case (*3). We have

∆ + 2 (h; u 1 , u 2 ; u 3 ; Z) = (A 1 (x), β(x)) + R 2 ≥0 in this case. Note that we have A 1 (x) ≥ 1: namely, β(x) = -1/i for some i, 1 ≤ i ≤ ω(x) such that ǫ(x) = 1 + ω(x) ≤ iA 1 (x) + ω(x) -i + 1, so A 1 (x) ≥ 1.
Suppose that 1 ≤ A 1 (x) < 2. By Lemma 7.20, x is good or x ′ satisfies again the assumption of the proposition with A 1 (x ′ ) = A 1 (x) -1 < 1: a contradiction with the previous remark. Induction on ⌊A 1 (x)⌋ concludes the proof.

Monic expansions: blowing up a closed point.

In this section, we control the behavior of the secondary invariant γ(x) (Definition 7.19) by blowing up a closed point. By Proposition 7.21 we may furthermore assume that γ(x) ≥ 1. At this point, we connect the proof with the equal characteristic proof given in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 3. Namely, this control is considered in Lemmas I.8.3 and I.8.8 (resp. Lemmas I.8.7 and I.8.9) [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 3 when x satisfies condition (*1) or (*2) (resp. condition (*3)). The proof relies on the definition of the form

in α h = Z p -G p-1 α Z + F p,Z,α ∈ (gr α S)[Z]
in Lemma 7.15(4)(5) w.r.t. the initial face σ 2,in of ∆ 2 (h; u 1 , u 2 ; u 3 ; Z), where (u 1 , u 2 ; u 3 ; Z) are well 2-adapted coordinates at x. notations used in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF]. The corresponding notation for F p,Z,α is

F p,Z,α = U a(1) 1 U a(2) 2 φ 0 U ω(x) 3
+ j∈J 0 U ω(x)-j 3 Φ j (U 1 , U 2 ) ( 7
a(j) = pd j , j = 1, 2, 0 = φ 0 ∈ k(x) and Φ j (U 1 , U 2 ) ∈ k(x)[U 1 , U 2 ].
By Definition 7.17, we have Φ j (U 1 , U 2 ) = 0 for some j 0 = 0. When x satisfies condition (*3), the notation is the same except that φ 0 and Φ j (U 1 , U 2 ) are replaced respectively by

U 2 φ 0 , φ 0 ∈ U -1 2 k(x)[U 1 , U 2 , U 3 ] 1 , and by U 2 Φ j (U 1 , U 2 ) with Φ j (U 1 , U 2 ) ∈ U -1 2 k(x)[U 1 , U 2
] (Definition I.8.6.1). We have a(2) = 0 in these formulae in cases (*1) and (*3).

Similarly, the corresponding notation for G α is

G p α = U a(1) 1 U a(2) 2 cl B(x)ω(x) (H(x) -1 g p ) (7.55)
when x satisfies condition (*1) or (*2). When x satisfies condition (*3), we have

G p α = U a(1) 1 cl 1+B(x)ω(x) (H(x) -1 g p ). ( 7

.56)

The numerical invariants β(x) and B(x) are denoted respectively by β3(x) and B3(x) in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] when x satisfies condition (*3). The statement "κ(x) ≤ 1" in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] stands for "x is resolved" in this article. The vector spaces cl µ 0 ,ω(x) J ([30] Definitions I.8.2.3 and I.8.6.3) are determined by the initial form polynomial in α h. The proofs of the following lemmas are almost entirely based on the numerical Lemmas I.8.2.2 and I.8.6.2 in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] which are characteristic free. We simply refer to their counterpart in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] except when they do not immediately adapt to our characteristic free setting.

Assume that (κ(x) = 2, x satisfies condition (*) and γ(x) ≥ 1). Let π : X ′ -→ X be the blowing up along x and x ′ ∈ π -1 (x). We denote by

d := [k(x ′ ) : k(x)].
Proof. We already know from Proposition 7.11(ii) that x ′ is resolved or (κ(x ′ ) = 2 and x ′ satisfies condition (*)). Note that we have

B(x) > 1 ⇔ τ ′ (x) = 1.
Namely, we have < U 3 >⊆ Vdir(x) by Definition 7.10, so

τ ′ (x) = 1 ⇔ H -1 F p,Z ∈< U ω(x) 3 >⇔ B(x) > 1,
where the left hand side equivalence is true because ∆ S (h; u 1 , u 2 , u 3 ; Z) is minimal.

If B(x) = 1, then x is of type (T0), (T2) or (T3) as defined along the proof of Proposition 7.11. What follows has been proved in the course of that proof: for type (T0), x is good; for type (T3), x ′ is resolved by Theorem 3.13 since Vdir(x) =< U 3 , U 1 >; for type (T2), x is good or (

d 1 + d 2 ∈ N, d 2 ∈ N, B(x) = C(x) = 1)
. In this situation, we have κ(x ′ ) = 2, x ′ satisfies condition (*) and there exist well 2-adapted coordinates (u ′ 1 , u ′ 2 ; u ′ 3 ; Z ′ ) at x ′ such that A 1 (x ′ ) = 0 and one of the following holds:

• x ′ is in case (*1) β(x ′ ) = i + 1 i , i ≡ 0 modp, p ≤ i ≤ ω(x); (7.58)
• x ′ is in case (*1) and

β(x ′ ) = ω(x) ω(x) -1 ; (7.59) • x ′ is in case (*3) and β(x ′ ) = 1.
See the discussion in the proof of Proposition 7.11: these three situations correspond respectively to I = {0}, I = {ω(x)} and I = ∅ therein. When (7.59) holds with ω(x) = p, we have (3'); otherwise, we have (3)(4). Note that γ(x ′ ) = γ(x) = 2 here.

If B(x) > 1, statement (1) is easily deduced from the characteristic free Proposition 2.18 as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF]. The rest of the proof relies on the characteristic free transformation formula [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF](4) on p.1918 and numerical Lemma I.8.2.2 and is identical to that of I.8.3(1)(2)(ii)(iv)-(vi). If x ′ satisfies condition (*3), note that (4) is an equivalent formulation of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma I.8.3 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF].

If B(x) > 1, the proof is identical to that of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma I.8.7(b)(b')(d)(i)-(iii)(v): this relies on the numerical Lemma I.8.6.2 and characteristic free transformation formula for cl µ 0 ,ω(x) J (Definition I.8.6.3). As observed before stating this lemma, a mistake in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] I.8.7.8 (case 2, B(x) ∈ N) has to be amended at this point. Namely, the bounds (3)(4) on p.1929 only hold when G = µ -1 2 ∂F ∂U 2 = 0 with notations as in there. The correct bounds are thus no better than those given in I.8.7.8 case 3:

β(x ′ ) ≤ deg U 2 Ψ j 1 (U 1 , U 2 ) j 1 d + 1 p , β3(x ′ ) ≤ deg U 2 Ψ j 1 (U 1 , U 2 ) j 1 d + 1 p - 1 p a , (7.65)
where a := ord p ω(x): this gives (1) of the present lemma. We note however that the bounds (3)(3')(4)(4') on p.1929-1930 are correct if d ≥ 2 (this relies on Lemma 6.7(2), statement "d = 1 if equality holds"). This proves that γ(x ′ ) ≤ γ(x) if k(x ′ ) = k(x). There remains to prove (2) and (3) (resp. ( 3)) of the present lemma for d = 1 (resp. for d ≥ 2).

First assume that d ≥ 2, i.e. k(x ′ ) = k(x). The conclusion follows trivially from (1) if β(x) ≥ 1, so we may assume that β(x) < 1.

The proof involves picking some element G ∈ cl µ 0 ,ω(x) , G = 0 [30] middle of p. 1930 and computing the order of its transform. This is done after possibly performing the Tschirnhausen transformation described in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] I.8.3.6. We discuss according to the set J 0 in (7.61):

Case 1: J 0 pN. Arguing as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] I.8.7.7, we get

β(x ′ ) ≤ deg U 2 Ψ j 1 (U 1 , U 2 ) j 1 d - 1 j 1 < β(x) d .
Case 2: J 0 ⊆ pN and B(x) ∈ N. By [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] (4) on p.1930, we get . The bound is:

β(x) -β(x ′ ) ≥ 1 - 1 d β(x) - 1 pd > 1 p 1 - 2 d ≥ 0.
β(x ′ ) ≤ deg U 2 Ψ j 1 (U 1 , U 2 ) j 1 d
as in case 2 with the same conclusion.

Assume that k(x ′ ) = k(x). By the independence statement in Theorem 7.18, it can be assumed that x ′ is the origin of the chart. We build upon (7.60) and connect the proof with [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] I.8.7.5. First note that x ′ satisfies condition (*3) if and only if µ = 0, since ∆ S (h; u 1 , u 2 , u 3 ; Z) is minimal. In this situation one gets easily β(x ′ ) ≤ β(x) from Proposition 2.18 as in case 3 of [30] I.8.7.5. This completes the proof when x ′ satisfies condition (*3). Assume now that x ′ satisfies condition (*1), so c = 0 in (7.60). Note to begin with that we have

deg U 2 Ψ j (U 1 , U 2 ) -1 j ≤ β(x) =⇒ deg U 2 Ψ j (U 1 , U 2 ) j ≤ γ(x) (7.68)
for each j ∈ J 0 in (7.60). We again consider the same cases 1 to 4 as for k(x ′ ) = k(x): Case 1: J 0 pN. Arguing as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] I.8.7.7, we get

β(x ′ ) ≤ deg U 2 Ψ j 1 (U 1 , U 2 ) j 1 ≤ γ(x).
Case 2: J 0 ⊆ pN and B(x) ∈ N. Same as in case 1 by [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] (3') on p.1929.

Case 3: J 0 ⊆ pN, B(x) ∈ N and G = U -pd Lemma 7.27. With notations as above, assume that x r satisfies condition (*2) for every r ≥ 0. Then there exists r 0 ≥ 0 such that C(x r ) = 0 for every r ≥ r 0 .

Proof. We consider the points y := (A 1 (x), A 2 (x) + a(x)), y ′ := (A 1 (x) + a ′ (x), A 2 (x)) ∈ ∆ 2 (u 1 , u 2 ; u 3 ; Z), where (u 1 , u 2 ; u 3 ; Z) are well 2-adapted coordinates. By standard arguments on combinatorial blowing ups, we have c(x 1 ) < c(x) for the lexicographical ordering whenever C(x) > 0, where c(x) := (C(x) = min{a(x), a ′ (x)}, max{a(x), a ′ (x)}).

Since these numbers belong to 1 ω(x)! N 2 , we get C(x r ) = 0 for all r >> 0. Proposition 7.28. With notations as above, there exists r 0 ≥ 0 such that x r 0 is resolved or γ(x r 0 ) ≤ 2.

Proof. Let (u 1 , u 2 ; u 3 ; Z) be well 2-adapted coordinates at x. We will name point "at infinity" for simplicity the origin x ′ of the second chart of the blowing up, i.e.

x ′ := (Z/u 2 , u 1 /u 2 , u 2 , u 3 /u 2 ). (7.73)

The notion is unambiguous if E = div(u 1 ), that is if x satisfies condition (*1) or (*3). If x satisfies condition (*2), the point "at infinity" furthermore depends on the numbering of u 1 , u 2 , where E = div(u 1 u 2 ). We may assume that γ(x) ≥ 3 for the whole proof. Note that the special situations described in Lemma 7.22 (3') and in Lemma 7.24(1') occur only when γ(x) ≤ 2. We may thus disregard them in this proof. To prove the proposition, it is sufficient to prove that there exists r ≥ 1 such that x r is resolved or γ(x r ) < γ(x). We first bound γ(x 1 ) in terms of γ(x) at a nonrational point or at a point "at infinity". Assume that k(x 1 ) = k(x). We apply Lemma 7.22 [START_REF] Abhyankar | Ramification theoretic methods in algebraic geometry[END_REF] Assume that x 1 = x ′ and k(x 1 ) = k(x). If x satisfies condition (*1) or (*3), the independence statement in Theorem 7.18 shows that we may actually assume that x 1 = (Z/u 1 , u 1 , u 2 /u 1 , u 3 /u 1 ). If x is in case (*1), then x 1 is resolved or satisfies again condition (*1) with β(x 1 ) ≤ β(x) by Lemma 7.22 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF].

If x is in case (*3), then x 1 is resolved or satisfies one of conditions (*1) or (*3). In the latter case, we have β(x 1 ) ≤ β(x) by Lemma 7.24(3); in the former case, we have γ(x 1 ) ≤ γ(x) except if "x satisfies the assumptions of Lemma 7.24(2)". (7.77) This situation occurs only when β(x) = γ(x) -1/ω(x) and gives β(x 1 ) = γ(x) + 1/ω(x), γ(x 1 ) = γ(x) + 1.

We first prove the proposition when x satisfies either condition (*1) or (condition (*3) with β(x) < γ(x) -1/ω(x)). By the above considerations, we are done except possibly if x 1 satisfies again condition (*1) or (*3) with Up to the end of this chapter, "resolved" stands for "resolved for (p, ω(x), 3)" (Remark 5.7). Definition 8.1. (Monic expansion for κ(x) ≥ 3). Assume that κ(x) ≥ 3. We say that x satisfies condition (**) if there exists well adapted coordinates (u 1 , u 2 , u 3 ; Z) at x such that the following conditions are fulfilled: (i) 1 + ω(x) = 0 mod(p);

(ii) E = div(u 1 ) (resp. E = div(u 1 u 2 )), and v := (d 1 , d 2 , (1 + ω(x))/p) is the only vertex (resp. is a vertex) of ∆ S (h; u 1 , u 2 , u 3 ; Z) in the region x 1 = d 1 , with the usual convention d 2 = 0 when div(u 2 ) ⊂ E.

Assume κ(x) = 4, we say that x satisfies condition (T**) (for "towards (**)") if there exists well adapted coordinates (u 1 , u 2 , u 3 ; Z) at x such that one of the following conditions is fulfilled: (i) ǫ(x) = ω(x), div(u 1 ) ⊆ E and Vdir(x) =< U 1 >;

(ii) ǫ(x) = ω(x), div(u 1 u 2 ) ⊆ E and v := (d 1 + ω(x)/p, d 2 , d 3 ) is the only vertex of ∆ S (h; u 1 , u 2 , u 3 ; Z) in the region x 2 = d 2 ;

(iii) E = div(u 1 u 2 ) and v := (d 1 + ω(x)/p, d 2 , 1/p) is the only vertex of ∆ S (h; u 1 , u 2 , u 3 ; Z) in the region x 2 = d 2 .

When x satisfies any of (**) or (T**), we simply say that "h has a monic expansion for (u 1 , u 2 , u 3 ; Z)". In cases (**) and (T**)(iii), the nonexceptional variable u 3 will usually be denoted v. Remark 8.2. First, let us remark that, by Definitions 2.68 and 5.1,

κ(x) = 3 ⇒ H -1 ∂F p,Z ∂U 3 ∈ k(x)[U 1 , U 2 ], (8.2)
Indeed, this is clear when E =div(u 1 u 2 ). When E =div(u 1 ), then Vdir(x) ⊂< U 1 , U 3 > and H -1 ∂F p,Z ∂U 2 ⊂< U ω(x) 1

>, (8.2) follows. If x satisfies (i)(ii) or ((iii) with ǫ(x) = ω(x)) above for (T**), we have κ(x) ≤ 2 or κ(x) = 4. On the other hand, one may have (iii) with κ(x) = 3 if ǫ(x) = 1 + ω(x). We however claim that τ ′ (x) = 3 in this situation. Namely, W.l.o.g. it can be assumed that U 3 ∈ Vdir(x). By (iii), we then have As a consequence, it is sufficient for our purpose to check (i)(ii) or (iii) in order to check (T**), since x is already resolved if κ(x) ≤ 3.

8.1 Preliminaries: transverse case. Proof. Take Y 0 := {x} in (8.1) and assume that x 1 is very near to x. Since U 1 ∈ Vdir(x), we have G = 0. Let u ′ j := u j /u 1 , j = 2, 3. By Theorem 3.13, we have x 1 = (X ′ := Z/u 1 , u 1 , v := u ′ 2 + γ 1 , u ′ 3 ), E ′ = div(u 1 ), k(x 1 ) = k(x), Lemma 8.5. Assume that κ(x) = 3. Then x is good, or there exist well adapted coordinates (u 1 , u 2 , u 3 ; Z) at x and an expansion (8.3) such that one of the following properties holds.

( (2) we have E = div(u 1 u 2 ), τ ′ (x) = 1 and x satisfies condition (**) (Definition 8.1).

Proof. We always take Y 0 := {x} in (8.1) and discuss according to x 1 . It can be assumed that ι(x 1 ) ≥ ι(x) (in particular ω(x 1 ) = ω(x)).

First suppose that x 1 = x ′ . By Proposition 2.18, (u ′ 1 , u 2 , u ′ 3 ; Z ′ ) are well adapted coordinates at x ′ . Since ǫ(x ′ ) ≥ ω(x) by assumption, we deduce that deg U 2 Φ i+1 ≤ 1, 0 ≤ i ≤ ω(x). Similarly, Φ i+1 ∈ k(x)[U 1 ] for ω(x)i ≡ 0 modp (resp. for ω(x)i ≡ 0 modp, i = ω(x)) because ω(x ′ ) = ω(x) (resp. because κ(x ′ ) > 2). Therefore (8.13) holds if ι(x ′ ) ≥ ι(x).

Assume now that x 1 = x ′ . By Theorem 3.13, x 1 is resolved if < U 1 , U 3 >⊆ Vdir(x).

If (E = div(u 1 u 2 ) and τ ′ (x) = 2), it can thus be assumed by symmetry on u 1 , u 2 that Vdir(x) =< U 3 , λ 1 U 1 + U 2 >, λ 1 = 0. Then x is resolved by Proposition 8.3. Since x 1 is very near to x, it can be assumed from now on that Vdir(x) =< U 3 > . (8.14) We get in (8.3): G = 0 and Φ i+1 = 0 for ω(x)i ≡ 0 modp. By (8.4), we furthermore have c = 0 and ω(x) + 1 ≡ 0 modp. (8.15) If E = div(u 1 u 2 ), we therefore have (2) and the proof is complete.

Since ι(x 1 ) ≥ (p, ω(x), 3) is assumed, (8.26) reads

U -pd ′ 1 1 ( ∂F p,X ′ ,W ′ ∂v , ∂F p,X ′ ,W ′ ∂w ) = (w ω(x) )
when E ′ = div(u 1 ). If (8.26) is achieved by ∂ ∂v , we then have ǫ(x 1 ) = ω(x) and x 1 satisfies the assumptions of Lemma 7.1; hence x 1 is resolved. Otherwise (8.26) gives

U -pd ′ 1 1 V -pd ′ 2 F p,Z ′ ,W ′ = (w 1+ω(x) ),
for E ′ = div(u 1 ) (so d ′ 2 = 0) or E ′ = div(u 1 v). This proves that x 1 satisfies condition (**).

• Assume that k = 1. By Theorem 3.13 and (8.22), we have

x 1 = (X ′ := Z/u 1 , u 1 , v := u ′ 2 + γ 1 , w := u ′ 3 + γ 2 ), E ′ = div(u 1 )
where γ 1 , γ 2 ∈ S are preimages of λ 1 , λ 2 . Assume that ǫ(x 1 ) = ω(x). As κ(x 1 ) > 2, d ′ 1 ∈ N, by Proposition 3.9(v), x 1 satisfies the assumptions of Lemma 8.4 and the conclusion follows.

Assume now that ǫ(x 1 ) = 1 + ω(x). Let (u 1 , v, w; Z ′ ) be well adapted coordinates at x 1 . By Proposition 3.9(v) and (8.22), we have Vdir(x 1 )+ < U 1 >=< U 1 , V, W > . This is a contradiction with Definition 5.1, since κ(x 1 ) ≥ 3 by assumption.

• Assume that k = 2. By Theorem 3.13 and (8.23), we have

x 1 = (X ′ := Z/u 1 , u 1 , v := u ′ 2 + γ 1 , u ′ 3 ), E ′ = div(u 1 u ′ 3 ), k(x 1 ) = k(x),
where γ 1 ∈ S is a preimage of λ 1 . By assumption, there exists

Φ := ω(x) i=0 Φ i (U 1 , U 2 )U ω(x)-i 3 ∈ J(F p,Z , E, m S )
with Φ i ∈ k(x)[U 1 , U 2 ] i and Φ ω(x) = c(λ 1 U 1 + U 2 ) ω(x) , c = 0. Applying Proposition 3.9(v) (with W ′ := div(u 1 ) ⊂ SpecS ′ ), we have We thus deduce that m(x 1 ) ≤ 1 + ω(x) < p, hence x 1 is resolved.

(Φ(1, v -λ 1 , u ′ 3 )) ⊆ J(F p,Z,W ′ , E ′ , W ′ ) ⊆ k(x)[u ′ 2 , u ′ 3 ] (v,u ′ 3 ) . ( 8
• Assume that k = 3. If ω(x) < p, we may assume to begin with that δ(x) = 1 arguing as in (8.28) sqq. Let

x ′ := (Z ′ := X/u 3 , v 1 := u 1 /u 3 , v 2 := u 2 /u 3 , u 3 ), E ′ := div(v 1 v 2 u 3 ).

First assume that x 1 = x ′ . We have x 1 = (Z/u 1 , u 1 , v := u ′ 2 + γ 1 , w := P (u ′ 3 )),

where γ 1 ∈ S is a preimage of λ 1 and P (t) ∈ S[t] is a unitary polynomial whose reduction P (t) ∈ k(x)[t] is irreducible. Let (u 1 , v, w; Z ′ 1 ) be well adapted coordinates. Applying Proposition 3.9(v) (with W ′ := div(u 1 ) ⊂ SpecS ′ ), we have

J(F p,Z ′ 1 ,W ′ , E ′ , W ′ ) = (v ω(x) ) ⊆ k(x 1 )[u ′ 2 , u ′ 3 ] (v,w) . (8.29)
The conclusion follows as for type (T0): x 1 satisfies condition (**) or x 1 is resolved by Lemma 7.1. The latter holds if ω(x) < p. Assume now that x 1 = x ′ . By Proposition 2.18, (v 1 , v 2 , u 3 ; Z ′ ) are well adapted coordinates at x ′ . We deduce that ǫ(x ′ ) = ω(x). Furthermore, (8.24) implies that

J(F p,Z ′ , E ′ , m S ′ ) ≡< (λ 1 V 1 + V 2 ) ω(x) > mod(U 3 ) ∩ G(m S ′ ) ǫ(x ′ ) .
(8.30)

Suppose that Vdir(x ′ ) is not skew. By (8.30), we have τ ′ (x ′ ) = 3, hence x ′ is resolved.

We prove that Proposition 9.1 holds in this situation.

If β(x) > 0, we have Vdir(x) =< U 1 > and get ι(x ′ ) ≤ (p, ω(x), 2) after blowing up, so x is resolved by blowing up along Y.

If β(x) ≤ 0, we blow up along x. By Proposition 9.3 below (proof in case (T**)(ii)), we get x ′ resolved or (x ′ satisfies condition (**) with ω(x) ≥ p) except if x ′ = (Z/u 3 , u ′ 1 := u 1 /u 3 , u ′ 2 := u 2 /u 3 , u 3 ) is the point on the strict transform Y ′ of Y, E ′ = div(u ′ 1 u ′ 2 u 3 ). We now have Vdir(x ′ ) =< U ′ 1 , U ′ 2 > or Vdir(x ′ ) =< λ 1 U ′ 1 + U ′ 2 >, λ 1 = 0. Blowing up along Y ′ gives x ′′ resolved or (x ′′ satisfies (**) with ω(x) ≥ p), arguing as in the proof of Proposition 9.4 below, assumption (2). Case (ii): Y = V (Z, u 1 , u j ), E = div(u 1 u 2 u 3 ), j = 2 or j = 3. Assumption (1) (resp. assumption (2)) of Proposition 9.4 is equivalent to A j (x) > 1 (resp. to: j = 3 and A 2 (x) > 0). By symmetry, there remains to deal with the case Y = V (Z, u 1 , u 3 ) with A 2 (x) = 0, A 3 (x) = 1. There is an expansion ord (u 2 ,u 3 ) f i i = C(x) < 1, since γ(x) = 1 is assumed here. We consider two cases: C(x) > 0 and C(x) = 0, arguing as above in case (i)(iii): C(x) > 0 corresponding to β(x) > 0, C(x) = 0 to case β(x) = 0 (i). Blowing up along Y, we get respectively x resolved; x ′ resolved or (x ′ satisfies (**) with ω(x) ≥ p). Proposition 9.1 holds in any case.

u -pd 1 1 u -pd
This proposition leads to:

Corollary 9.2. Assume that ω(x) < p and either κ(x) = 4, or (κ(x) = 3 and τ ′ (x) = 2). Then x is resolved.

Proof. Indeed, by Propositions 8.9 and 8.11, there exists an independent sequence of local blowing ups (8.1) along µ such that x r is resolved or x r satisfies condition (T**). In the last case, apply Proposition 9.1.

If Q = 0, the reader sees that x ′ satisfies (T**)(ii) or (T**)(iii) if ι(x ′ ) ≥ (p, ω(x), 3). The difficult case is Q = 0. By (8.20), we have

V (T F p,Z , E, m S ) = H -1 ∂T F p,Z ∂U 3 ⊆< U ω(x) 1
> .

This gives ∂Q ∂U 3 = 0, i.e. Q ∈ k(x)[U 1 , U 2 , U p 3 ] in both cases G = 0 and G = 0. Expand again

Q = i 0 i=0 U ω(x)-i 1 Q i (U 2 , U p
3 ), Q i 0 (U 2 , U p 3 ) = 0. (9.5)

If i 0 = 0, we reduce to Q = 0 after possibly picking new well adapted coordinates (u 1 , u 2 , v; Z ′ ) at x.

If i 0 ≥ 1, we apply Proposition 3.9(v) to those elements of J(F p,Z , E, m S ) of the form:

U -pd 1 1 U -pd 2 2 D • F p,Z = λ D U 3 U ω(x) 1 + U 2 i 0 i=0 U ω(x)-i 1 Q i,D (U 2 , U p 3 ),
where D ∈ {U 3 )), we get ω(x ′ ) ≤ ω(x) with strict equality if k(x ′ ) = k(x). If k(x ′ ) = k(x), it can be assumed w.l.o.g. that x ′ = (Z/u 2 , u 1 /u 2 , u 2 , u 3 /u 2 ). Then ι(x ′ ) ≤ (p, ω(x), 2) by (9.4)-(9.5) and the conclusion follows. Case 2: Vdir(x) =< λ 1 U 1 + U 2 >, λ 1 = 0. We now have G = 0 and expand

U -pd 1 1 U -pd 2 2 F p,Z = U 3 (λ 1 U 1 + U 2 ) ω(x) + U 2 Q, Q ∈ k(x)[U 1 , U 2 , U p 3 ] ω(x) .
If Q = 0, as H -1 ∂T F p,Z ∂U 3 ⊆< U ω(x) 1

>, we expand again

Q = i 0 i=0 U pi 3 Q ω(x)-i (U 1 , U 2 ), Q ω(x)-i 0 (U 1 , U 2 ) = 0.
Since (u 1 , u 2 , u 3 ; Z) are well adapted coordinates, we have

U pd 1 1 U pd 2 +1 2 Q ω(x)-i 0 (U 1 , U 2 ) ∈ G(m S ) p .
If i 0 = 0, we argue as in the proof of Proposition 8.11, cf. (8.41) sqq.: after possibly picking new well adapted coordinates (u 1 , u 2 , v; Z ′ ) at x, it can be assumed that U 1 divides Q = Q ω(x) [U 1 , U 2 ]. We get ω(x ′ ) < ω(x) if Q = 0; if Q = 0, we obtain ι(x ′ ) ≤ (p, ω(x), 2) or x ′ satisfies the assumptions of Lemma 7.1 (Lemma 7.2 if ω(x) = 1), so x ′ resolved. In particular, the proof is complete if ω(x) < p.

If i 0 ≥ 1, arguing as in case 1, we obtain ω(x ′ ) < ω(x) except possibly if k(x ′ ) = k(x) and a(1) := pd 1 , a(2) := pd 2 + 1, a(3) := 0,

F 0 := Q ω(x)-i 0 [U 1 , U 2 ]
satisfies the assumptions of Lemma 7.3 with λ = λ -1

1 . Then it can be assumed w.l.o.g. that x ′ = (Z/u 1 , u 1 , γ 1 + u 2 /u 1 , u 3 /u 1 ), where γ 1 ∈ S is a unit with residue λ 1 . We obtain ι(x ′ ) ≤ (p, ω(x), 2) or x ′ satisfies the assumptions of Lemma 8.4. Then x ′ is resolved and this concludes the proof. Proposition 9.4. Let x be in the case (T**) of Definition 8.1 and Y ⊂ (X , x) be a permissible curve of the first kind, η(Y) ⊂ div(u 1 ), with generic point y. Let π : X ′ -→ (X , x)

be the blowing up along Y and x ′ ∈ π -1 (x), ι(x ′ ) ≥ (p, ω(x), 3). Assume furthermore that one of the following extra assumptions holds:

(1) Vdir(y) =< U 1 >;

(2) Y = V (Z, u 1 , u 3 ) and x satisfies (T**)(ii) or (iii),

where (u 1 , u 2 , u 3 ; Z) are well adapted coordinates. Then one of the following holds:

(i) x ′ is resolved, or (x ′ satisfies (**) with ω(x) ≥ p);

(ii) x ′ maps to the strict transform of div(u 1 ) and satisfies (T**).

Proof. As Y has normal crossings with E, we can choose in any case well adapted coordinates (u 1 , u 2 , u 3 ; Z) at x such that Y = V (Z, u 1 , u i ), i = 2 or i = 3. Let us see the case where Y = V (Z, u 1 , u 2 ), up to renumbering u 2 , u 3 . As Y is a permissible curve of the first kind, we have, with the usual convention d 3 = 0 when div(u 3 ) ⊂ E:

U -pd 1 1 U -pd 2 2 U -pd 3 3 F p,Z ∈ k(x)[U 1 , U 2 ] ǫ(x)
• if τ ′ (x) = 2, let also X ′ → (X , x) be the blowing up along x. W.l.o.g. we have Vdir(x) =< U 1 + α 1 U 3 , U 2 + α 2 U 3 >, α 1 , α 2 ∈ k(x), (9.8) where div(u 1 u 2 ) ⊆ E, and E = div(u 1 u 2 u 3 ) if (α 1 , α 2 ) = (0, 0). As H(x) = (1),

< { ∂F p,Z ∂λ l } l∈Λ 0 >⊆ k(x)[U 1 + α 1 U 3 , U 2 + α 2 U 3 ], (9.9)
where (λ l ) l∈Λ 0 is an absolute p-basis of S/m S (see beginning of section ??). By Theorem 3.13, we have k(x 1 ) = k(x) and x 1 has for parameters (Z/u 3 , u 1 /u 3 + γ 1 , u 2 /u 3 + γ 2 , u 3 ) where γ i has residue α i ∈ S, i = 1, 2. By Proposition 3.9(v),

U -p 3 ∂F p,Z ∂λ l ∈ J(F p,Z ′ ,W , E 1 , W ), l ∈ Λ 0 , (9.10)
where W =div(u 3 ). If α 1 α 2 = 0, as κ(x 1 ) > 2, we therefore have We now assume that α 1 = 0. If α 2 = 0, we derive a contradiction in a similar way: by (9.10), the coefficient of degree 0 in U 1 in F p,Z must be zero; Lemma 7.3(ii) applied to the term of minimal degree d in U 1 of F p,Z gives again a contradiction, since 0 < d < p. This proves that Vdir(x) =< U 1 , U 2 >, F p,Z ∈ k(x)[U 1 , U 2 ] modG(m S ) p . By Proposition 2.18, we have δ(x 1 ) = 1 and may iterate. By Proposition 3.17, this process is finite or the curve Y := V (Z, u 1 , u 2 ) is permissible of the first kind. Since Vdir(x) =< U 1 , U 2 >, blowing up along Y then completes the proof.

• if τ ′ (x) = 1, it can be assumed that (9.7) has the form inh = Z p + λ(U 1 + α 2 U 2 + α 3 U 3 ) p , λ ∈ k(x) p , (9.11)

with div(u 1 ) ⊆ E, and div(u j ) ⊆ E if α j = 0, j = 2, 3. If α 2 α 3 = 0, let X ′ → (X , x) be the blowing up along x. We get a contradiction with κ(x 1 ) ≥ 3 unless λ ∈ k(x 1 ) p ; but then δ(x 1 ) = 1 implies that x 1 satisfies the assumptions of Lemma 7.1 from which the conclusion follows. We now assume that α 3 = 0.

If α 2 = 0, let also X ′ → (X , x) be the blowing up along x. The previous argument works in the same way unless x 1 = (Z/u 3 , u 1 /u 3 , u 2 /u 3 , u 3 ). Then x 1 satisfies again (9.11) for some α 3 ∈ k(x) and we may iterate. By Proposition 3.17, this process is finite or the curve Y := V (Z, u 1 , u 2 ) is permissible of the first kind and we blow up along Y. But then k(x 1 ) = k(x), and this gives a contradiction with κ(x 1 ) ≥ 3. Therefore the Lemma is proved unless inh = Z p + λU p 1 , λ ∈ k(x) p , div(u 1 ) ⊆ E. (9.12)

We now define a refinement C of the function x → (m(x), ω(x)), cf. chapter 6. Let π : X ′ → (X , x) be the blowing up along a permissible center of the first kind Y ⊆ div(u 1 ), x ′ ∈ π -1 (x). We set: C(x ′ ) < C(x) ⇔ x ′ satisfies the conclusion of the lemma. By Theorem 3.13, we have C(x ′ ) < C(x) unless x ′ ∈ div(u ′ 1 ), where div(u ′ 1 ) ⊆ E ′ is the strict transform of div(u 1 ). Otherwise, we let C(x ′ ) = C(x).

With notations as in chapter 6, we claim that div(u 1 ) has maximal contact for the condition C (Definition 6.1). To see this, suppose that C(x ′ ) = C(x). Note that δ(x ′ ) > 1 or x ′ satisfies the assumptions of Lemma 7.1 if λ ∈ k(x ′ ) p : a contradiction. If δ(x ′ ) = 1 and λ ∈ k(x ′ ) p , we get an expansion inh ′ = Z ′ p + F p,Z ′ , 0 = F p,Z ′ ∈ k(x)[U ′ 1 , . . . , U ′ e ′ ] p , where (u ′ 1 , u ′ 2 , u ′ 3 ; Z ′ ) are well adapted coordinates at x ′ , and the leading coefficient of F p,Z ′ in U ′ 1 is λU ′ 1 p . Since C(x ′ ) = C(x) is assumed, we actually have inh ′ = Z ′ p + λU ′ 1 p by (9.12) and the claim is proved. The conclusion now follows from Theorem 6.3.

Proposition 9.11. Let µ be a valuation of L = k(X ) centered at x. There exists a finite and independent composition of local permissible blowing ups of the first kind:

(X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ), (9.13) is the projection. By Proposition 3.9(iv), we have H(x ′ ) = u ǫ(y)-p H(x)S ′ . Therefore: ord u H(x ′ ) p = d ′ j ′ for some j ′ , 1 ≤ j ′ ≤ e ′ 0 := e 0 (x ′ ).

The claim follows immediately. We now define Ω(x) ⊂ (X , x) to be the Zariski closure of the set: Ω • (x) := {y ∈ X : m(y) = p, ω(y) > 0 and ∀j, 1 ≤ j ≤ e 0 , y ∈ div(u j )}.

By Proposition 3.26, Ω(x) is a (possibly empty) curve. Note that (1) Ω(x ′ ) is the strict transform of Ω(x) in (X ′ , x ′ ) if Y satisfies (9.14), and

(2) Ω(x) = ∅ if e 0 ≥ 2 or if d ′′ (x) = 0.

We consider two cases: Case 1: Ω(x) = ∅. This implies that any permissible center of the first kind Y satisfies (9.14). As we are done if d ′′ (x) = 0, by the above claim, there exists j, e 0 < j ≤ e such that div(u j ) has maximal contact for the condition C. By Theorem 6.3, we obtain a sequence (9.13) such that C(x r ) < C(x). Case 2: Ω(x) = ∅. Consider the quadratic sequence along µ. By the above claim and (1), we either obtain C(x r ) < C(x) (in particular if we reach case 1), or achieve that Ω(x r 1 ) is irreducible for some r 1 ≥ 0; by Proposition 3.17, it can be furthermore assumed that Ω(x r 1 ) is permissible of the first kind when the latter holds. Let then y 1 ∈ (X r 1 , x r 1 ) be the generic point of Ω(x r 1 ). By (2), we also have: e 0 (x r ) = e 0 = 1 and d ′′ (x r 1 ) ≥ 1. (9.16)

Let π 1 : X ′ → (X r 1 , x r 1 ) be the blowing up along Ω(x r 1 ) and x ′ ∈ π -1 1 (x). Since d ′ (x ′ ) ≤ d ′ (x r 1 ) = d ′ (x), we have C(x ′ ) < C(x) or are done by (1) and case 1 unless d ′ (x ′ ) = d ′ (x), e ′ 0 := e 0 (x ′ ) = 1 and d ′′ (x ′ ) ≥ 1.

Then π 1 restricts to a finite morphism Ω(x ′ ) -→ Ω(x r 1 ). (9.17)

We now iterate this construction: this constructs a sequence

(X r 1 , x r 1 ) ← (X r 2 , x r 2 ) ← • • • ← (X r k , x r k ) ← • • •
where x r i ∈ X r i is the center of µ. If C(x r k ) = C(x), there is an induced two-dimensional quadratic sequence

(X r 1 , y 1 ) ← (X r 2 , y 2 ) ← • • • ← (X r k , y k ) ← • • •
where y k ∈ (X r k , x r k ) is the generic point of the permissible curve Ω(x r k ) by (9.17). By two-dimensional resolution, we have (m(y k ), ω(y k )) < (p, ω(x)) for k >> 0: a contradiction with permissibility. Therefore the above sequence achieves C(x r k ) < C(x) for some k ≥ 0 and the proof is complete. 9.2.2 Proof of Proposition 9.5.

From now on, we assume that (E)' is satisfied.

Definition 9.12. (Preparation). Assume that x is in case (**) (Definition 8.1). We define

pr : {(x 1 , x 2 , x 3 ) ∈ R 3 ≥0 | x 3 < 1+ω(x) p } -→ R 2 ≥0 , (x 1 , x 2 , x 3 ) → x := 1 1+ω(x) p -x 3 (x 1 -d 1 , x 2 -d 2 ) . (9.18)
as the translation by the vector (-d 1 , -d 2 , 0) followed by projection from the point (0, 0, 1+ω(x) p ) over the (x 1 , x 2 )-plane, followed by the homothety of ratio p 1+ω(x) . We will write ∆ 2 (h; u 1 , u 2 ; v; Z), even ∆ 2 if no confusion is possible instead of pr∆ S (h; u 1 , u 2 , v; Z) for short. Let x be a vertex of ∆ 2 . We say that x is a left vertex if its ordinate is bigger or equal to the ordinate of the vertex of biggest ordinate of the side of slope -1.

x neither satisfies (i) nor (ii). In particular γ(x) ≥ 1. We first claim that there exists r 0 ≥ 0 such that x r 0 is resolved or γ(x r ) = 1 for all r ≥ r 0 .

(9.33) By Proposition 9.20(i), we have γ(x 1 ) ≤ γ(x); by Proposition 9.20(iii), inequality is strict if:

E = div(u 1 ), E 1 = div(u 1,1 u 2,1 )
provided γ(x) ≥ 2, β(x) = 2. In case β(x) = 2, we obtain C(x 1 ) ≤ 1. Then any further occurrence of E r = div(u 1,r ) along the algorithm will satisfy β(x r ) < 2 by Proposition 9.20(ii)-(iv). Therefore it can be assumed that E and E i have the same number of irreducible components for every i ≥ 0 in order to prove (9.33) (note that we are done if (2) is applied). If E = div(u 1 ), we reach (i) or k(x i ) = k(x) for i >> 0 by Proposition 9.20(iv). The claim follows from Corollary 3.20.

If E = div(u 1 u 2 ), we get (9.33) by standard arguments on combinatorial blowing ups.

To conclude the proof, we may hence assume that (E = div(u 1 ), β(x) = 1) or (E = div(u 1 u 2 ), C(x) < 1).

When (E = div(u 1 ), β(x) = 1), this is stable by blowing up or yields E 1 = div(u 1,1 u 2,1 ) (Proposition 9.18(3) and Proposition 9.20(iii)). Stability ends after finitely many steps by Proposition 9.20(iv) and Corollary 3.20.

When (E = div(u 1 u 2 ), C(x) < 1), this is stable by blowing up or yields (i) (Proposition 9.20(ii)). Stability ends up in (ii) for r >> 0 by standard arguments on combinatorial blowing ups. Proposition 9.22. Assume that x satisfies conditions (**) and (E)' together with one of the following: (i) E = div(u 1 ), β(x) < 1;

(ii) E = div(u 1 u 2 ), A 1 (x) < 1, C(x) < 1 2 , β(x) < 1 -1 1+ω(x) ;

(iii) E = div(u 1 u 2 ) and C(x) = 0.

Then x is resolved for (p, ω(x), 3).

If x 1 = x ′ , Proposition 9.20(iii) gives

A 1 (x ′ ) = A 1 (x), C(x ′ ) < 1 2
, β(x ′ ) = β(x) + A 1 (x) -1 < β(x).

Therefore x ′ satisfies again assumption (ii) of the proposition together with (9.34) and c(x ′ ) < c(x).

If E = div(u 1 ) and κ(x) = 4, x is resolved by Propositions 9.18(1)(2) and 9.19. Therefore the proposition holds by induction on c(x) under the extra assumption (9.34).

Assume now that x satisfies assumption (i) with A 1 (x) ≥ 1. In particular ǫ(x) = 1 + ω(x) and V ∈ Vdir(x) by Proposition 9.16. Furthermore, We are done by Theorem 3.13 if Vdir(x) =< V, U 1 >. Otherwise we have A 1 (x) > 1 or β(x) > 0. Since V ∈ Vdir(x), the only point which may be ω-near x is the point x ′ := (Z ′ , u ′ 1 , u ′ 2 , v ′ ) = (Z/u 1 , u 1 , u 2 , v/u 1 ), E ′ = div(u 1 ). (9.36) These are well adapted coordinates. If A 1 (x) > 1, we have

β(x ′ ) = β(x), A 1 (x ′ ) = A 1 (x) -1 > 0, d ′ 1 = d 1 + 1 + ω(x) p -1.
Then x ′ satisfies again conditions (**) and (E)' by (9.35). By induction on A 1 (x), we reduce to A 1 (x) = 1, since A 1 (x) < 1 is (9.34).

If A 1 (x) = 1, expand f p,Z = u pd 1 1 (γv 1+ω(x) + 1≤i≤1+ω(x)

γ i v 1+ω(x)-i u i 1 u a 2 (i) 2
+ f 1 ), with f 1 ∈ (v, u 1 ) 2+ω(x) , γ ∈ S invertible, γ i ∈ S invertible or zero, γ i 0 invertible for some i 0 with a 2 (i 0 ) = i 0 β(x) < i 0 . We get f p,Z ′ = u ′ 1 pd 1 +1+ω(x)-p (γv ′ 1+ω(x) + 1≤i≤1+ω(x)

γ j v ′ 1+ω(x)-i u ′ 2 a 2 (i) +u ′ 1 f ′ 1 ), f ′ 1 ∈ S ′ .

9. 2 . 1

 21 An extra assumption on the singular locus. . . . . . . . 281 9.2.2 Proof of Proposition ??. . . . . . . . . . . . . . . . . . 287 1 Introduction.

( 4 . 1 )

 41 reduction of Theorem 1.1 to its Local Uniformization form along valuations; (4.2) reduction of Local Uniformization to Theorem 1.5; (4.3) the normal crossings condition (E) can be achieved (Corollary 4.19).

Definition 2 . 20 .

 220 Let Y ⊂ X be an integral closed subscheme with generic point y. We say that Y is Hironaka-permissible at x ∈ Y if m(y) = m(x) and Y is regular at x. Definition 2.21. Let Y ⊂ X be an integral closed subscheme with generic point y. We say that Y is Hironaka-permissible with respect to E at x ∈ Y if Y ⊆ Sing m X , i.e. m(y) = m(x) = m, and W := η(Y) has normal crossings with E at s := η(x).

tion 2 .

 2 40), D(W ) is generated as a G(W )-module by those derivations w.r.t. constants Der(k, G(W )) ≃ ( G(W )) Λ , viz. (2.39), together with the finite family

  and ω(x) > 0 in what follows since Theorem 2.81 takes care of the case ω(x) = 0. Two different kinds of permissible blowing ups are required. Permissibility behaves well with respect to regular base change (Theorem 3.8). A permissible center is permissible on a nonempty Zariski open set (Theorem 3.22). None of these is true for permissible centers of a fixed kind. Furhermore, by Example 3.6 we need both kinds of permissible blowing ups. Definition 3.1. Let Y ⊂ X be an integral closed subscheme with generic point y. We say that Y is permissible of the first kind at x if m(y) = m(x) = p and the following conditions hold: (i) Y is Hironaka-permissible w.r.t. E at x (Definition 2.21);

. 4 ) 3 . 3 .

 433 Proposition Let Y be permissible of the first kind at x ∈ Y. Then for any well adapted coordinates (u 1 , . . . , u n ; Z) at x such that

  with generic point y. Let π : X → X be any composition of Hironaka-permissible blowing ups with X regular. Since y is an isolated point of Sing p X , the map π factors through the blowing up π 0 along Y above y. Define a nonempty Zariski open subset U ⊆ Y by:

57 )

 57 be a sequence of such local blowing ups and centers withx r ∈ Y r Z r (ϕ) := {ϕ r (ξ)} ⊂ X r . (3.58) Note that the local ring O Xr,ϕr(ξ) is independent of r ≥ 0. In particular, m(ϕ r (ξ)), ǫ(ϕ r (ξ)) and ω(ϕ r (ξ)) are independent of r ≥ 0. An important case of such sequences is when taking Y r = {x r } for every r ≥ 0; then (3.57) is called the quadratic sequence along ϕ. In any case, given a sequence (3.57), we let d(ϕ) := min r≥0 {dimO Xr,xr }.

2 m 2

 22 Let nowe r := dim k(xr) I r + m Sr Sr ≥ e -1, t r := e r -(e -1) ≥ 0 for r ≥ 0. It can be assumed w.l.o.g. that (u e+1 , . . . , u e+t 0 ) ⊆ I 0 . We have e r+1 ≥ e r for every r ≥ 0 and let e ∞ := max r≥0 {e r }. It can be assumed w.l.o.g. that e 0 = e ∞ . Since l 0 = k(x r ) and M r = aN for every r ≥ 0, the ring morphism S r → O v |K factors through Ŝr to a surjective morphism φr : Ŝr → O v |K .

. 66 )

 66 By Lemma 3.21 below, there exists r ≥ 1 and a writing f r = u m+r e g r , g r ∈ (u e )S r , ord m Sr g r = 1.

1 WrF

 1 satisfies a relation(3.73) with associated integer r1 = r 1 -1. Iterating r 1 times this procedure, we get some ( Xr 1 , xr 1 ) with initial formin Wr hr = Zp r + F p, Zr, Wr ∈ G( Wr )[ Zr ], G( Wr ) = Sr / Ĩr [{ Ũj,r } j∈J ] with Ũj,r = u -r 1e U j , j ∈ J. We have Φr := H -p, Zr, Wr ) ⊆ G( Wr ) ǫ(z) , cl 1 Φr ∈ (u e )G(W 0 ) ǫ(z) .

Theorem 3 . 22 .

 322 Let Y ⊂ (X , x) be an integral closed subscheme with generic point y. The setΩ(Y) := {y ′ ∈ Y : (m(y ′ ), ω(y ′ ), κ(y ′ )) = (m(y), ω(y), κ(y))} ⊆ Y contains a nonempty Zariski open subset of Y.Let furthermore Z ⊃ Y be an integral closed subscheme with generic point z such that Z is permissible (of the first or second kind) at y. The setPerm(Y, Z) := {y ′ ∈ Y : Z is permissible at y ′ } ⊆ Ycontains a nonempty Zariski open subset of Y.

. 78 )

 78 where γ(i, a) denotes the image in k(y). Let B 0 := {(i, a) ∈ B : ∃(i, a) ∈ B, i = p or (i = p and a ∈ N J )}.

Case 2 .

 2 Suppose on the contrary that B 0 = ∅. By (3.78), we have in m Ss h = Z p + that δ(y) ∈ N, ω(y) = ǫ(y) and κ(y) ≥ 2.(3.82)

  Since Y (p,a) is excellent, its regular locus is a nonempty Zariski open set. We deduce that U is a nonempty Zariski open subset of Y. For y ′ ∈ U 1 and (p, a) ∈ B, we denote by λ y ′ (p, a) ∈ k(y ′ ) the residue of γ(p, a). The property "η -1 (p,a) (y ′ ) red is a regular point of Y (p,a) " is equivalently characterized as follows: either (a) λ y ′ (p, a) ∈ k(y ′ ) p , or (b) there exists δ y ′ (p, a) ∈ O Y,y ′ such that v y ′ (p, a) := γ(p, a)δ y ′ (p, a) p is a regular parameter at y ′ .

Corollary 3 . 23 .

 323 With notations as above, the functionι : X → {1, . . . , p} × N × {0, 1, ≥ 2}, y → (m(y), ω(y), κ(y))is a constructible function on X . In particular, it takes finitely many distinct values.Proof. This follows from the previous theorem and Noetherian induction on X .Remark 3.24. The constructible sets X p,a := {y ∈ X : (m(y), ω(y)) ≥ (p, a)}, a ∈ N are not in general Zariski closed (Example 3.25 below). See next proposition for closedness of the set X p,1 .

Lemma 4 . 5 .

 45 Let A be a reduced excellent Noetherian domain of dimension three and X -→ SpecA, Y -→ SpecA be projective birational morphisms. Denote by ρ : Y • • • -→ X the birational correspondence and F ⊂ Y its fundamental locus (i.e. the complement of the largest open set of definition). There exists a sequence

Step 3 :

 3 achieving (i) in Theorem 1.1 with π projective for X = SpecA affine. The Riemann-Zariski space of valuations Zar(X ) := {v valuation of K : A ⊆ O v } is quasi-compact by [89] Theorem 40 on p.113 and Noetherianity of A. The assumption on v in (LU) means that v is a closed point of Zar(X ). Regularity

Remark 4 . 7 .

 47 Corollary 1.3 can be strengthened in the obvious way: given any proper and flat O-scheme Y with generic fiber Y F = Σ and an open set U ⊆ SpecO, there exists a proper and flat O-scheme X isomorphic to Y above U and regular away from U.

  (7.8.3)(vii) and Remark (7.8.4)(i), soK := Tot( Â) = c i=1 Ki , Ki = QF ( Â/ Pi )and the Pi 's are minimal primes. Let v be an extension of v to, say K1 , after possibly renumbering. Note that dimO v ≥ 1 and that inequality is strict in general. We may also choose v with k v|k v algebraic (hence k v|k algebraic) by composing again if necessary (this means that v is a closed point in the Riemann-Zariski space of valuations Zar(Spec  P1 ) := {w valuation of K1 :  P1 ⊆ O w } see [89] Theorem 38 p. 111). We have r ≤ d := dim( Â/ P1 ).

  8.3)(v).There are inclusions  ⊂ T ′ P ′ ⊆ O Ŷ,ŷ . By (4.10)-(4.11), the right-hand side inclusion satisfies P ′ O Ŷ,ŷ = m ŷ , so O Ŷ,ŷ = T ′ P ′ by Zariski's Main Theorem [73] Theorem 1 p. 41 and the proof is complete. Proposition 4.10. Theorem 1.5 implies Theorem 1.1.

Remark 5 . 3 .

 53 y) and Vdir(y) =< U 1 > . The emblematic cases of κ(x) = 2 are:

  [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Theorem I.4, equation (3) page 1962). As B(x) ≥ 1, we have a ≥ b.(6.10)

  is homogeneous of degree iB(x). Note that b ≤ 1: indeed, b > 1 would give a ≥ b > 1 and in (6.11) deg(Φ p-1 ) < p -1 or deg(Φ p ) < p, which contradicts ord x (h) = p. Furthermore, we may assume b < 1 or a = b = 1. (6.13)

Case 3 : 1 1 1 ∂F

 311 J 0 ⊆ pN, B(x) ∈ N and G = U -pd ∂F p,Z,α ∂U 2 . Amending[START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] I.8.7.8 as in(7.65), we obtain the bound β(x ′ ) ≤ β(x)/d except possibly if j 1 = p a ; in this case, we leta ′ := max{b : U b p a 1 Ψ p a (U 1 , U 2 ) ∈ (k(x)[U 1 , U 2 ]) p b } < a (7.66)and obtain the bound:β(x ′ ) ≤ max{p a ′ -a , β(x)} (resp. β(x ′ ) < β(x)) (7.67)from Lemma 6.7(2) (resp. ibid. with degF ≥ 2 if β(x) > 1/p).Case 4: J 0 ⊆ pN, B(x) ∈ N and U -pd 1

H - 1 ∂T 1 +

 11 U 2 Φ(U 1 , U 2 , U 3 ), with λ = 0 and Φ ∈ k(x)[U 1 , U 2 ].It is then obvious that τ ′ (x) = 3, by Theorem 3.13, x is resolved.

Let (u 1 1 U -pd 2 2 F 3 Φ 4 ) 8 . 3 .

 1123483 , u 2 , u 3 ; Z) be well adapted coordinates at x, where κ(x) = 3. In particular, we have ǫ(x) = ω(x) + 1. The initial form polynomialin m S h = Z p -G p-1 Z + F p,Z ∈ G(m S )[Z] has H -1 G p ⊂ k(x)[U 1 , . . . , U e ] ω(x)+1 and an expansion U -pd 1 i+1 (U 1 , U 2 ),(8.3)withU 3 ∈ Vdir(x), c ∈ k(x) and Φ i+1 ∈ k(x)[U 1 , U 2 ] i+1 , 0 ≤ i ≤ ω(x). Since κ(x) = 3, by (8.2), we have    (ω(x) + 1 ≡ 0 modp and c = 0), or Φ i+1 (U 1 , U 2 ) = 0 for some i ≤ ω(x) -2, ω(x)i ≡ 0 modp .(8.Proposition Assume that κ(x) = 3, E = div(u 1 u 2 ) andVdir(x) =< U 3 , λ 1 U 1 + U 2 >, λ 1 = 0.Then x is resolved.

Φ 1 , λ 1 , λ 2 ∈

 112 i+1 ∈ k(x)[U 1 ], 0 ≤ i ≤ ω(x) -1, and Φ ω(x)+1 = (λ 1 U 1 + λ 2 U 2 )U ω(x)Furthermore (Φ i+1 = 0 for every i ≥ 0) or (x 1 = x ′ in (8.1)), whereY 0 := {x} and x ′ := (Z ′ := Z/u 2 , u ′ 1 := u 1 /u 2 , u 2 , u ′ 3 := u 3 /u 2 );

. 27 )

 27 Therefore x 1 satisfies condition (**) since E ′ = div(u 1 u ′ 3 ), c = 0. Assume now that ω(x) < p. By Lemma 7.3(ii), we haved 1 , d 2 ∈ N, d 3 ∈ N, pd 1 + pd 2 + ω(x) = p. (8.28) If d j ≥ 1, j = 1, 2, 3, the center Y j := V (Z, u j ) is Hironaka-permissible w.r.t. E. Blowing up finitely many times, we reduce to the case d 3 = 0, 0 < d 1 , d 2 < 1. By (8.28), we thus have pδ(x) = p(d 1 + d 2 ) + ω(x) = p, ω(x) ≤ p -2.

2

 2

2 u -pd 3 3 f 3 ,

 233 f i ∈ S with γ ∈ S a unit. Let f i ∈ S/(u 1 ) be the residue of f i . Then min 1≤i≤ω(x)

  F p,Z ∈ k(x) p [U 1 , U 2 , U 3 ]. In particular, 0 < d := deg U 1 F p,Z < p, F p,Z := i=d i=0 U i 1 Φ i (U 2 , U 3 ) since ∆ S (h; u 1 , u 2 , u 3 ; Z) is minimal. Lemma 7.3(ii) applied to the term U d 1 Φ d (U 2 , U 3 ) of F p,Z ,after a relabeling U 3 ↔ U 1 , gives a contradiction with (9.8), since d = d ≡ 0 modp, d corresponds to a in Lemma 7.3(ii).

  [START_REF] Bravo | Singularities in positive characteristic, stratification and simplification of the singular locus[END_REF] and ǫ(x) ≥ ω(x) ≥ p. We get d ′ 1 ≥ d 1 : either x ′ is on the strict transform of div(u 1 ), or we can take u = u 1 , so ord W H(x) p ≥ d 1 in (9.15). We getd ′ (x ′ ) = max{0, 1d ′ 1 } ≤ max{0, 1d 1 } = d ′ (x).If equality holds, (9.15) implies that min{1, d ′ 1 } = ord u H(x ′ )/p, i.e.

  We have m(y) = m(x), ǫ(y) = ǫ(x) where Y = V (Z, u 1 , v) with generic point y, so Y is permissible of first kind. Let us blow up along Y.

  .3) Given φ ∈ S and a rational number d ≤ ord m S φ, we denote by cl d φ the initial form of φ in gr m S S ≃ S/m S [U 1 , . . . , U n ] (resp. the null form) if d = ord m S φ (resp. otherwise). Similarly, if I ⊆ S and d ≤ ord m S I, we denote

  and (2.14), we get in x h = (Z -λU x ) m , i.e. x is a solvable vertex: a contradiction. Therefore ∆ S s J (h; {u j } j∈J ; Z) has no solvable vertex, hence is minimal by the second statement in Proposition 2.7. The last statement is a rewriting of Definition 2.11.

  Definition 2.6 then shows that x ′ is solvable if and only if x is solvable. Since ∆

S (h; u 1 , . . . , u n ; Z) is minimal, the polyhedron ∆ S ′ (h ′ ; u ′ 1 , . . . , u ′ n ; Z ′ ) is also minimal by Proposition 2.7.

  Since we are dealing with hypersurface singularities, the latter condition can be stated in terms of the multiplicity function m, viz. (2.1) sqq. Introducing a normal crossings divisor E leads to an additional transverseness requirement for the center. This leads to Definitions 2.20 and 2.21 below.

Definition 2.19. A r.s.p. (u 1 , . . . , u n ) of S is said to be adapted to

  view of Definition 2.25 and Proposition 2.26, this follows from Proposition 2.29 (ii)(iii) applied to the local rings S (u j ) and S s , s := η(y).

	This corollary allows us to define the following invariant:
	Definition 2.31. Let (u 1 , . . . , u n ) be a r.s.p. of S which is adapted to the
	normal crossings divisor

  be the discriminant of P where ϕ1 , • • • , ϕ d are the roots of P in a suitable extension of S[a 1 , • • • , a d ].As a polynomial in the ϕ i , D P is homogeneous of degree d(d -1). By the theorem on symmetric functions, D P can be expressed as a homogeneous polynomial in a 1 , . . . , a d (the elementary symmetric functions) where a i has degree i.

		1 , • • • , a d indetermi-
	nates. Let
	D P = Disc Z (P ) :=	(ϕ i -ϕ j ) 2 ,
	i<j

  u 1 , . . . , u n ; Z) for 1 ≤ i ≤ p. We deduce the formula

	cl α,p(p-1)δα(h;u 1 ,...,un;Z) D = Disc Z (in α h).	(2.23)
	On the other hand, in α h has a multiple root over an algebraic closure of
	QF (gr α S) if and only if i 0 (α) = p by Proposition 2.29 (i). When this holds,
	we are in case (1) of this theorem.

  Let (γ l ) l∈Λ be a family of units in O W whose residue (λ l ) l∈Λ is an absolute p-basis of k. There exists a unique ring isomorphism φ : O

				39)
	as k-vector spaces. We allow Λ infinite in this construction. Note the strict
	inclusion	Vect k ((	∂ ∂λ l	) l∈Λ ) Der(k)
	when Λ is infinite.
	The following is Cohen's Structure Theorem stated in a constructive way
	[65] Theorem 28.3 and Lemma 1 on p. 216.
	Proposition 2.40. (I.S. Cohen)

  chapter 1 II.3.3.1 and II.3.3.2; proof of II.5.4.2(iv); Theorem II.5.6.

  and 2.23. Since W has normal crossings with E at x, W := η( Ỹ) has normal crossings with Ẽ at x. Since Y is permissible at x, we have m(y) = p. Any generic point ỹ of Ỹ has m(ỹ) = p by Theorem 2.74(1), and Ỹ itself is irreducible by

	Proposition 2.28. Theorem 2.74(2) applies to ỹ (with n(y) = ñ(y)) and to	x
	and states that

  has the required properties.

Proof of Corollary 1.2: A is excellent by

[START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF]

(7.8.3)(iii). Proof of Corollary 1.3: let Y be any projective O-scheme with generic fiber Y F = Σ, e.g. clearing denominators in Σ. By generic flatness

[START_REF] Grothendieck | Éléments de géométrie algébrique IV-2[END_REF]

(6.9.1), there exists U ⊆ SpecO such that s -1 (U) is flat over U. Apply Theorem 1.1 to the Zariski closure of s -1 (U) in Y, where s : Y -→ SpecO is the structure morphism.

  .53) By Definition 3.1, Y is permissible of the first kind at x if and only if (x satisfies condition (*1) or (*2)) and A 1 (x) ≥ 1. By Proposition 3.7, Y is permissible of the second kind at x only if x satisfies condition (*3) and A 1 (x) ≥ 1 by (7.53). Conversely, Definition 3.5(i) is satisfied because m

1 1

 1 ∂F p,Z,α ∂U 2 . In this situation, equality in(7.68) implies deg U 2 Ψ j (U 1 , U 2 ) ∈ pN. Therefore deg U 2 ∂Ψ j ∂U 2 ≤ deg U 2 Ψ j (U 1 , U 2 ) -2in (7.60) and we get the same bound as in case 1.

  and Lemma 7.24(1) with d ≥ 2. Note that for α > 1, we have if and only if α = d = 2. If x is in case (*1) or (*2), we deduce that x 1 is resolved or γ(x 1 ) < γ(x). (7.75) For α ∈ N, α ≥ 3, we have similarly If x is in case (*3), we deduce from Lemma 7.24(1) that (7.75) also holds. Assume that x 1 = x ′ is the point at infinity (7.73). By Lemma 7.25, x 1 is resolved or satisfies condition (*2). If x is in case (*1), Lemma 7.25(1) gives

	1 + d and equality holds α γ(x 1 ) ≤ 1 + α d + 1 p β(x) ≤ ⌈α⌉ < α. 2 < γ(x)	(7.74) (7.76)

by

(7.74)

, since β(x) > 2.

If x is in case (*2), Lemma 7.25

[START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]

gives C(x 1 ) ≤ C(x), so γ(x 1 ) ≤ γ(x). If x is in case (*3), then Lemma 7.25(3) similarly gives

γ(x 1 ) ≤ 1 + 1 + β(x) 2 < 1 + ⌊β(x)⌋ = γ(x) since β(x) ≥ 2.

The conclusion is again (7.75).

  , { ∂ ∂λ l } l∈Λ 0 }. By Lemma 6.7(2) (applied to F := Q i 0 (U 2 , U p

	1	∂ ∂U 1 , U 2	∂ ∂U 2

  Hironaka Resolution, 3 Hironaka-permissible blowing ups, see also Definition 2.20, 3 Hironaka-permissible with respect to E, Definition 2.21, 30 initial face of the polyhedron, 10 Initial forms, Definition 2.3 Initial forms defined by a weight vector α, 21 maximal contact, weak maximal contact, Definition 6.1, 164 monic expansion in case (**) or (T**), h has a monic expansion for (u 1 , u 2 , u 3 ; Z), 256 multiplicity m(x), Definition 2.67, 71 ω-near, 295 very near x r is very near x, Definition 5.4, 162 p-monomial, Definition 2.38, 49 permissible of the first kind, Definition 3.1, 89 permissible of the second kind, Definition 3.5, 92 purely inseparable, Remark 2.35, 40 quadratic sequence along a formal arc, equations (3.57)(3.58), 118 Resolution of Singularities conjecture, 1 resolved x is resolved for..., Definition 5.4, 162 Riemann-Zariski space of valuations, 143 skew κ(x) = 4, Vdir(x) is skew, 263 solvable vertices, Definition 2.6, 23 support of a formal arc, Definition 3.16, 118 truncation operator S/m S -linear truncation operator T , Definition 2.59, 64 well parametrized formal arc, Definition 3.16, 118

4 Application to Resolution in dimension three.

In this chapter, we deduce Theorem 1.1 from Theorem 1.5 and prove corollaries 1.2 and 1.3. Achieving condition (E) allows us to use all results from the previous chapters.

While Theorem 1.1 is global in nature for it states the existence of a proper morphism resolving singularities of X , Theorem 1.5 is very local: it only deals with valuations and the existence of a model which is regular at their center. Deducing the former from the latter goes back to O. Zariski Fundamental Theorem [START_REF] Zariski | Reduction of the singularities of algebraic three dimensional varieties[END_REF] p.539 on patching Local Uniformizations. Zariski proved: Proposition 4.1. (Zariski) Let K be a function field in three variables over an algebraically closed firld k of characteristic zero. Let N be a set of valuations of K, trivial on k. Let Σ be a projective model of K|k. If there exists a resolving system of N consisting of two projective models V and V ′ , then there also exists a resolving model for N (i.e. a projective model of K|k on which every valuation of N has a regular center). Proposition 4.6 below states the appropriate version of the Fundamental Theorem in the category of quasi-excellent reduced and separated Noetherian schemes of dimension at most three. Once the appropriate definitions have been set, the proof goes along the same line as Zariski's. Zariski could not state this more general result because the main notions (schemes, proper morphisms and quasi-excellence) were not defined at the time. Furthermore, his proof relies on the next three propositions which were only known for varieties of characteristic zero at the time.

We also remark that Zariski's Fundamental Theorem has been enhanced by the first author in [START_REF] Cossart | Modèle projectif régulier et désingularisation[END_REF] in the context of algebraic varieties over arbitrary ground fields. This enhancement is essential to obtain (ii) in Theorem 1.1.

All results in this chapter are extensions of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF]. The proofs are based on the following three characteristic free results which can be found respectively in [START_REF] Abhyankar | On the valuations centered in a domain[END_REF] Theorem 3, a special case of [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF] Theorem 0.3 (with B = ∅) and [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] Proposition 4.2: Proposition 4.2. (Abhyankar) Let (R, m) and (R ′ , m ′ ) be regular twodimensional local domains with a common quotient field and such that

Corollary 5.6. Theorems 1.1 and 1.5 hold true.

Proof. Theorem 1.1 has been reduced to Theorem 1.5 for residually algebraic valuations, Propositions 4.6 and 4.10. By Corollary 4.19, it can be furthermore assumed that condition (E) is satisfied. Theorem 1.5 is then an immediate consequence of [START_REF] Cossart | Resolution of Singularities of Threefolds in Mixed Characteristics. Case of small multiplicity[END_REF] Main Theorem 1.3 (m(x) < p), Theorem 2.81 ((m(x), ω(x)) = (p, 0)) and Theorem 5.5.

Remark 5.7. Let µ be a valuation of L = k(X ) centered at x and consider an independent sequence of local blowing ups (Definition 2.77)

For example, the quadratic sequence along µ is an independent sequence.

Then x is resolved for (p, ω(x), a) if for every µ, there exists some r = r(µ) ≥ 0 such that x r is resolved for (p, ω(x), a) (the converse follows from Definition 5.4 with r(µ) = 0 for every µ). This fact is used all along the next chapters, vid. chapter 7 for a = 2 and chapter 8 for a = 3. Proposition 5.8. With assumptions as in Theorem 5.5, assume furthermore that Max(inh) = Dir(inh), where inh ∈ k(x)[U 1 , U 2 , U 3 , Z] p is the initial form of h (Proposition 2.55). Then κ(x) ≥ 2 and x is resolved for (p, ω(x), 2).

Proof. By Remark 2.56, the assumption holds only if p = 2 and

with [k(x) 2 (λ 1 , λ 2 ) : k(x) 2 ] = 4 up to a linear change of variables. We have H(x) = (1), ω(x) = ǫ(x) = 2 and κ(x) = 4 (resp. κ(x) = 2) if E = div(u 1 u 2 u 3 ) (resp. otherwise). Since

we have τ ′ (x) = 3. Let X ′ → (X , x) be the blowing up along x and x ′ ∈ π -1 (x). Since τ ′ (x) = 3, we have ι(x ′ ) ≤ (2, 2, 1) by Theorem 3.13.

(1) a ′ = a 2 + a 3 + deg(F ), e ′ ≤ 1 + ⌊ degF d ⌋; if equality holds, then degF d ∈ N, a ′ /p ∈ N, e ′ /p ∈ N, and

(2) if a 3 = 0, then e ′ ≤ max{degF, 1}. Equality holds only if degF ≤ 1 or d = 1.

Proof. We suppose neither a 2 maximal nor a 3 maximal, i.e. we may have F (0, V 3 ) = 0 or F (V 2 , 0) = 0. The proof is identical to [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] II.5.3.2 on p. 1862. Note that it is not necessary to assume R excellent. Now we follow [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] chapter 4. Consider the blowing up π : X ′ → (X , x) at x and let x ′ ∈ π -1 (x) be a closed point, with d := [k(x ′ ) : k(x)]. Following [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Theorem I.4 on p.1962, we have: Proposition 6.8. With hypotheses and notations as above, assume that x is in case 1-2. Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x and assume furthermore that

If C(x ′ ) = C(x), we have:

and there exist well adapted coordinates (u ′ 1 := u 1 /u 2 , u 2 , v; Z ′ ) at x ′ such that the following holds:

(1) if x ′ = (Z/u 2 , u ′ 1 , u 2 , u 3 /u 2 ), then x ′ is again in case 1-2 and

(2) if x ′ = (Z/u 2 , u ′ 1 , u 2 , u 3 /u 2 ), then x ′ is in case 1 or 3. We have

Furthermore, x ′ is in case 3 only if k(x ′ ) is inseparable over k(x) (in particular p divides d).

Remark 6.9. The case where

is denoted 1* when x is in case 1, resp. 2* when x is in case 2. The monomial Φ p-1 corresponds to the vertex v 0 defined in Notation 6.5.

Proof. Statement (1): by Proposition 2.18, (u ′ 1 , u 2 , u 3 /u 2 ; Z/u 2 ) are well adapted coordinates at x ′ . Furthermore, ∆ 2 (x ′ ) = l 1 (∆ 2 (x)) + R 2 >0 , where l 1 is the affine transformation R 2 -→ R 2 , l 1 (b, c) := (b + c -1, c). These transformation laws are the classical transformations of the characteristic polyhedron of a surface singularity and give statement [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF].

For [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF], we define

when a > b ū1 := u 1 u 2 mod(u 2 ), ū3 :

2 ) when a = b.

(6.21)

we have v mod(u ′ 1 , u 2 ) = P ( ū3 ) ∈ k(x)[ ū3 ], P irreducible. In the extreme case a = b = 1 (6.13), we have in m S (h) = in x (h), τ (x) ≥ 2. The existence of x ′ implies d 2 = 0, β(x) = 1 and U 3 mod (U 1 , U 2 ) is in the ideal of the directrix of in x (h). The end of the proof is left to the reader. From now on, we assume 1 > b. Let H Finally, γ(x ′ ) ≤ γ(x) is a trivial consequence of the Definitions (6.7) and (6.9) except if (x is in case 3, β 2 (x) < 0 and C(x) < 0). But then β 2 (x) = -1/i for some i, 1 ≤ i ≤ ω(x) and (3) gives C(x ′ ) ≤ C(x)β 2 (x) < 1, so γ(x ′ ) ≤ 1 as required. Remark 6.14. With the hypotheses of Proposition 6.13 above, when γ(x) ≥ 2, x is in case 1 or 3 and γ(x ′ ) = γ(x), then we have γ(x) = 2 and: [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] x is in case 1 and β(x) = 2, C(x) = 1,

(2) x is in case 3 and 1 ≤ β(x) < 2, β(x) -C(x) ≥ 1.

We now go ahead to prove Theorem 6.3. The key lemma to reach the case γ(x) = 1 goes as follows: Lemma 6.15. Assume that div(u 1 ) has weak contact maximal for C. Let µ be a valuation of L = k(X ) centered at x and consider the quadratic sequence (6.1) along µ, i.e. with Y i = {x i } for every i ≥ 0.

Assume that one of the following holds:

(1) x is in case 1 with: β(x) = 2 and

has Φ p,α,1 = 0 with notations as in (6.14), where σ 2 := {(A 2 (x), 2)};

(2) x is in case 3 with β(x) = 1.

Assume furthermore that x 1 = (Z ′ := Z/u 3 , u ′ 1 := u 1 /u 3 , u ′ 2 := u 2 /u 3 , u 3 ), C(x 1 ) = C(x) and γ(x 1 ) = 2. Then C(x 2 ) < C(x) or γ(x 2 ) = 1 or (x 1 is in case 1* and x 2 in case 2* of Remark 6.9 with β(x 2 ) < 2).

Proof. Note that x 1 is in case 2 with γ(x 1 ) = 2 by assumption. By Proposition 6.13, we get A 2 (x 1 ) = A 2 (x) and respectively:

(1) C(x) = C(x 1 ) = 1, β 2 (x) = 0, A 3 (x 1 ) = B(x) -1 = A 2 (x), β(x 1 ) = A 2 (x) + 1; has F ′′ 1 of the form:

which leads to a point (d ′ 1 + (ω(x) -1)/p, d ′ 1 + d ′ 2 + d ′ 3 + A 2 (x 1 )+A 3 (x 1 )+ω(x) p , a p ) with a = 0 or 1.

Note that, when Φ p-1 = 0, by Proposition 6.8 (6.22), we get the claim (6.35) except when x 1 is in case 2*.

In this last case, x 2 is in case 2* and

In this case we come back to the argument of the proof of Proposition 6.8: let i 0 :=sup{i|F ′ i = 0} > 0, we have

An eventual translation on Y may only add ap-th power to

When i 0 = 0 mod p or deg(F ′ i 0 ) < i 0 we get the inequalities (6.35) by Lemma 6.7 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF]. When i 0 = 0 mod p and deg(F ′ i 0 ) = i 0 , Lemma 6.7 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] gives the last assertion of Lemma 6. [START_REF] Cano | Desingularization strategies for three-dimensional vector fields[END_REF].

Until the end of the proof, we assume Φ p-1 = 0 at x 1 . We get the inequalities (6.35) provided

Indeed no p-th power may pollute

Under assumption [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF], when this fails to hold, we have (6.37) with λ 2 = 0 and

by the above calculations. In case p = 2, by Lemma 6.7 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF], ord v F ′′ 1 + p-th power≤ 1: we have the inequalities (6.35). When F ∈ k(x) (ref. line above (6.37)), once again, by Lemma 6.7 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF], ord vF ′′ 1 + p-th power≤ 1. So we have just to look at the case p ≥ 3, deg(F )=1, the latter implies a i = A i (x 1 ) ∈ N, i = 2, 3. We deduce that d 2 + A 2 (x)+1 p ∈ N, which in turn implies that

with notations as in Lemma 6.7 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF], applying U 3

Then equality is strict in Lemma 6.7 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] and the conclusion follows.

Under assumption [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF], note that since β 2 (x) = -1 we necessarily have

> .

In the former case, the proof is parallel to that under assumption [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF], exchanging the roles of U ′ 2 , U 3 . In the latter case, as e = (1+A 2 (x))(p-1) p = 0, we conclude from Proposition 6.8 (6.33) with Φ p-1 = 0. Proposition 6. [START_REF] Cano | Reduction of the singularities of codimension one singular foliations in dimension three[END_REF]. Assume that div(u 1 ) has weak maximal contact for C. Let µ be a valuation of L = k(X ) centered at x and consider the quadratic sequence (6.1) along µ, i.e. with Y i = {x i } for every i ≥ 0.

If C(x i ) = C(x) for every i ≥ 0, one of the following is true:

(i) γ(x i ) = 1 for every i >> 0, or (ii) there exists a formal arc (Definition 3.16) ϕ : SpecO → (X , x) with l|k(x) finite algebraic, support Z := Z(ϕ) with η(Z) ⊆ div(u 1 ), η(Z) not an intersection of components of E, whose strict transform passes through x i for every i ≥ 0.

Proof. Note that (ii) fails to hold if and only if: for every i ≥ 0, there exists i ′ > i such that either k(x i ′ ) = k(x i ) (i.e. some of Proposition 6.8, 6.11 applies to x i ′ with d ≥ 2) or x i ′ is in case 2. Assume therefore that (ii) does not hold. By Propositions 6.8, 6.11 and 6.13, we have γ(x i+1 ) ≤ γ(x i ) for every i ≥ 0 and inequality is strict for i ′ as above if γ(x i ′ ) ≥ 3. W.l.o.g. it can be assumed that γ(x i ) = 2 for every i ≥ 0. This implies by (6.7) (6.9):

The conclusion follows easily. Proposition 6.17. Assume that div(u 1 ) has maximal contact for C and that γ(x) = 1. Let µ be valuation of L = k(X ) centered at x. There exists a finite and independent composition of local permissible blowing ups of the first kind:

where x i ∈ X i is the center of µ, such that C(x r ) < C(x) or x r is resolved for m(x) = p.

Proof. We may assume that C(x i ) = C(x) for every i ≥ 1 for the resolution process to be defined below; we will either derive a contradiction or prove that x r is resolved for m(x) = p for some r ≥ 0. By Propositions 6.8, 6.11 and 6.13, we have γ(x i ) = 1 for all i ≥ 0. This implies by (6.7) (6.9):

• when x i is in case 1:

• when x i is in case 2: ⌊C(x i )⌋ = 0, thus 0 ≤ C(x i ) < 1,

• when x i is in case 3: ⌊β(x i )⌋ ≤ 0, thus C(x i ) ≤ β(x i ) < 1. Suppose that i ≥ 1 and that A 2 (x i-1 ) < 1 and (x i-1 is in case 2 =⇒ β(x i-1 ) < 1). (6.39)

Then we consider the quadratic sequence (6.1) along µ. In every case, we have

where inequality is strict except if either Proposition 6.13 applies, or (x i-1 is in case 1 with β(x i-1 ) = 1). If Proposition 6.13 applies, we have

This proves in particular that (6.39) holds at x i ′ for every i ′ ≥ i. W.l.o.g. it can be assumed that, when (6.39) occurs, then i = 1. If x is in case 1 with β(x) = 1 and k(x 1 ) = k(x), then β(x 1 ) < 1 by Proposition 6.8; if Proposition 6.13 applies to x, then β(x 1 ) < β(x). In other terms, we have (A 2 (x 1 ), β(x 1 )) < (A 2 (x), β(x))

for the lexicographical ordering except possibly if x is in case 1 with β(x) = 1 and k(x 1 ) = k(x). So in the sequence (6.1), we may assume that x i is in case 1 with

Assume that x is in case 2 with d 1 + ω(x)/p < 1 and x is blown up. If x ′ := x 1 is in case 3, we have:

by Theorem 3.13 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF]. Let (u ′ 1 := u 1 /u 2 , u 2 , v ′ ; Z ′ ) be well adapted coordinates at x ′ , so we have

Therefore x ′ is resolved for m(x) = p by blowing up codimension one centers of the form

Algorithm: if x is in case 2 and Y j := V (Z, u 1 , u j ) is permissible for some of j = 2, 3, blow up along Y j , in case where both Y 2 and Y 3 are permissible, take j such that div(u j ) is "younger" that div(u j ′ ) {j, j ′ } = {2, 3}, i.e. let i 0 the index such that div (u j ′ ) is the strict transform of the exceptional divisor of (X i 0 , x i 0 ) ← (X i 0 +1 , x i 0 +1 ), the projection of div(u j ) on X i 0 is x i 0 or a curve; otherwise blow up along x. We claim that this algorithm succeeds, i.e. produces x r in case 1, cf. (a), or x r resolved for m(x) = p. The proof is different for small values of ω(x): (b) proof when d 1 + ω(x)/p < 1. Let x be in case 2. We may assume that (6.42) holds. (b1) if d 1 + d j + ω(x)/p < 1, j = 2, 3, the algorithm blows up along x. By the above Remark 6.18, it can be assumed that x 1 = (Z/u 2 , u 1 /u 2 , u 2 , u 3 /u 2 ) up to renumbering u 2 , u 3 . We obtain

Assumption (b1) is stable by blowing up and can possibly repeat only finitely many times. (b2) by the above Remark 6.18, the algorithm succeeds or produces an infinite sequence of points in case 2. By (6.41), any subsequence of blowing ups along curves is finite, so, for every r >> 0, the blowing ups are centered at closed points, by the argument of the proof of Proposition 6.16 (d-1), C(x r ) = 0 for every r >> 0. Take r = 0 to begin with and assume w.l.o.g. that x is blown up. The extra assumption (6.39) holds if 0 ≤ A 2 (x), A 3 (x) < 1. Up to renumbering u 2 , u 3 , we may furthermore assume by (6.42) that

1 and the algorithm blows up along a curve (A 3 (x 1 ) ≥ 1), note that

This proves that any further blowing up at a closed point either satisfies: some of (b1) or (6.39), or satisfies again (6.43) with a smaller value of (A 2 (x), d 3 ) for the lexicographical ordering. Induction on (A 2 (x), d 3 ) completes the proof for x in case 2 (vid. the same argument in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] 1.7.4 on p. 1968). Let now x be in case 3. We are done unless x 1 is again in case 3. Then, as γ(x) = 1 (⇒ C(x) ≤ β(x) < 1):

Therefore the algorithm reaches (6.39) after finitely many steps. This completes the proof of (b). (c) proof when d 1 + ω(x)/p ≥ 1. By the above Remark 6.18, we may assume that 0

We have c ′ (x) ≥ 1 if the extra assumption (6.39) does not hold. Applying Propositions 6.8, 6.11 and 6.13, we obtain:

, notations of (6.44), then

). Note that blowing up along the curve

• if x is in case 3 and x 1 is in case 2 (resp. in case 3), then, as

Induction on c ′ (x) completes the proof.

7 Projection Theorem: very transverse case, resolution of κ(x) = 2.

In this chapter, we prove Theorem 5.5 when κ(x) = 2 (Definition 5.1). This is restated as Theorem 7.26 at the end of this chapter. Assume that a valuation µ of L = k(X ) centered at x is given. We consider finite sequences of local blowing ups along µ:

with Hironaka-permissible centers Y i ⊂ (X i , x i ), where x i , 0 ≤ i ≤ r, denotes the center of µ, see (5.3) and following comments. Also recall the definition of "resolved" and "good" (Definition 5.4) and Remark 5.7 about the logical scheme of the proof of Theorem 5.5.

Unfortunetaly, in (7.1), it may happen that, for some x i , 1 ≤ i ≤ r, we have κ(x i ) > 2. In the next subsection 7.1, we study points x i such that Let Y := V (Z, u 1 , u 3 ) ⊂ X and y ∈ X be its generic point. If Y is permissible of the first kind, i.e. m(y) = p and ǫ(y) = ǫ(x), we take Y 0 := Y in (7.1). By Theorem 3.13, we have ι(x 1 ) ≤ (p, ω(x), 1) unless

) is minimal, and we deduce that

and the lemma holds.

Assume now that Y is not permissible of the first kind. We take Y 0 := {x} in (7.1). If ι(x 1 ) ≥ (p, ω(x), 2), x 1 belongs to the strict transform of div(u 1 ) by Theorem 3.13.

If

) is minimal by Proposition 2.18 and we deduce as above that

, we deduce that x 1 satisfies again the assumptions of the lemma if ι(x 1 ) ≥ (p, ω(x), 2).

The conclusion of Proposition 3.17(2.b) is not satisfied by the formal arc Ŷ → X . Iterating, we deduce from Proposition 3.17(1) that one of the following three properties is satisfied for some r ≥ 1:

(1) ι(x r ) ≤ (p, ω(x), 1);

(2) x r belongs to the strict transform Y r of Y in X r and Y r is permissible of the first kind at x r , or

(3) x r does not belong to Y r .

The lemma holds when (1) is satisfied; it has been proved above that the lemma also holds when (2) is satisfied. If ( 3) is satisfied, it can be assumed w.l.o.g. that r = 1. We claim that x 1 satisfies the conclusion of the lemma if

To prove the claim, first note that there exists a unitary polynomial P (t) ∈ S[t], whose reduction P (t) ∈ k(x)[t] is irreducible, P (t) = t, and

3 ) and

We have

) by Proposition 3.9(iv) and

where θ denotes the image in

) be the residue of u 2 /u 3 . Since v ′ is solvable, we have:

2 is not a p th -power, since v was not a solvable vertex. We deduce that

) and this proves that ω(x ′ ) ≤ 1. This concludes the proof of the claim.

To conclude, take

, where d ′ 3 := d 1 +d 2 +d 3 -1+ǫ(x)/p. Therefore ǫ(x ′ ) ≤ 2 and we are done unless ω(x) = 2, ι(x ′ ) ≥ (p, 2, 2) and E = div(u 1 u 2 u 3 ), which we assume from now on.

We have

) and the initial form polynomial has an expan-

) p and (λ 1 , λ 2 ) = (0, 0). Moreover, we have

If τ (x 1 ) = 3, we take Y 1 := {x 1 } in (7.1) and obtain ι(x 2 ) ≤ (p, 2, 1). We conclude by analyzing the cases τ (x 1 ) ≤ 2. By [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] II.1.5 p.1888, this implies that λ 1 = 0. Therefore λ 2 = 0 and we get

By Lemma 7.3 below with (i, ω) = (1, 2), we have p ≥ 3 and

consider a finite monomial expansion (2.5):

The polyhedron assumption on h gives

and that at least one of these inequalities is strict. Now f i,Z ′ = u -i 3 f i,Z and one deduces that

By (7.4), we have i(d 1 + d ′ 3 + 1) ∈ N, so (7.5) actually implies that

does not satisfy (resp. satisfies) µ ′ + µ 2 = 0. This concludes the proof.

The following lemma extends the previous result when ω(x) = 1. Lemma 7.2. Lemma 7.1 remains valid when ǫ(x) = ω(x) = 1 and div(u 1 ) ⊆ E ⊆ div(u 1 u 2 ), all other assumptions being otherwise unchanged.

Proof. Let Y := V (Z, u 1 , u 2 ) ⊂ X and y be its generic point. Arguing as in (7.2) above, any vertex of ∆ S (h; u 1 , u 2 , u 3 ; Z) is induced by f p,Z . By Proposition 2.12, we have δ(y

Therefore Y is permissible of the first kind and we take Y 0 := Y in (7.1). By Theorem 3.13, we have ι(x 1 ) ≤ (p, ω(x), 1) unless

) and the lemma holds.

Given an integer α ∈ N, we denote by α ∈ {0, . . . , p -1} the remainder of α modulo p. The following elementary lemma is useful.

By (7.16), we have

It can be assumed that

) are well prepared at x ′ , this leads to the same cases 1 and 2 as above. One concludes applying Lemma 7.1 or [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma II.3.3 on p.1892, exactly as above.

If ω(x) = 1, then (X ′ , x ′ ) satisfies the assumption of Lemma 7.2 or there is an expansion

) are well adapted coordinates at x ′ . With notations as in Lemma 7.3 applied with a 3 = 0, i = 1, we let a 1 := pd 1 , a 2 := p(d 1 -1) + 1.

Let

2 ) ⊂ X ′ with generic point y ′ . By (7.17), any vertex of ∆ S ′ (h ′ ; u ′ 1 , u ′ 2 , u ′ 3 ; Z ′ ) is induced by f p,Z ′ and we have δ(y ′ ) = 2d 1 -1+2/p = δ(x ′ ), so Y ′ is permissible of the first kind at x ′ .

Either we have not the conditions ( 7.7), blowing up Y ′ then gives ι(x ′′ ) ≤ (p, ω(x), 1) by Theorem 3.13, where x ′′ is the center of µ. Or we blow up up consecutively Y ′ 1 , then Y ′ 2 , and iterating, we reduce to the case

Proposition 7.7. Assume that ǫ(x) = ω(x) ≥ 2, κ(x) ≥ 2 and E = div(u 1 ). Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x. Assume furthermore that the initial form polynomial in E h = Z p + U pd 1 1 F , F ∈ S/(u 1 ), of Lemma 7.5 satisfies the following two conditions:

By Proposition 2.18, ∆ S ′ (h ′ ; u 1 , u ′ 2 , u ′ 3 ; Z ′ ) is minimal. We deduce that ǫ(x ′ ) ≤ ǫ(x); if x 1 is very near x, we have ǫ(x 1 ) = ǫ(x) = ω(x 1 ) and

Moreover Proposition 3.9(v) implies that

We deduce that κ(x 1 ) = 1 (so ι(x 1 ) < ι(x)) if G ′ = 0. Otherwise we have Vdir(x 1 ) ≡< U ′ 2 , U ′ 3 > modU 1 , so x 1 satisfies again the assumptions of the proposition. The proposition then follows from Corollary 3.20.

Remark 7.9. All local blowing ups considered in this section are permissible of the first kind except when p ≥ 3 and ω(x) ≤ 2 (proof of Lemma 7.1 for ω(x) = 2, proof of Lemma 7.6 for ω(x) = 1).

Reduction to monic expansions.

In this section, we further reduce the proof of the Projection Theorem to those points with κ(x) = 2 satisfying condition (*) below. To begin with, let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates and

be the corresponding initial form. If κ(x) = 2, we have div(u

Definition 7.10. Assume that κ(x) = 2. We say that x satisfies condition (*) if there exist well adapted coordinates (u 1 , u 2 , u 3 ; Z) such that one of the following properties is satisfied:

) when condition (i) holds. Condition (*) is labeled (*3) when condition (ii) holds. Proposition 7.11. Assume that κ(x) = 2. Let µ be a valuation of L = k(X ) centered at x and consider the quadratic sequence

The following holds:

(i) there exists r ≥ 0 such that x r is resolved or (ι(x r ) = ι(x) and x r satisfies condition (*));

(ii) if x satisfies condition (*), then x 1 is resolved or (ι(x 1 ) = ι(x) and x 1 satisfies again condition (*));

(iii) if ω(x) ≡ 0 modp, then x is good.

Proof. We first prove together (i) and (ii) by a casuistic analysis. The discussion goes according to the value of τ ′ (x) and subdivides in the different situations ω(x) = ǫ(x) and ω(x) = ǫ(x) -1.

• Assume that τ ′ (x) = 3. Then ι(x 1 ) < ι(x) by Theorem 3.13, so x is good and there is nothing more to be proved.

• Assume that τ ′ (x) = 1 and ω(x) = ǫ(x). We may pick well adapted coordinates (u 1 , u 2 , u 3 ; Z) such that U 3 ∈ Vdir(x), so

We deduce that ω(x) ≡ 0 modp and x satisfies condition (*1) or (*2). This proves that (i) holds with r = 0.

To prove (ii), we may assume that ι(x 1 ) ≥ ι(x) (in particular ω(x 1 ) = ω(x)). There is an expansion (7.22) with

By Theorem 3.13, x 1 lies on the strict transform of div(u 3 ). Let

) is minimal by Proposition 2.18. One deduces from (7.23) that

This proves that (ι(x ′ ) = ι(x) and x ′ satisfies condition (*2)), so (ii) holds.

If x 1 = x ′ , there exists a monic polynomial P (t) ∈ S[t], whose reduction

We have

2 . Note that we may furthermore assume that P (t) = t if E = div(u 1 u 2 ) by symmetry between u 1 and u 2 , i.e. c pd 2 = 0 (and (7.23), it can be assumed w.l.o.g. that ord (u 1 ) φ 1 > d ′

1 . The initial form in E 1 h 1 of Lemma 7.5 is then of the form:

We have ǫ(x 1 ) = ω(x) and

.

Therefore ι(x 1 ) = ι(x) and x 1 satisfies condition (*1), so (ii) holds.

Case 2: d ′ 1 ∈ N and λc pd 2 ∈ k(x 1 ) p . It can be assumed w.l.o.g. that

where

be well adapted coordinates at x 1 , so the initial form in E 1 h 1 of Lemma 7.5 is now of the form:

205

If ǫ(x 1 ) = ω(x), then x 1 satisfies the assumptions of Lemma 7.1, so x 1 is resolved. Otherwise we have ǫ(x 1 ) = 1 + ω(x) and

Then there exist well adapted coordinates of the form (u 1 , v ′ 2 , v 3 ; Z ′ ) at x 1 satisfying Definition 7.10, so ι(x 1 ) = ι(x) and x 1 satisfies condition (*3).

• Assume that τ ′ (x) = 1 and ω(x) = ǫ(x) -1. By Definition 5.1, we then have H -1 ∂F p,Z ∂U 2 = (0), therefore

>, (7.25) so x satisfies condition (*3). This proves that (i) holds.

To prove (ii), we may assume that ι(x 1 ) ≥ ι(x). By (7.25), there is an expansion (7.22) with

26) This furthermore implies that ω(x) ≡ 0 modp, so Φ 0 = 0. By Theorem 3.13, x 1 lies on the strict transform of div(u 3 ). Note that we may furthermore assume that λ = 1 and deg

in (7.26): this is achieved by possibly changing u 2 to γ 0 u 2 + γu 1 , γ 0 γ ∈ S a unit, then picking again well prepared coordinates. Let

If x 1 = x ′ , the proof is identical to that in the case when ω(x) = ǫ(x): one gets (ι(x ′ ) = ι(x) and x ′ satisfies condition (*2)), so (ii) holds.

If

)/p and use the same notation as in the case ω(x) = ǫ(x). We have E 1 = div(u 1 ) and consider three cases.

and the initial form in E 1 h 1 of Lemma 7.5 is of the form:

206

If Φ = 0, we either have ǫ(x ′ 1 ) = ω(x), so x ′ 1 satisfies the assumptions of Lemma 7.1 and x 1 is resolved; or ǫ(x ′ 1 ) = 1 + ω(x) and

Then ι(x ′ 1 ) = ι(x) and x ′ 1 satisfies condition (*3). If Φ = 0, we have ǫ(x ′ 1 ) = ω(x) and

If U 2 U 3 divides Φ, then x 1 is good by Proposition 7.8; otherwise Φ is monic in U 2 or in U 3 , so ι(x ′ 1 ) = ι(x) and x ′ 1 satisfies condition (*1). If

. With notations as in (7.24) sqq., we get ǫ(x ′ ) = ω(x) and

where deg

p ) < ω(x) by (7.27). Therefore ι(x 1 ) = ι(x) and x 1 satisfies condition (*1).

Case 3: d ′ 1 ∈ N and c ∈ k(x 1 ) p . It can be assumed w.l.o.g. that

where γ 1 ∈ S 1 is a preimage of c 1/p ∈ k(x 1 ) and

be well adapted coordinates. We have

where

Then x is good (if τ ′ (x ′ ) = 3), or (τ ′ (x ′ ) = 2 and x ′ is of type (T2) or (T3)). This concludes the proof of the claim. In particular, we have proved (ii).

We now prove (i). Suppose on the contrary that for every r ≥ 0, x r does not satisfy condition (*). The above proof shows that x r is resolved for some r ≥ 0 or there exists r 0 ≥ 0 such that for every r ≥ r 0 , we have

where k ∈ {1, 3} is independent of r. If k = 1, we derive a contradiction from Corollary 3.20.

with the following property: for every i ≥ 0, we have ι(x i ) = ι(x) and the strict transform in (X i , x i ) of the formal curve Ŷ = (Z -φ, u 1 , û3 ) ⊂ X is nonempty.

Note that the conclusion of Proposition 3.17(2) applied to the formal arc ϕ : Ŷ → X does not hold. To see this, note that ibid.(2.b) implies that Z r 0 (ϕ) is an irreducible component of E r 0 ; by ibid.(2.c) we have ǫ(x r 0 ) = 1: a contradiction, since it is assumed (from the beginning of this proof) that ω(x) ≥ 2.

Therefore the conclusion of Proposition 3.17(1) holds. Let (u ′ 1 , u ′ 2 , u ′ 3 ; Z ′ ) be well adapted coordinates at x r 0 , where Y := (Z ′ , u ′ 1 , u ′ 3 ) ⊂ (X r 0 , x r 0 ) is permissible of the first kind at x 0 . Since Vdir(x r 0 ) =< U ′ 1 , U ′ 3 >, x r 0 is good by Theorem 3.13, hence x is good.

To prove (iii), it can be assumed by (i) that x satisfies condition (*). Suppose that ǫ(x) = ω(x). Then J(F p,Z , E, x) contains no monic polynomial in U 3 , since ω(x) ≡ 0 modp. So ǫ(x) = ω(x) + 1. It has been proved above that τ ′ (x) = 1 =⇒ ω(x) ≡ 0 modp.

We deduce that τ ′ (x) ≥ 2. Therefore x r is resolved for some r ≥ 0 or

Definition 7.14. With notations as above, we define a convex set:

is convex. Note that ∆ 2 (h; u 1 , u 2 ; u 3 ; Z) will have in general points with negative ordinate when (*3) holds. We now prove some basic properties of ∆ 2 (h; u 1 , u 2 ; u 3 ; Z). The situation is different and somewhat simpler when (*1) or (*2) holds.

Lemma 7.15. With notations as above, the following holds:

(1) there exists a = (a 1 , a 2 , a 3 ) ∈ ∆ S (h;

(4) assume that x satisfies condition (*1) or (*2) (resp. condition (*3)). Let

be a compact face. The topological closure σ of

(5) assume that x satisfies condition (*3) and let

Proof. Let V be the set of all vertices of ∆ S (h; u 1 , u 2 , u 3 ; Z) and

We claim that

Since ω(x)/p ≥ 1, we deduce that Y := V (Z, u 3 ) ⊂ Sing p X by Proposition 2.10: a contradiction with assumption (E).

In order to associate relevant combinatorial data to ∆ 2 (h; u 1 , u 2 ; u 3 ; Z), some minimizing process on the u 3 coordinate is required. This process is similar to that used in Definition 2.6 and Proposition 2.7.

Definition 7.17. Let x satisfy condition (*), (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x satisfying Definition 7.10 and y = (y

where Φ ∈ gr α S and λ, c ∈ k(x).

We say that (u 1 , u 2 ; u 3 ; Z) are well 2-adapted if furthermore the polygon ∆ 2 (h; u 1 , u 2 ; u 3 ; Z) (∆ + 2 (h; u 1 , u 2 ; u 3 ; Z) in case (*3)) has no 2-solvable vertex.

Theorem 7.18. With notations as above, there exist well 2-adapted coordinates. Furthermore, the polygon ∆ + 2 (h; u 1 , u 2 ; u 3 ; Z) is independent of the well 2-adapted coordinates (u 1 , u 2 ; u 3 ; Z). For such (u 1 , u 2 ; u 3 ; Z), let

Proof. Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates and assume on the contrary that (u 1 , u 2 ; u 3 ; Z) are not well 2-adapted. Let y ∈ N 2 be a 2-solvable vertex of ∆ 2 (h; u 1 , u 2 ; u 3 ; Z) with y 1 +y 2 minimal (and

given by Definition 7.17. Since y is a vertex of ∆ 2 (h; u 1 , u 2 ; u 3 ; Z), we have c = 0, so γ is a unit. We let

with notations as in Definition 7.17.

Let now y ′ = y be a vertex of ∆ 2 (h; u 1 , u 2 ;

>0 define the corresponding edge

given by Lemma 7.15 [START_REF] Abhyankar | Ramification theoretic methods in algebraic geometry[END_REF]. In particular we have

This implies that in α ′ h is unchanged when computed in ∆ S (h; u 1 , u 2 , u 3 ; Z) or in ∆ S (h; u 1 , u 2 , u ′ 3 ; Z), i.e. obtained by substituting the variable U 3 by the variable U ′ 3 . Therefore σ ′ is again an edge of ∆ S (h; u 1 , u 2 , u ′ 3 ; Z). If x satisfies condition (*1) or (*2), we deduce that

If x satisfies condition (*3), we obtain

then y ∈ {(1, 0), (0, 1)} because 2-solvable vertices have integer coordinates. By Definition 7.17 and Definition 7.10, we have

). Otherwise, it can be assumed w.l.o.g. that d = 0 by substituting u 2 by u ′ 2 = u 2 + δu 1 , where δ ∈ S is a preimage of d ∈ k(x). Note that this substitution does not change the requirements in Definition 7.10 and we thus get U ′ 3 ∈ Vdir(x) as required. By (7.45), we now have

.

Iterating this construction, we deduce that there exists a sequence (finite or infinite) of 2-solvable vertices (y (i) ) i≥0 , y (0) := y and corresponding well adapted coordinates (u 1 , u 2 , u

)

2 is chosen to be minimal at each step, we have y

This proves the existence of well 2-adapted coordinates when S = Ŝ.

Let now (u 1 , u 2 ; u 3 ; Z) and (u ′ 1 , u ′ 2 ; u ′ 3 ; Z ′ ) be two sets of well 2-adapted coordinates. We assume of course that div(u j ) = div(u ′ j ), j = 1, 2, in case (*2). To prove that ∆

2 (h; u 1 , u 2 ; u 3 ; Z) and let α ∈ R 3 >0 be given by Lemma 7.15(4) w.r.t. the face σ 2 := y. Since y ∈ ∆ + 2 (h; u 1 , u 2 ; u 3 ; Z), we have

. We deduce that in α h is unchanged when computed w.r.t. the coordinates (u ′ 1 , u ′ 2 ; u 3 ; Z). This implies furthermore that y

) for every α = α(y). Otherwise, there is an expansion

with Σ finite, δ, γ(x) ∈ S units and µ α (u

This is a contradiction since (u ′ 1 , u ′ 2 ; u ′ 3 ; Z ′ ) are well 2-adapted coordinates, so we get

Let now (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x satisfying Definition 7.10. Applying finitely many times the above algorithm and (7.46), as ∆ 2 (h; u 1 , u 2 , u 3 ; Z) = ∅ by Lemma 7.15(3), it can be assumed w.l.o.g. that

,

Definition 7.19. Let x satisfy condition (*) and (u 1 , u 2 ; u 3 ; Z) be well 2adapted coordinates. We let

Finally, we define γ(x) ∈ N by:

. Lemma 7.20. Assume that κ(x) = 2 and x satisfies condition (*). Let (u 1 , u 2 ; u 3 ; Z) be well 2-adapted coordinates and assume furthermore that

))

if x satisfies condition (*1) or (*2) (resp. condition (*3)). Let π : X ′ → X be the blowing up along Y := V (Z, u 1 , u 3 ) ⊂ X and x ′ ∈ π -1 (x). Then x ′ is resolved or the following holds:

and x ′ satisfies again condition (*1) or (*2) (resp. (*3)); the coordinates

-1 and we have:

Proof. By Theorem 7.18, the curve Y is permissible since A 1 (x) ≥ 1. Since U 3 ∈ Vdir(x) by Definition of well 2-prepared coordinates, x is then good except possibly if Vdir(x) =< U 3 > by Theorem 3.13; in this case, we have

)) is a nonsolvable vertex. We may assume that x ′ is very near x.

If x satisfies condition (*1) (resp. (*2)), then κ(x ′ ) = 2 and x ′ satisfies again condition (*1) (resp. (*2)).

If x satisfies condition (*3) and ǫ(x ′ ) = ǫ(x), then κ(x ′ ) = 2 and x ′ satisfies again condition (*3).

If x satisfies condition (*3) and ǫ(x ′ ) = ω(x), then x ′ satisfies the assumptions of Lemma 7.1, so x is good if

Therefore i < ω(x) since (i 0 , i) = (ω(x), ω(x)); then x ′ is good by Proposition 7.8, so x is good.

),

Lemma 7.22. With notations as above, assume that x is in case (*1) or (*2). Let (u 1 , u 2 ; u 3 ; Z) be well 2-adapted coordinates at x and assume furthermore that

Then x ′ is resolved or (κ(x ′ ) = 2, x ′ satisfies again condition (*) with

and there exist well 2-adapted coordinates

, then x ′ satisfies condition (*1) or (*3), and either (3') below holds or ( 3)-( 4) below hold;

(3') the point x satisfies condition (*2) with

where d 1 , d 2 ∈ N, λ, µ, c p ∈ k(x), λµc p = 0 and µ -1 c p ∈ k(x) p up to change of well 2-adapted coordinates; furthermore, x ′ satisfies condition (*1), k(x ′ ) = k(x) and we have

)

(4) we have

We prove here that the bound in Lemma 7.22(3) is sharp when x ′ satisfies either condition (*1) or (*3).

where

, where

, we get

where

In both cases we get β(x ′ ) = C + 1/p. Note that the above argument also works for (a = 1 and x ′ satisfies condition (*1)).

We now turn to the (*3)-version of the previous lemma. We point out that the situation J 0 ⊂ pN has not been correctly analyzed in the proof of [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma I.8.7. Namely, the bound (3') (ibid. p. 1929) may fail (case 2 on p.1930 when d = 1) unlike stated therein; the same mistake occurs in I.8.7.5 case 1.

We review and amend the corresponding statements in Lemma 7.24(2) below. Denote F p,Z,α as in Lemma 7.15 [START_REF] Abhyankar | Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry[END_REF]. Adapting notations of (7.54), there is an expansion (7.60) where µ, c ∈ k(x) (µ = 0 if B(x) > 1), and bounds:

The subset J 0 ⊆ {1, . . . , ω(x)} is defined by

(7.61) Lemma 7.24. Assume that x satisfies condition (*3). Let (u 1 , u 2 ; u 3 ; Z) be well 2-adapted coordinates at x and assume furthermore that

Then x ′ is resolved or (κ(x ′ ) = 2, x ′ satisfies condition (*1) or (*3) with

and there exist well 2-adapted coordinates

up to change of well 2adapted coordinates; furthermore x ′ satisfies condition (*1) and (7.57) holds at x ′ with λ ′ = 0 and (d

and (7.60) reads

) ω(x) (7.62)

; furthermore, we have

and x ′ satisfies condition (*1) with

Proof. We already know from Proposition 7.11(ii) that x ′ is resolved or (κ(x ′ ) = 2 and x ′ satisfies condition (*)). Note that we have

> .

If τ ′ (x) ≥ 2, we certainly have B(x) = 1 and x is of type (T1) or (T4) as defined along the proof of Proposition 7.11. For type (T4), x ′ is resolved by Theorem 3.13 since Vdir(x) =< U 3 , U 1 >. For type (T1), note that we have β(x) = 1, hence γ(x) = 2. The following holds: x is good or κ(x ′ ) = 2, x ′ satisfies condition (*) and there exist well 2-adapted coordinates

See the discussion along the course of the proof of Proposition 7.11: these two situations correspond respectively to case 1 and case 2 therein. This proves that x ′ is resolved or (γ(x ′ ) = γ(x) = 2 and (1)(3) hold) when τ ′ (x) = 2. Assume now that (B(x) = 1 and τ ′ (x) = 1). The argument in the proof of Proposition 7.11, viz. (7.26)-(7.27), gives

, and Φ i monic in T 2 whenever Φ i = 0. Similarly, we have

with notations as in (2.42). After possibly changing Z with Zφ, φ ∈ S, it can thus be assumed that

, where γ ∈ S is a preimage of λ. The proof now goes on along the same lines as that of the case B(x) = 1 in the previous lemma: x ′ is resolved or x ′ satisfies condition (*1), A 1 (x ′ ) = 0 and one of (7.58)-(7.59) holds (in particular γ(x ′ ) = 2). When (7.59) holds with ω(x) = p, we have (1)'; otherwise, we have (1), (3) being pointless.

For (2), note that x ′ satisfies the assumptions of Proposition 7.8 (so x is good) if c i = 0 for some i < ω(x)/p. Otherwise, we have

). (7.64) By Definition 7.19, we also have

By assumption, γ(x) = 1, so β(x) < 1 and we get

We deduce that i 1 = i = ω(x). By (7.64), this implies that

)

and the conclusion follows. This is the special situation considered in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma I.8.7(b).

Case 4:

. We now have

2 ) for j ∈ J 0 and must take

Arguing as in the case (B(x) = 1 and τ ′ (x) = 1), we obtain the same bound as in case 1 except possibly if

where k := B(x)γ(x) ∈ N. Define:

is a vertex which is not 2-solvable and we get β(x ′ ) ≤ γ(x). Otherwise, we may assume w.l.o.g. that Q = 0 after changing Z with Zφ, φ ∈ S, which gives (7.62). One concludes as in the case (B(x) = 1 and τ ′ (x) = 1) above.

We now consider the remaining point "at infinity" for the blowing up π : X ′ -→ X along x. Lemma 7.25. With notations as above, assume that x satisfies condition (*). Let (u 1 , u 2 ; u 3 ; Z) be well 2-adapted coordinates at x and assume furthermore that

and the following holds:

Proof. This relies on the characteristic free Proposition 2.18. The argument in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemmas I.8.8 and I.8.9 gives all statements before "γ(x ′ ) ≤ γ(x)". Moreover equations (2) on p.1933 and (2) on p.1934 give:

Assume that x is in case (*1) or (*3). We have

This proves (3); if x satisfies condition (*1), then β 2 (x) ≥ 0 and the conclusion follows from (7.70). If x satisfies condition (*2), we have

is a trivial consequence of Definition 7.19 except if (x is in case (*3) and C(x) < 0). But then we have β 2 (x) = -1/i for some i, 1 ≤ i ≤ ω(x) by Lemma 7.15 and Corollary 7.16. Therefore 3) and we get γ(x ′ ) ≤ 1.

Monic expansions: the algorithm.

In this chapter, we prove Theorem 5.5 when κ(x) = 2. This is restated as Theorem 7.26 below. The strategy of the proof has much in common with the one used for Theorem 6.3 or for Embedded Resolution of Singularities for surfaces [START_REF] Cossart | Resolution of surface singularities[END_REF]: roughly speaking, the invariant γ(x) is in general nonincreasing by blowing up a point x, and drops at a nonrational exceptional point or exceptional point "at infinity" x ′ . Infinite chains of rational points not "at infinity" do not occur by Corollary 3.20. This general idea is illustrated by the proof of Proposition 7.28 below which provides the logical scheme of the proof.

Considering however the precise behaviour of the invariant γ(x) under blowing up, the situation turns out to be more complicated than expected. Two phenomena contribute: on the one hand, the directrix vector space Vdir(x) is not well-behaved under blowing up; on the other hand, γ(x) does not necessarily drop at a nonrational exceptional point or exceptional point "at infinity" and may also increase in some special situations (Lemma 7.24(1')(2)). These phenomena make the proof very intricate when γ(x) = 2, especially when p = 2. One is then driven to a step by step proof where the main difficulty is to avoid loops (Propositions 7.31 to 7.37). We also emphasize that most of these intricacies actually occur when S is equicharacteristic with algebraically closed residue field.

Let µ be a valuation of L = k(X ) centered at x and consider the quadratic sequence

along µ. We will show that x r is resolved for some r ≥ 0, hence x is good as explained in Remark 5.7.

Theorem 7.26. Projection Theorem 5.5 holds when κ(x) = 2. One may take all local blowing ups in (5.3) permissible (of the first kind or second kind) if p = 2 or if ω(x) ≥ 3.

Proof. By Proposition 7.11, it can be assumed that ω(x) ≡ 0 modp and that x r satisfies condition (*) for every r ≥ 0. Under these assumptions, the invariant γ(x r ) ∈ N is defined for r ≥ 0 (Definition 7. [START_REF] Cossart | Desingularization of embedded excellent surfaces[END_REF]. By Proposition 7.28 below, there exists r 0 ≥ 0 such that either x r 0 is resolved or γ(x r 0 ) ≤ 2.

If γ(x r 0 ) = 0, then x r 0 is resolved by Proposition 7.21. Suppose that γ(x r 0 ) = 1. If x r 0 satisfies condition (*1) (resp. (*2)), then x r 0 is resolved by Proposition 7.29(1) (resp. Proposition 7.30) below. If x r 0 satisfies condition (*3) and β(x) < 1 -1/ω(x) (resp. and (k(x 1 ) = k(x) and γ(x 1 ) = γ(x)). Iterating, we conclude from Corollary 3.20 that x r is resolved or γ(x r ) < γ(x) for some r ≥ 1.

Assume now that x satisfies condition (*2). By the above considerations and Lemma 7.22(4), we are done except possibly if x 1 satisfies again condition (*2). Iterating, we conclude from Lemma 7.27 above that x r is resolved or γ(x r ) < γ(x) for some r ≥ 1.

Assume finally that x satisfies condition (*3) with β(x) = γ(x) -1/ω(x). By the above considerations, we are done except possibly if k(x 1 ) = k(x) and ( 1) or (2) below holds:

(1) x 1 satisfies again condition (*3) with β(x 1 ) = β(x);

(2) x 1 satisfies condition (*1) with β(x 1 ) = γ(x) + 1/ω(x), viz. (7.77).

Suppose that (2) holds; we now review the above proof with this extra assumption in mind. Since β(x 1 ) > 3, β(x 1 ) = 4, (7.74) or (7.76) applied to the point x 1 give the stronger

We conclude that either x 2 is resolved, either γ(x 2 ) < γ(x), or x 2 satisfies again condition (*1) with β(x 2 ) ≤ β(x 1 ). If the latter inequality is strict, we have β(x 2 ) ≤ γ(x) and we are thus already done. Otherwise x 2 satisfies again [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF].

Summing up, there exists r 0 ≥ 0 such that either x r 0 is resolved, either γ(x r 0 ) < γ(x), or (x r satisfies one and the same property (1) or (2) above for every r ≥ r 0 ). Iterating, we conclude again by Corollary 3.20.

Proposition 7.29. Assume that κ(x) = 2 and one of the following properties holds:

(1) x satisfies condition (*1) with γ(x) = 1;

(2) x satisfies condition (*2) with β(x) < 1;

(3) x satisfies condition (*3) with β(x) < 1 -1/ω(x).

Then x is good.

in any case. If (x satisfies condition (1) with A 1 (x) = 0), then x is good by Proposition 7.8. Applying repeatedly Lemma 7.20 if A 1 (x) ≥ 1, it can be assumed w.l.o.g. that 0

To prove the proposition, we first claim: x 1 is resolved or (x 1 satisfies again the assumptions of the proposition and c(x 1 ) ≤ c(x) for the lexicographical ordering), where c(x) := (A 1 (x), β(x)).

If x 1 belongs to the first chart, i.e. x 1 is distinct from the point x ′ at infinity (7.73), we apply Lemma 7.22 and Lemma 7.24. Note that the special situations described in Lemma 7.22(3') and in Lemma 7.24(1')(2) do not occur under the assumptions of the proposition, so we may also disregard them in this proof. We obtain that x 1 is resolved or x 1 satisfies again condition (*) with

.79)

Assume that x 1 belongs to the first chart and x satisfies (1). We have C(x) ≤ β(x) ≤ 1. If k(x 1 ) = k(x), it can be assumed that x 1 is the origin of the chart by the independence statement in Theorem 7. [START_REF] Cossart | Sur le polyèdre caractéristique d'une singularité[END_REF]. By Lemma 7.22(1) we have β(x 1 ) ≤ β(x) and the claim follows. Note that we obtain c(x 1 ) = c(x) only if β(x) = 1 by (7.79), in which case x 1 satisfies again (1). If k(x 1 ) = k(x), the claim follows from Lemma 7.22(4) with strict inequality c(x 1 ) < c(x).

Assume that x 1 belongs to the first chart and x satisfies (2). Since β(x) < 1, inequality is strict in (7.79). The claim also follows from Lemma 7.22(1)(4) with strict inequality c(x 1 ) < c(x).

Assume that x 1 belongs to the first chart and x satisfies (3). Note that if x 1 satisfies condition (*1), then x 1 satisfies again the assumptions of the proposition since Lemma 7.24(2) does not occur for β(x) < 1 -1/ω(x); this is also true if x 1 satisfies condition (*3) by Lemma 7.24(3) (note that p = ω(x) = 2 does not occur: (7.78) gives A 1 (x) = 1/2 while (3) gives β(x) = 0, a contradiction with B(x) ≥ 1). The claim now follows with strict inequality c(x 1 ) < c(x) by (7.79).

Assume that x 1 = x ′ (point at infinity (7.73)). Turning to Lemma 7.25, x ′ is resolved or x ′ satisfies condition (*2) with

by (7.78). This proves the claim with c(x 1 ) < c(x) in this case.

Summing up, we have proved the claim with strict inequality c(x 1 ) < c(x) except possibly if both x and x 1 are in case (*1), k(x 1 ) = k(x) and β(x 1 ) = β(x) = 1. One concludes the proof again by Corollary 3.20.

Proposition 7.30. Assume that κ(x) = 2, x satisfies condition (*2) and γ(x) = 1. Then x is good.

Proof. By Lemma 7.22(4), x 1 is resolved or satisfies the assumptions of Proposition 7.29(1) or (3) if x 1 is not a point at infinity. Therefore x 1 is resolved in this case. If x 1 is the origin of a chart, then x 1 is resolved or satisfies again the assumptions of this proposition by Lemma 7.25 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF].

Applying Lemma 7.27, it can thus be assumed that C(x) = 0. Applying repeatedly Lemma 7.20 if

Then β(x) = A 2 (x) < 1 and the conclusion follows from Proposition 7.29(2).

Proposition 7.31. Assume that κ(x) = 2 and one of the following properties holds:

(i) x satisfies condition (*1) with β(x) < 2;

(ii) x satisfies condition (*3), β(x) = 1 -1/ω(x) and (p, ω(x)) = (2, 2).

Then x is good.

Proof. Note that the special situations described in Lemma 7.24(1')(2) do occur here.

Assume that x 1 belongs to the first chart. Under assumption (i), x 1 is resolved or x 1 satisfies condition (*1) or (*3); note that the latter occurs only if k(x 1 ) is an inseparable extension of k(x) (in particular d ≥ p) and d 1 ∈ N. Then by Lemma 7.22(4):

, so x 1 is resolved by 7.29 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF]. When x 1 satisfies condition (*1), by Lemma 7.22(4), x 1 satisfies again assumption (i) of this proposition with k(x 1 ) = k(x) by Lemma 7.22(4), or is resolved by Proposition 7.29(1)(3).

Under assumption (ii), x 1 is resolved or x 1 satisfies condition (*1) or (*3). If x is as stated in Lemma 7.24(1'), then x 1 is resolved or satisfies assumption (i) with β(x 1 ) = p/(p -1) < 2, since (p, ω(x)) = (2, 2).

Otherwise we may apply Lemma 7.24(1)-( 3): if x 1 satisfies condition (*1), we get β(x 1 ) ≤ 1 + 1/p, β(x 1 ) ≤ 1 if k(x ′ ) = k(x), from Lemma 7.24(1); if x 1 satisfies condition (*3), we get β(x 1 ) ≤ β(x), strict inequality if k(x ′ ) = k(x), from Lemma 7.24(2)(3). By Proposition 7.29(1)(3), x 1 is resolved or satisfies again the assumptions of the proposition with k(x 1 ) = k(x). Assume that x 1 = x ′ is the point at infinity. Then x 1 is resolved or x 1 satisfies condition (*2) with C(x 1 ) < 1 by Lemma 7.25(1)(3); therefore x 1 is resolved in any case by Proposition 7.30.

One concludes the proof again by Corollary 3.20.

Lemma 7.32. Assume that κ(x) = 2 and one of the following properties holds:

(i) x satisfies condition (*1) with β(x) = 2;

(ii) x satisfies condition (*3) with β(x) < 2.

Let (u 1 , u 2 ; u 3 ; Z) be well 2-adapted coordinates at x and

be the point at infinity. Then x ′ is resolved or (x ′ satisfies condition (*2) with C(x ′ ) = 1 and the following respectively hold:)

Proof. By Lemma 7.25, x ′ is resolved or x ′ satisfies condition (*2).

Under assumption (i), Lemma 7.25(1) furthermore gives C(x ′ ) ≤ 1; if C(x ′ ) < 1, we are done by Proposition 7.30. If C(x ′ ) = 1, Lemma 7.25 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] implies that C(x) = 1; moreover

(7.80)

We now prove that x ′ is resolved unless (p = 2 and d 1 ∈ N). To prove this, it is sufficient to prove that any possible x 2 in (7.72) is resolved when

notations as in Lemma 7.15(4) w.r.t. the face σ 2,in of ∆ 2 (h ′ ; u ′ 1 , u 2 ; u ′ 3 ; Z ′ ). We expand

where d ′ 2 := d 1 + ω(x)/p -1 and

we have by definition

be the points "at infinity". If

so the latter holds if and only if (p = 2 and d 1 ∈ N) as required.

Under assumption (ii), we are done by Proposition 7.

and that

The proof is now a variation of that under assumption (i) and we explain now how it is to be adapted. To begin with, (7.81) holds with

Note in particular that we have

x ′′ 2 }, we apply Lemma 7.25:

2 ) = 0) by (7.83) (resp. by (7.85)). Therefore x 2 is resolved in any case by Proposition 7.30.

If

, then x 2 is resolved by the same argument as under assumption (i).

If

, we first note that x ′ is not as specified in Lemma 7.22(3'): since C(x) < 1, we have A 1 (x ′ ) = A 1 (x) > 0. Applying then Lemma 7.22(3)(4), the argument used under assumption (i) gives x 2 resolved or

Lemma 7.33. Assume that κ(x) = 2 and x has one of the following properties:

(i) x satisfies condition (*1), β(x) = 2 and, given well 2-adapted coordinates (u 1 , u 2 ; u 3 ; Z), the polynomial in α h = Z p -G p-1 α Z + F p,Z,α , where

notations as in Lemma 7.15(4) w.r.t. the face

has µ i = 0 for some i with 1 ≤ i ≤ p -1;

(ii) x satisfies condition (*3) and β(x) < 2 -1/p.

Then x is good.

Proof. We again consider three cases.

Assume that x 1 = x ′ is the point at infinity. We review the proof of Lemma 7.32 with our extra assumptions and claim that x ′ is resolved.

Under assumption (i), we get 1 ≤ i 2 ≤ p -1 in (7.82) by (7.86). Turning to (A) and (B) in the proof of Lemma 7.32, note that (A) does not hold since µ 1 = 0 in (7.86). Finally if (B) holds, then β(x 2 ) ≤ 1 -1/(p -1) because 1 ≤ i 2 ≤ p -1. Therefore x 2 is resolved by Proposition 7.29 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF].

Under assumption (ii), note that (7.84) is strengthened to (7.85). We also get β(x 2 ) ≤ 1 -1/(p -1) if (B) holds, so x 2 is resolved by Proposition 7.29 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF].

Assume that k(x 1 ) = k(x). If x 1 satisfies condition (*1), Lemma 7.22(4) and Lemma 7.24(1) give β(x) < 2 in any case. Therefore x 1 is resolved by Proposition 7.31(i).

If x 1 satisfies condition (*3), the same conclusion holds under assumption (i) except possibly if C(x) = d = 2. By (7.86), we then get β(x 1 ) ≤ 1 -1/(p -1) and x 1 is resolved by Proposition 7.29 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF]. Under assumption (ii), x 1 satisfies again the assumption (ii) in this lemma with β(x 1 ) < β(x) by Lemma 7.24 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF].

Assume that x 1 = x ′ and k(x 1 ) = k(x). The independence statement in Theorem 7.18 reduces to

Note that the extra assumption (7.86) is unaffected by this coordinate change.

Under assumption (i), Lemma 7.22 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] shows that x 1 is resolved or x 1 satisfies again condition (*1) with β(x 1 ) ≤ β(x) = 2. By Proposition 7.31(i), x 1 is resolved unless equality holds. In this case, we have

and x 1 satisfies again assumption (i) of this lemma.

Under assumption (ii), Lemma 7.24 shows that x 1 is resolved or satisfies condition (*1) or (*3). If one of Lemma 7.24(1')(2) applies, we have γ(x) = 1 and x 1 satisfies condition (*1) with β(x 1 ) ≤ 2. We are done if inequality is strict by Proposition 7.31(i); otherwise ω(x) = p = 2 and x 1 satisfies (i) of this lemma.

Any other situation yields γ(x 1 ) ≤ γ(x). If x 1 satisfies condition (*3), then x 1 satisfies again (ii) of this lemma with β(x 1 ) ≤ β(x) by Lemma 7.24 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF]. If x 1 satisfies condition (*1), we have β(x 1 ) ≤ 2. We are done if inequality is strict by Proposition 7.31(i).

Assume then that (x 1 satisfies condition (*1) and β(x 1 ) = 2). We argue as in the proof of Lemma 7.24. Denote F p,Z,α as in Lemma 7.15 [START_REF] Abhyankar | Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry[END_REF]. We have:

where µ ∈ k(x) and

By assumption (ii), we have

Note that for j ∈ J 0 , we then have deg U 2 Ψ j (U 1 , U 2 ) ≤ 2j, and inequality is strict if j ≥ p. If min J 0 ≥ p, arguing as in the proof of Lemma 7.24 (B(x) > 1, cases 1 to 4), we then get β(x 1 ) < 2: a contradiction. This proves that 1 ≤ j 0 := min J 0 ≤ p -1. (7.88)

, where (u 1 , u ′ 2 ; u ′ 3 ; Z ′ ) are well 2-adapted coordinates. With notations as in Lemma 7.15(4), the initial form polynomial in α ′ h ′ w.r.t. the face σ ′ 2 = y ′ satisfies an equation (7.86), say

) (7.88). Therefore x 1 satisfies assumption (i) in this lemma.

Summing up, the following has been proved: if x satisfies (i), then x 1 is resolved or (k(x 1 ) = k(x) and x 1 satisfies again (i)). If x satisfies (ii), then x 1 is resolved or x 1 satisfies (i) or (ii); if (ii) holds, then β(x 1 ) ≤ β(x) and inequality is strict if k(x 1 ) = k(x).

Consider the quadratic sequence (7.72). By the previous considerations, there exists r 0 ≥ 0 such that either x r 0 is resolved, or (x r satisfies one and the same assumption in the lemma with k(x r ) = k(x r 0 ) for every r ≥ r 0 ). One concludes the proof again by Corollary 3.20.

We will now conclude the proof of Theorem 7.26. Note the interesting extra twist for p = 2. Proposition 7.34. Assume that κ(x) = 2 and one of the following properties holds:

(i) x satisfies condition (*1) with β(x) = 2;

(ii) x satisfies condition (*3) and β(x) < 2 -1/ω(x).

Then x is good.

Proof. This is a variation on the two previous lemmas. Note that we may disregard the special case stated in Lemma 7.24(1') in this proof.

Assume that x 1 = x ′ is the point at infinity. By Lemma 7.32, x ′ is resolved under assumption (i) (resp. (ii)) if p ≥ 3 (resp. if p = 2). Reviewing the proof of Lemma 7.32, we are done except possibly when (A) or (B) stated therein hold. If (A) holds, then x 2 is resolved by Lemma 7.33(i). If (B) holds, x 2 satisfies condition (*3) with β(x 2 ) ≤ 1 + 1/p. If p ≥ 3 or if (p = 2 and β(x 2 ) < 3/2), we have β(x 2 ) < 2 -1/p and the conclusion follows from Lemma 7.33(ii). Therefore x ′ is resolved or p = 2 and β(x 2 ) = 3/2.

In the special case p = ω(x) = 2, an explicit computation gives β(x 2 ) ≤ 1 if x 2 satisfies condition (*3) (cf. (ii) of proof of Lemma 7.35 below), so x ′ is resolved. This proves that x 2 is resolved or satisfies again the assumptions of the proposition in any case.

Assume that k(x 1 ) = k(x). Under assumption (i), x 1 is resolved or Assume that x 1 = x ′ and k(x 1 ) = k(x). We may assume once again that x 1 is the origin of the first chart of the blowing up.

Under assumption (i), x 1 is resolved or x 1 satisfies again assumption (i): same proof as in Lemma 7.33(i).

Under assumption (ii), x 1 is resolved or satisfies again one of (i)(ii): same proof as in Lemma 7.33(ii). If x 1 satisfies again (ii), we have β(x 1 ) ≤ β(x) by Lemma 7.24 [START_REF] Abhyankar | On the valuations centered in a domain[END_REF]. Summing up, it has been proved that x 1 is resolved or x 1 satisfies again the assumptions of the proposition. Under assumption (i), x 1 is resolved or one of the following properties holds:

(1) k(x 1 ) = k(x) and x 1 satisfies again (i);

(2) p = 2 and x 1 satisfies (ii) with β(x 1 ) = 3/2;

(3) p = 2 and x 2 satisfies (ii) with β(x 2 ) = 3/2.

Under assumption (ii), x 1 is resolved or one of the following properties holds:

(1') k(x 1 ) = k(x) and x 1 satisfies (i);

(2') k(x 1 ) = k(x) and x 1 satisfies again (ii) with β(x 1 ) ≤ β(x).

Consider the quadratic sequence (7.72) and suppose that (2) (resp. ( 3)) above occurs. Suppose that event (1') occurs again at x r for r ≥ 1 (resp. for r ≥ 2). By (2') and Lemma 7.33(ii), we may assume that β(x r ) = 3/2, so x r is resolved by Lemma 7.35 below. Therefore there exists r 0 ≥ 0 such that either x r 0 is resolved, or (x r satisfies one and the same assumption (i) or (ii) with k(x r ) = k(x r 0 ) for every r ≥ r 0 ). The proof now concludes once again by Corollary 3.20. Lemma 7.35. Assume that p = 2, κ(x) = 2 and x satisfies condition (*3) with β(x) = 3/2. If x 1 satisfies condition (*1), then x 1 is resolved.

Proof. We argue as in the proof of Lemma 7.33 (7.87) and (7.89): we have β(x 1 ) = 2 and, since β(x) = 3/2, there exist well 2-adapted coordinates (u 1 , u ′

and get µ ′ 2 = 0 in (7.90). Since β(x 1 ) = 2, we nevertheless obtain µ ′ j 0 = 0 for some j 0 ≥ 3 in (7.90). In other terms, we may assume that one of the following assumptions holds:

2 = 0 and µ ′ j 0 = 0 for some j 0 ≥ 3. We consider three cases and review again the proof of Lemma 7.33: Assume that x 2 = x ′ 1 is the point at infinity. Situation (A) has been solved in Lemma 7.33(i). Situation (B) does not hold by [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] proof of I.8.3: equality β(x 3 ) = 3/2 is achieved only in the situation of ibid. I.8.3.6 case 2. This implies (µ ′ j = 0 for 1 ≤ j ≤ 2 a -1, and µ ′ 2 a = 0): a contradiction with (i) and (iii) above. This also implies B(x) = A 1 (x) + β(x) ∈ N viz. [30] I.8.3.4 (so A 1 (x) ∈ N since β(x) = 2), and

[30] I.8.3.5 where d ′ 1 ∈ N here: a contradiction with (ii). One gets β(x 3 ) < 3/2 (actually: β(x 3 ) ≤ 1 if x 3 satisfies condition (*3)), so x ′ is resolved by Lemma 7.33(ii). Assume that k(x 2 ) = k(x 1 ). Then x 2 is resolved. Assume that x 2 = x ′ 1 and k(x 2 ) = k(x 1 ). Then x 2 is resolved or x 2 satisfies again (7.90) with µ ′ j = 0 for some j ≥ 1, j ≤ 2 if a ≥ 2. Iterating, the conclusion follows again from Corollary 3.20.

Proposition 7.36. Assume that κ(x) = 2, x satisfies condition (*2) with γ(x) = 2. Then x is good. Proof. By Lemma 7.22, x 1 is resolved or satisfies again condition (*) with γ(x 1 ) ≤ 2.

If x 1 satisfies condition (*1), then x 1 is resolved by Proposition 7.31(i) or by Proposition 7.34(i).

If x 1 satisfies condition (*3), we have β(x 1 ) < 2 -1/ω(x) by Lemma 7.22 [START_REF] Abhyankar | Ramification theoretic methods in algebraic geometry[END_REF]. Therefore x 1 is resolved by Proposition 7.34(ii).

If x 1 satisfies condition (*2) and γ(x 1 ) = 1, x 1 is resolved by Proposition 7.30. Therefore x 1 is resolved or satisfies again the assumptions of the lemma. The conclusion follows from Lemma 7.27. Proposition 7.37. Assume that κ(x) = 2, x satisfies condition (*3) with β(x) = 2 -1/ω(x). Then x is good.

Proof. This is now a variation of Proposition 7.28. By Lemma 7.24, x 1 is resolved or satisfies again condition (*) with γ(x 1 ) ≤ 2 except in the special situation specified in Lemma 7.24 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. Applying the previous lemmas, we are done except possibly if k(x 1 ) = k(x) and ( 1) or (2) below holds:

(1) x 1 satisfies again condition (*3) with β(x 1 ) = β(x) = 2 -1/ω(x);

(2) x 1 satisfies condition (*1) with β(x 1 ) = 2 + 1/ω(x).

Suppose that (2) holds; by Lemma 7.22(1)(4) and Lemma 7.25(2), x 2 is resolved (γ(x 2 ) ≤ 2, β(x 2 ) < 2 -1/ω(x) if x 2 satisfies condition (*3)) or satisfies again (2) with k(x 2 ) = k(x 1 ). In both cases (1)(2), we conclude once more by Corollary 3.20.

8 Projection Theorem: transverse and tangent cases, reduction of κ(x) = 3, 4 to monic expansions.

In this chapter and the next one, we prove Theorem 5.5 when κ(x) = 3, 4 (Definition 5.1). This is restated as Theorem 9.6 below. The structure of the proof is similar to that of Theorem 7.26: first getting a stable form for the equation of in m S h (i.e. monic expansions, Definition 8.1 below), then introducing a projected polygon with secondary invariant γ(x). Two important differences with κ(x) = 2 arise. On the one hand, no simple reduction works for each of κ(x) = 3, 4 separately and we have to deal with both cases at the same time. On the other hand, the monic case is resolved by blowing up Hironaka-permissible centers Y ⊂ X which are not necessarily permissible in the sense of Definitions 3.1 and 3.5.

Given a valuation µ of L = k(X ) centered at x, we consider finite sequences of local blowing ups along µ:

with Hironaka-permissible centers Y i ⊂ (X i , x i ), viz. (5.3).

where γ 1 ∈ S is a preimage of λ 1 . By assumption,

= 0 or homogeneous of degree i, c i ∈ k(x) and c i = 0 for some i = ω(x). Let (u 1 , v, u ′ 3 ; Z ′ ) be well adapted coordinates at x 1 . Applying Proposition 3.9(v) (with W ′ := div(u 1 ) ⊂ SpecS ′ ), we have

Since ω(x) = ω(x 1 ) and κ(x 1 ) ≥ 3 are assumed, we have

If ǫ(x 1 ) = ǫ(x), we get Vdir(x 1 )+ < U 1 >=< U 1 , V, U ′ 3 > by (8.6), so κ(x 1 ) = 2 by Definition 5.1: a contradiction. Therefore ǫ(x 1 ) = ω(x). Let

We deduce from (8.6) that

The proof is now a variation of that of Proposition 7.6, τ ′ (x) = 1. We treat first the case d ′ 1 ∈ N. We state the case d ′ 1 ∈ N in the following lemma for further use: the assumptions are satisfied by (8.7)-(8.8) and this will conclude the proof.

Let us blow up along x 1 , let x 2 be a point very near to x 1 . As κ(x 1 ) > 2 is assumed, we have < U 1 >=Vdir(x 1 ). By (8.8), we have ω(x) ≥ 2, all this implies that x 2 is rational over x 1 . Obviously, x 2 is not the point of parameters (Z ′ /u ′ 3 , u 1 /u ′ 3 , v/u ′ 3 , u ′ 3 ). After an eventual translation on u ′ 3 and maybe on Z ′ to get well adapted coordinates at x 1 , after some abuse of notations, we may assume that x 2 is the point of parameters (Y, u 1 , u 2 , w) := (Z ′ /v, u 1 /v, v, u ′ 3 /v). With W ′′ :=div(u 1 ), we get

, we are done by Lemma 7.1.

Lemma 8.4. Assume that ǫ(x) = ω(x) and E = div(u 1 ). Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x. Assume furthermore that the initial form polynomial

where

. Then x is resolved.

Proof. It can be assumed that κ(x) = 4, i.e. Vdir(x) =< U 1 >. We then review the proof of Proposition 7.6 for τ ′ (x) = 1, cases 1 and 2. We take Y 0 := {x} in (8.1). Case 2 of loc.cit. gives ι(x 1 ) ≤ (p, ω(x), 2) after blowing up x, hence x 1 is resolved. Similarly, case 1 yields ι(x 1 ) ≤ (p, ω(x), 2) or after possibly changing well adapted coordinates:

with ω(x) ≡ 0 modp and

The case ω(x) = 1 is dealt with as in Proposition 7.6: If ω(x) = 1, then (X ′ , x ′ ) satisfies the assumption of Lemma 7.2 or there is an expansion (7.17)

As in Proposition 7.6,

is is permissible of the first kind at x ′ and either blowing up Y ′ then gives ι(x ′′ ) ≤ (p, ω(x), 1) by Theorem 3.13, where x ′′ is the center of µ, or we blow up up consecutively Y ′ 1 , then Y ′ 2 , and iterating, we reduce to the case

Assume that ω(x) ≥ 2. Let E 1 := div(u ′ 1 ) ⊂ SpecS ′ be the strict transform of E. We get an expansion

It can be furthermore assumed that κ(x 1 ) = 4. By Lemma 7.3(ii) applied with a(3) = 0, we have Vdir(x 1 ) =< U ′ 1 > or Vdir(x 1 ) =< U ′ 1 , U 2 > since pd 1 = 0 is assumed in this lemma. We take Y 1 := {x 1 } in (8.1) and first consider the point

. By (8.11), we obtain ω(x ′′ ) < ω(x) (resp. τ ′ (x ′′ ) = 3) if ω(x) ≥ 3 (resp. if ω(x) = 2), so x ′′ is resolved in any case. By Theorem 3.13 it can therefore be assumed that Vdir(x 1 ) =< U ′ 1 > . (8.12)

Applying again (8.11), we obtain

Once again, we obtain ι(x 2 ) ≤ (p, ω(x), 2) or after possibly changing well adapted coordinates:

It is now clear that (8.11)-(8.12) are stable by blowing up. Iterating, we obtain that x r is resolved for some r ≥ 1 in (8.1) or there exists a formal curve Ŷ = V ( Ẑ, u 1 , û3 ) whose strict transform passes through all points x r , r ≥ 1. By Proposition 3.17 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF], it can be assumed that Y = V (Z, u 1 , u 3 ) is permissible of the first kind. Then x is resolved by blowing up Y and the conclusion follows.

Assume now that E = div(u 1 ). Let I := {i : Φ i+1 = 0}. To conclude the proof, we will prove that

Let i ∈ I. By (8.14) and (8.15), we have

There is an expansion

where U 1 does not divide Ψ i+1 . By (8.14), we have

As pb 0 ≤ 1 and b 0 ∈ N, i ∈ I: (8.13) holds for i = 0. After possibly changing Z with Z -φ, φ ∈ S, it can be assumed that pd 1 + a i ≡ 0 modp or µ i ∈ k(x) p .

If I = {0}, κ(x 1 ) > 2 implies that ǫ(x 1 ) = ω(x 1 ): x 1 satisfies the assumptions of Lemma 7.1 (or of Lemma 7.2) and the conclusion follows.

Suppose that i ≥ 1 in what follows. We can take a unitary polynomial P (t) ∈ S[t], whose reduction P (t) ∈ k(x)[t] is irreducible and

where equality holds only if a i = 1 and k(x 1 ) = k(x) by Lemma 6.7 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. In particular we have I ⊂ pN. Since k(x 1 ) = k(x), it can be assumed w.l.o.g. that P (t) = t and

for every i ≥ 0 after possibly changing well adapted coordinates (including i = 0, cf. above). Then (u 1 , u ′ 2 , u ′ 3 ; X ′ ) are well adapted coordinates at x 1 by Proposition 2.18. We obtain: ǫ(x 1 ) = ω(x) and

. But then κ(x 1 ) ≤ 2: a contradiction. This completes the proof when E = div(u 1 ).

Preliminaries: tangent case.

Let (u 1 , u 2 , u 3 ; Z) be well adapted coordinates at x, where κ(x) = 4. This splits into two different situations:

where

By Definition 2.68, we have

Definition 8.6. Assume that κ(x) = 4 and ǫ(x) = ω(x). We say that Vdir(x) is skew if for every subset J ⊆ {1, . . . , e}, we have

Assume that Vdir(x) is skew first note that e = 2 or e = 3. Elementary casuistics, similar to that performed in the proof of Proposition 7.11, yield the following types up to reordering exceptional variables:

Proposition 8.7. Assume that Vdir(x) is skew. Assume furthermore that

if x is of type (T2) above. Take (8.1) to be the quadratic sequence along µ.

Then there exists r ≥ 0 such that either x r is resolved or x r satisfies condition (**). If ω(x) < p, then x is resolved.

Proof. We discuss according to x 1 in (8.1), where x 0 = x is of type (Tk) for some k ∈ {0, 1, 2, 3}. It can be assumed w.l.o.g. that ι(x 1 ) ≥ (p, ω(x), 3). Let u ′ j := u j /u 1 , j = 2, 3. • Assume that k = 0. By (8.21), we have

(8.25) By Lemma 7.4, ω(x) ≡ 0 modp (in particular ω(x) ≥ p) and it can be assumed that

after possibly changing Z with Zφ, φ ∈ S. After possibly reordering exceptional variables, we have

where γ 1 , γ 2 ∈ S are preimages of λ 1 , λ 2 and P (t) ∈ S[t] is a unitary polynomial whose reduction P (t) ∈ k(x)[t] is irreducible. Applying Proposition 3.9(v) (with W ′ := div(u 1 ) ⊂ SpecS ′ ), we have

Suppose that Vdir(x ′ ) is skew. By (8.30), x ′ is of type (Tk) for some k ∈ {0, 1, 2, 3}. Furthermore if k = 2, then x ′ satisfies again the extra assumption in the proposition also by (8.30). We are already done if k ≤ 2, so we may assume again that x ′ is of type (T3) and iterate. In particular, we have e = 3. In case ω(x) < p, we again have d ′ j = d j , 1 ≤ j ≤ 3. By Proposition 3.17, it can be assumed that Y := V (Z, u 1 , u 2 ) is permissible of the first kind. Let π : X ′ → X be the blowing up along Y and x ′ 1 ∈ π -1 (x) satisfy ι(x ′ 1 ) ≥ (p, ω(x), 3). By Theorem 3.13, we have

where γ 1 ∈ S is a preimage of λ 1 . Then x ′ 1 satisfies condition (**). If ω(x) < p, then m(x ′ 1 ) < p and x is resolved.

Proposition 8.8. Assume that κ(x) = 4, ǫ(x) = ω(x) and E = div(u 1 u 2 ). Assume furthermore that the following properties are satisfied:

(ii) the polyhedron ∆ S (h; u 1 , u 2 , u 3 ; Z) has a vertex of the form

where (u 1 , u 2 , u 3 ; Z) are well adapted coordinates at x. Take (8.1) to be the quadratic sequence along µ. There exists r ≥ 0 such that either x r is resolved or x r satisfies condition (**). If ω(x) < p, then x is resolved.

Proof. Suppose that x 1 is very near to x. By (i) and Theorem 3.13, we have

and the polyhedron ∆ S ′ (h ′ ; u ′ 1 , u ′ 2 , u 3 ; Z ′ ) is minimal by Proposition 2.18. Since v 3 > 0 in (ii), v is induced by f p,Z by Theorem 2.36, and f p,Z has an expansion

To conclude the proof, we compute Vdir(x 1 ). First note that

by (i). If τ ′ (x 1 ) = 3, then x 1 is resolved by Theorem 3.13. Suppose that τ ′ (x 1 ) ≤ 2. This gives

Since λ ′ = 0, we have (λ ′ 1 , λ ′ 2 ) = (0, 0). We are done by Proposition 8.

where the extra assumption holds by (8.34), after permuting U ′ 2 and U 3 . Suppose finally that

We now apply Lemma 7.3(ii) to the U ′ 1 ω(x)-1 -term in (8.31), i.e. for the variables (U 3 , U ′ 2 , U ′ 1 ) respectively and i = 1. We deduce from (7.7) that

Turning back to (8.32), we get

This is a contradiction, since p ≥ 3, and the proof is complete.

Reduction to monic expansions (**) and (T**).

We can now conclude the reduction to monic expansions.

Proposition 8.9. Assume that κ(x) = 3. Let µ be a valuation of L = k(X ) centered at x. There exists a finite and independent sequence of local permissible blowing ups of the first kind (8.1) along µ such that one of the following holds for some r ≥ 0:

(i) x r is resolved or satisfies condition (T**);

(ii) x r satisfies condition (**).

Furthermore if ω(x) < p and τ ′ (x) = 2, (8. 35)

Proof. It can be assumed that the conclusions of Lemma 8.5(1) or (2) above hold.

If Φ i+1 = 0 for every i ≥ 0, then τ ′ (x) = 1 (so (8.35) does not hold) and x 1 satisfies condition (**) and we are done. Otherwise, we may furthermore assume that

with ι(x ′ ) ≥ ι(x). Note that when E = div(u 1 u 2 ), (8.13) marks an exceptional component div(u 1 ) of E.

If in (8.3) (c = 0 and ω(x) + 1 ≡ 0 modp), then x ′ satisfies condition (**) and we are done for ω(x) ≥ p. Otherwise (i.e. if either (8.35) holds or c = 0, or ω(x) + 1 ≡ 0 modp), we have E ′ = div(u ′ 1 u 2 ) and (u ′ 1 , u 2 , u ′ 3 ; Z ′ ) are well adapted coordinates at x ′ . Furthermore Vdir(x) =< U 1 , U 3 > either by (8.13) or by assumption if (8.35)

and consider two cases: Case 1: ǫ(x ′ ) = ω(x). We have κ(x ′ ) = 4 and

, then x ′ satisfies condition (T**) or x ′ satisfies the assumptions of Proposition 8.7 type (T3) respectively, and the proof is complete. We may thus furthermore assume that Vdir(x ′ ) =< U ′ 1 , U 2 > . (8.37)

Since κ(x) = 3, we have at this point:

where (u 1 , u 2 , u 3 ; Z) are well adapted coordinates at x. The proposition follows from Proposition 8.8 whose assumptions are satisfied by (8.37), (8.38).

Case 2: ǫ(x ′ ) = ǫ(x). Note that this implies λ 2 = 0 in (8.13). We again have κ(x ′ ) = 3 and may iterate:

We are done if τ ′ (x ′ ) = 3 and may iterate if τ ′ (x ′ ) = 2 as asserted.

Since the exceptional component div(u 1 ) of E has been marked (cf. beginning of the proof), the theorem holds except possibly if x r is in case 2 for every r ≥ 0. In this situation, we apply Proposition 3.17(1): w.l.o.g. it can be assumed that Y := V (Z, u 1 , u 3 ) is permissible of the first kind. Since Vdir(x) =< U 1 , U 3 >, it follows from Theorem 3.13 that x is resolved by blowing up Y. Lemma 8.10. Assume that κ(x) = 4 and ǫ(x) = ω(x). Let µ be a valuation of L = k(X ) centered at x. There exists a finite and independent sequence of local permissible blowing ups of the first kind (8.1) along µ such that one of the following holds for some r ≥ 0:

(i) x r is resolved or satisfies condition (T**);

(ii) x r satisfies condition (**).

If ω(x) < p, then (i) holds.

Proof. By Proposition 8.7, it can be assumed that one of the following conditions holds:

(1) Vdir(x) is skew and satisfies condition (T2);

(2) div(u 1 u 2 ) ⊆ E and Vdir(x) =< U 1 , U 2 >. Take (8.1) to be the quadratic sequence along µ. Under assumption (1), we have E = div(u 1 u 2 u 3 ) and Vdir(x) =< U 1 , λ 2 U 2 + U 3 >, λ 2 = 0 up to renumbering variables. By Proposition 8.7, it can be assumed that

By Theorem 3.13, we have

where γ ∈ S is a preimage of λ 2 . Let W ′ := div(u 2 ) ⊂ SpecS ′ and (u ′ 1 , u 2 , v; Z ′ ) be well adapted coordinates at x 1 . By Proposition 3.9(v), we have .39) In this last situation, the conclusion follows in each of the following possible cases:

• x 1 satisfies the assumptions of Proposition 8.8 by (8.39

, we are also done by Proposition 8.9 if κ(x 1 ) = 3, since τ ′ (x 1 ) ≥ 2. Assume finally that κ(x 1 ) = 4, i.e.

Similarly, x 1 satisfies condition (T**) unless Vdir(x 1 ) =< U ′ 1 , U 2 >. The conclusion then follows again from Proposition 8.8.

Under assumption (2), it can be assumed that

, where

By Proposition 2.18, (u ′ 1 , u ′ 2 , u 3 ; Z ′ ) are well adapted coordinates at x ′ . We get ǫ(x ′ ) = ω(x) and

, then x ′ is resolved by Theorem 3.13. Otherwise, x ′ satisfies again the assumptions of the proposition, with (1) up to renumbering variables or (2) above.

Iterating, the proof concludes as in the proof of Proposition 8.9: x is resolved or the curve Y := V (Z, u 1 , u 2 ) is permissible of the first kind; then x is resolved by blowing up Y, since Vdir(x) =< U 1 , U 2 >. Proposition 8.11. Assume that κ(x) = 4. Let µ be a valuation of L = k(X ) centered at x. There exists a finite and independent sequence of local permissible blowing ups of the first kind (8.1) along µ such that one of the following holds for some r ≥ 0:

(i) x r is resolved or satisfies condition (T**);

(ii) x r satisfies condition (**).

If ω(x) < p, then (i) holds.

Proof. By Lemma 8.10, we are done if ǫ(x) = ω(x). Otherwise one of the following conditions holds up to reordering exceptional variables:

Take (8.1) to be the quadratic sequence along µ. We may always assume that ι(x 1 ) ≥ (p, ω(x), 3) and ǫ(x 1 ) = 1 + ω(x) (8.40) in this proof. Let

where

x ′ , we have ǫ(x ′ ) = ω(x): a contradiction with (8.40). This concludes the proof under assumption (1) by Theorem 3.13.

Assume that x 1 = x ′ . Under assumption (2), we can take a unitary polynomial P (t) ∈ S[t], whose reduction P (t) ∈ k(x)[t] is irreducible, and

Let W ′ := div(u 1 ) ⊂ SpecS ′ and (u 1 , v, w; Z ′ ) be well adapted coordinates at x 1 , Z ′ := X ′φ, φ ∈ S ′ . By Proposition 3.9(v), we deduce that

where

If ω(x) < p, assumption (2) reads:

If Φ = 0, this is a contradiction since then κ(x 1 ) = 2 by (8.41). After possibly performing a linear change of coordinates in u 3 , then picking again well adapted coordinates, we reduce to:

also a contradiction, since ω(x 1 ) = ω(x) is assumed.

If ω(x) ≥ p, we may then furthermore assume that ǫ(x 1 ) = ǫ(x) by (8.40), so κ(x 1 ) = 3 by (8.41). We conclude by Proposition 8.9.

Under assumption (3), we define a refinement C of the function x → (m(x), ω(x)), cf. chapter 6. Let π : X ′ → (X , x) be the blowing up along a permissible center of the first kind Y ⊆ div(u 1 ), x 1 ∈ π -1 (x). We set: C(x 1 ) < C(x) ⇔ x 1 satisfies the conclusion of the proposition. By Theorem 3.13, we have C(x 1 ) < C(x) unless x 1 belongs to the strict transform div(u ′ 1 )

With notations as in chapter 6, we claim that div(u 1 ) has maximal contact for the condition C (Definition 6.1). To see this, suppose that C(x 1 ) = C(x) and apply Proposition 8.9, Lemma 8.10 and ( 1) and ( 2) above. It can be assumed that ǫ(x 1 ) = ǫ(x), κ(x 1 ) ≥ 3 and Y = {x}.

If ω(x) ≥ p, we are done unless x 1 satisfies again (3) and the claim is proved; if ω(x) < p, we must still check that the situation κ(x 1 ) = 3, τ ′ (x 1 ) = 1 does not occur. By assumption (3), (8.20) with G = 0 gives an expansion

after blowing up, where (u ′ 1 , u 2 , v ′ ; Z ′ ) are well adapted coordinates at x ′ : a contradiction with κ(x 1 ) = 3, τ ′ (x 1 ) = 1. This concludes the proof of the claim when ω(x) < p. The proposition now follows from Theorem 6.3. 9 Resolution of κ(x) = 3, 4 with monic expansions.

In this chapter, we prove projection Theorem 5.5 in the case where κ(x) ≥ 3.

Up to the end of this chapter, "resolved" stands for "resolved for (p, ω(x), 3)" (Remark 5.7).

9.1 From (T**) to (**), resolution for ǫ(x) = ω(x) < p.

The purpose of this section is to reduce Theorem 5.5 for κ(x) = 3, 4 to points satisfying condition (**) in Definition 8.1. This reduction uses the concept of maximal contact with respect to a refinement x → C(x) of the function x → (m(x), ω(x)), see chapter 6. Let x be in the case (T**) of Definition 8.1, in particular, κ(x) = 4. We consider a finite sequence of local blowing ups along µ:

with permissible centers of the first kind Y i ⊂ (X i , x i ), where x i , 0 ≤ i ≤ r, denotes the center of µ. For 1 ≤ i, we define:

We prove the following proposition.

Proposition 9.1. Let x be in the case (T**) of Definition 8.1, and µ be a valuation of L = k(X ) centered at x. There exists a finite and independent sequence of permissible blowing ups of the first kind

where x i is the center of µ in X i , 0 ≤ i ≤ r, such that x r is resolved or (x r satisfies condition (**) and ω(x) ≥ p).

Proof. By Proposition 9.3 below, there is weak maximal contact (Definition 6.1) for the refinement C defined above. Furthermore nonresolved points created by blowing up along closed points satisfy condition (**) with ω(x) ≥ p (Proposition 9.3(i)). Theorem 6.3 does not apply directly since maximal contact does not necessarily hold. We must check that its proof remains valid when using only those blowing ups of the first kind which are well behaved w.r.t. C (Proposition 9.4 below).

By performing blowing ups at closed points, any curve Y ⊂div(u 1 ) of generic point y with ω(y) > 0 (a) either maps to an intersection of two components of E, i.e.

, for some u 3 . Furthermore, this curve is unique.

Suppose that there exists Y permissible satisfying (b). Then Y satisfies assumption (2) of Proposition 9.4 except possibly in case (T**)(i). Let W := η(Y) and expand:

with γ i ∈ S/(u 1 , u 3 ), γ 0 a unit. We are done by Proposition 9.4(1) if γ i = 0 for 1 ≤ i ≤ ω(x). Otherwise, blow up along x. There is nothing to prove except at the point

. Then x ′ is now in case (T**)(ii) and the conclusion follows from Proposition 9.4, assumption [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. From now on, we assume that all curves contained in div(u 1 ) with ω(y) > 0 satisfy (a). In particular, all permissible curves satisfy (a).

We use notations as in Proposition 6.17 and Notation 6.5. By performing blowing ups at closed points, we reach γ(x) = 1, i.e. either:

Proposition 9.3. Let x be in the case (T**) of Definition 8.1. Then div(u 1 ) has weak maximal contact (Definition 6.1) for the condition (T**) and κ(x) ≥ 3. More precisely, let π : X ′ -→ (X , x) be the blowing up along x and x ′ ∈ π -1 (x), with ι(x ′ ) ≥ (p, ω(x), 3):

(i) if x ′ is not on the strict transform of div(u 1 ), then x ′ is resolved or satisfies (**) with ω(x) ≥ p;

(ii) if x ′ is on the strict transform of div(u 1 ), then x ′ satisfies (T**).

Proof. In the case (T**)(i), the reader sees that < U 1 >= Vdir(x) and, if we blow up along x, any point x ′ with ι(x ′ ) ≥ (p, ω(x), 3) verifies (T**)(ii) or (iii).

In the case (T**)(ii) and not (i), we have

then, by Theorem 3.13, x ′ is very near to x only if x ′ = (Z/u 3 , u 1 /u 3 , u 2 /u 3 , u 3 ). Clearly ι(x ′ ) < (p, ω(x), 3) or x ′ satisfies (T**)(ii). Or we have

This is case (T3) of Proposition 8.7. Arguing as in its proof, cf. (8.29),

x ′ satisfies condition (**) with ω(x) ≥ p or x ′ is resolved by Lemma 7.1 except possibly if x ′ = (Z/u 3 , u 1 /u 3 , u 2 /u 3 , u 3 ). Then ι(x ′ ) < (p, ω(x), 3) or x ′ satisfies again (T**)(ii).

In the case (T**)(iii), we apply Lemma 7.1 when ǫ(x) = ω(x) ≥ 2 or Lemma 7.2 when ǫ(x) = ω(x) = 1: x is resolved for ι = (p, ω(x), 2). Assume that ǫ(x) = 1 + ω(x). By Remark 8.2, we may assume κ(x) = 4.

If

and in case of equality, ǫ(x ′ ) = ω(x) and x ′ satisfies (T**)(ii). In particular, we are done if Vdir(x) =< U 1 , U 2 > by Theorem 3.13. There remains to deal with the case

by Proposition 3.3. This implies ǫ(x) = ω(x).

If < U 1 >⊆ Vdir(x), we are done by Theorem 3.13 unless equality holds and x ′ = (Z ′ := Z/u 2 , u ′ 1 := u 1 /u 2 , u 2 , u 3 ). We may therefore assume that x satisfies (T**)(i). Note that (u ′ 1 , u 2 , u 3 ; Z ′ ) are well adapted coordinates at x ′ by Proposition 2.18. The proof is trivial under assumption (1) and we get x ′ resolved or (T**)(ii). Under assumption (2) (with u 2 , u 3 relabeled), we have E = div(u 1 u 2 u 3 ) and there is an expansion

with γ ∈ S a unit. We get x ′ resolved or (T**)(ii).

Finally if Vdir(x) =< λ 1 U 1 + U i >, λ 1 = 0, i = 2 or 3. Assumption (2) is true. So i = 3 and x is in case (T**)(ii) with E = div(u 1 u 2 u 3 ). We are done by Theorem 3.13 unless

where γ 1 ∈ S is a unit with residue λ 1 . Applying Proposition 3.9(v) (with

If ι(x 1 ) ≥ (p, ω(x), 3), (9.6) thus reads

where d ′ 1 := d 1 +d 3 +ω(x)/p-1, i.e. x ′ satisfies condition (**). This situation occurs only if

, we thus have Vdir(x) =< U 1 > by Proposition 3.3, in particular x satisfies (T**)(i) or (ii). We are done by Theorem 3.13 unless x ′ = (Z/u 3 , u 1 /u 3 , u 2 , u 3 ). The reader ends the proof easily as above, under either assumption (1) or (2): we get x ′ resolved or (T**)(ii).

If ǫ(x) = 1 + ω(x), x satisfies (T**)(iii) by definition. By Proposition 3.3, we have

>. Since Vdir(x) =< U 1 >, we are done by Theorem 3.13 unless x ′ = (Z/u 3 , u 1 /u 3 , u 2 , u 3 ). The reader ends the proof easily as before. 9.2 Resolution for (**), the end.

The purpose of this section is to prove the following proposition and theorem which end the proof of Projection Theorem 5.5. Proposition 9.5. Assume that x is in case (**) (Definition 8.1), then x is resolved for ι = (p, ω(x), 3).

Proof. This follows from Corollary 9.9 and Propositions 9.21 and 9.22 below. Theorem 9.6. Assume that κ(x) ≥ 3, then x is resolved.

Proof. By Propositions 8.9 and 8.11, it can be assumed that κ(x) ≥ 3, x satisfies (**) or (T**). By Proposition 9.1, the remaining case is when x satisfies (**). This case is just the assumption of Proposition 9.5.

An extra assumption on the singular locus.

The following extra assumption (E)' is used as a shortcut in order to ensure that certain exceptional curves on X are Hironaka-permissible and can be blown up in order to reduce ω(x). Such blowing up centers are not used in [START_REF] Cossart | Polyèdre caractéristique d'une singularité[END_REF] and the authors do not know if such blowing ups are relevant in dimension n ≥ 4. Definition 9.7. We say that (S, h, E) satisfies condition (E)' if it satisfies condition (E) and if

where η -1 (m S ) =: {x}. Proposition 9.11 below will show that we can attain condition (E)'. As stated after Definition 2.32, we have in any case Sing p X ⊆ η -1 (E) whenever (S, h, E) satisfies condition (E). Proposition 9.8. Let π : X ′ → X be a permissible blowing up (of the first or second kind) at x ∈ η -1 (m S ) and x ′ ∈ π -1 (x). If (S, h, E) satisfies condition (E)', then (S ′ , h ′ , E ′ ) satisfies again (E)' at x ′ .

Proof. This reduces to Proposition 2.34 if ω(x) ≤ p -1. Assume that ω(x) ≥ p, so we have d j ≥ 1, 1 ≤ j ≤ e, by assumption (E)'. Let Y ⊂ X be permissible with generic point y, W := η(Y) = V ({u j } j∈J ) ⊂ E and I(W )S ′ =: (u), where η ′ : (X ′ , x ′ ) -→ SpecS ′ is the projection. By Definition 3.1 or Proposition 3.7, we have ǫ(y) ≥ ω(x) ≥ p. Applying Proposition 3.9(iv), we have H(x ′ ) = u ǫ(y)-p H(x)S ′ , therefore

and the conclusion follows.

Corollary 9.9. It can be assumed that condition (E)' holds in the proof of Proposition 9.5 and Theorem 9.6.

Proof. All blowing ups used in the proofs of Propositions 8.9, 8.11 and 9.1 are permissible of the first kind.

Lemma 9.10. Assume that condition (E)' does not hold at x. Let µ be a valuation of L = k(X ) centered at x. There exists a finite and independent composition of local permissible blowing ups of the first kind:

where x i ∈ X i is the center of µ, such that x r is resolved or H(x r ) = (1).

Proof. By definition of condition (E)', ω(x) ≥ p. Since resolved means "resolved for (p, ω(x), 3)" in this section, it can be assumed that

for every i ≥ 0 along the process to be defined. Note that ord m S 1 H(x 1 ) > 0 is achieved by blowing up x if δ(x) > 1.

Assume now that δ(x) = 1, i.e. τ (x) ≥ 2 (Definition 2.67). Since κ(x) ≥ 3 and ǫ(x) = ω(x) = p, we actually have κ(x) = 4 and G = 0 as κ(x) > 1, i.e.

where (u 1 , u 2 , u 3 ; Z) are well adapted coordinates.

• if τ ′ (x) = 3, let X ′ → (X , x) be the blowing up along x. Then x is resolved by Theorem 3.13. where x i ∈ X i is the center of µ, such that x r is resolved or x r satisfies condition (E)'.

Proof. It can also be assumed that ω(x) ≥ p and that ω(x i ) = ω(x), κ(x i ) ≥ 3 for every i ≥ 0 along the process to be defined. By Lemma 9.10, we may assume that H(x) = (1) to begin with. Order

We define e 0 , 1 ≤ e 0 ≤ e, by: min{1, d e 0 } = min{1, d 1 } and d e 0 +1 < min{1, d 1 }.

The invariant is:

Note that d(x) = (0, 0) if and only if x satisfies condition (E)'.

Let π : X ′ → (X , x) be the blowing up along a permissible center of the first kind Y and x ′ ∈ π -1 (x): Y = x or Y is an irreducible curve of generic point y with ǫ(y) = ǫ(x). We refine the function x → (m(x), ω(x)), cf. chapter 6, by setting:

Otherwise, we let C(x ′ ) = C(x). To prove the proposition, it is sufficient to prove that there exists a sequence (9.13) such that C(x r ) < C(x). We claim the following: assume that η(Y) ⊂ div(u j ) for some j, 1 ≤ j ≤ e 0 .

(9.14)

, then x ′ belongs to the strict transform of div(u j ) for every j (resp. for some j) such that e 0 < j ≤ e.

To prove this claim, let W := η(Y) and I(W )S ′ =: (u), where

Let x be a vertex of ∆ 2 . Let pr -1 (x) the edge of ∆(h; u 1 , u 2 , v; Z) giving x by projection, this edge is defined by an equation α 1 x 1 + α 2 x 2 + α 3 x 3 = 1, α 1 α 2 α 3 > 0, as usual we define the monomial valuation v αx by

verifies one of the following:

∂V

is not proportional to an ω(x)-power, 3-or H -1 ∂F p,Z,x ∂V = λV ω(x) , λ ∈ k(x) * . We say that (Z, u 1 , u 2 , v) is totally prepared if (i) ∆ S (h; u 1 , u 2 , v; Z) is minimal, (ii) when pd 2 = 0 (f.i. when E = div(u 1 )), all the left vertices of ∆ 2 (h; u 1 , u 2 ; v; Z) are prepared, (iii) when pd 1 > 0 and pd 2 > 0 (⇔ E = div(u 1 u 2 ) when ω(x) ≥ p), all the vertices of ∆ 2 (h; u 1 , u 2 ; v; Z) are prepared. Proposition 9.13. Assume that x is in case (**) Definition 8.1. There exists v ∈ S, φ ∈ S such that (Zφ, u 1 , u 2 , v) is totally prepared. Furthermore x is resolved for m(x) = p if ∆ 2 (h; u 1 , u 2 ; v; Zφ) = ∅.

Proof. We apply a strategy similar to Hironaka's strategy of minimizing in [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF]. Let us start by a vertex x = (x 1 , x 2 ) not prepared. With the notations as above, we have in αx (h) = Z p + F p,Z,x , with

We take any invertible γ x ∈ S whose residue is λ ′ and we define w := v + γ x u x 1 1 u x 2 2 . Then (Z, u 1 , u 2 , w) is a regular system of parameters of S. ∆ 2 (h; u 1 , u 2 ; w; Z) ⊂ ∆ 2 (h; u 1 , u 2 ; v; Z). Furthermore, let y = (y 1 , y 2 ) another vertex of ∆ 2 (h; u 1 , u 2 ; w; Z), let

be an equation of the edge of ∆ S (h; u 1 , u 2 , v; Z) defined by y, of course v αy (u x 1 1 u x 2 2 ) > 1, so in αy (v) = in αy (w). In particular, y is still a vertex of ∆ 2 (h; u 1 , u 2 ; w; Z) and, if it was prepared for (u 1 , u 2 , v; Z), it is still prepared for (u 1 , u 2 , w; Z). Furthermore, if we make an eventual translation on Z ← Zφ, φ ∈ S to minimize ∆ S (h; u 1 , u 2 , w; Z), as in αy (v) = in αy (w), in the of expansion of in αy (h), we just change in αy (v) by in αy (w): we can choose φ with v αy (φ) > 1. So ∆ 2 (h; u 1 , u 2 ; w; Zφ) ⊂ ∆ 2 (h; u 1 , u 2 ; v; Z),

We apply this process to each x = (x 1 , x 2 ) to be prepared, starting by those of smallest modules. When this process is finite, we get the announced result.

When this process is infinite, we get φ, ψ ∈ Ŝ such that (u 1 , u 2 , v-ψ; Z-φ) is totally prepared. Let us remark that x is resolved if ∆ 2 (h; u 1 , u 2 ; w; Zφ) = ∅.

The contrary would mean that ∆(h; u 1 , u 2 ; w; Zφ) has only one vertex (d 1 , d 2 , 1+ω(x) p ): V (Zφ, w) would be a component of dimension two of the locus of multiplicity min{p, 1 + ω(x)}, η( V (Zφ, w)) E. This contradicts (E) if ω(x) ≥ p or if h is separable (assumption (ii) in Theorem 1.5). If ω(x) < p and h = Z p + f p,Z , charS = p, x is resolved for m(x) = p by a combinatorial algorithm, vid. proof of Theorem 2.81.

The remark above implies that, after a finite number of steps, we apply infinitely the process to vertices of smallest abscissa or (smallest ordinate and E = div(u 1 u 2 )) of ∆ 2 (h; u 1 , u 2 ;v; Z) and this smallest abscissa or smallest ordinate remains constant.

Let us study the very special case where x := (A, β) is the vertex of smallest abscissa of ∆ 2 and that the process dissolves it, creating a new vertex (A, β ′ ), β ′ > β infinitely times. This implies A, β, β ′ ∈ N.

Let α = (α 1 , 0, α 3 ), such that α 1 x 1 + α 3 x 3 = 1 is the equation of the non compact face of ∆ S (h; u 1 , u 2 , v; Z) whose image by pr is the non compact face x 1 = A of ∆ 2 . We get α 1 pd 1 + α 3 (1 + ω(x)) = p, α 1 Aα 3 = 0, and

Let C := Spec S (v,u 1 ) . By quasi-homogeneity and the uniqueness of the solution [START_REF] Hironaka | Characteristic polyhedra of singularities[END_REF] Corollary (4.1.1), there exists

Lemma 9.14. There exists

not contained in div(U 1 ). Since O C [U 1 , V ] is excellent and Noetherian, by [START_REF] Cossart | Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes[END_REF] Lemma 1.37, this component is algebraic and the conclusion follows.

When ω(x) < p, V (Z -Φ, V -Ψ) is the only component in the locus of multiplicity 1 + ω(x) of Ξ not contained in div(U 1 ): we conclude as above. This ends the proof of Lemma 9.14.

Let us remark that, if there exists another vertex x 1 which is already prepared, then

By applying Lemma 9.14, we see that there exists φ ∈ S and w ∈ S such that the vertex of smallest abscissa of ∆ 2 (h; u 1 , u 2 ; w; Zφ) is prepared.

The case where the process is infinite along points of smallest ordinates is, mutatis mutandis, the same: by applying the remark above, we see that, when E = div(u 1 u 2 ), there exists φ ∈ S and w ∈ S such that both the vertices of smallest abscissa and smallest ordinate of ∆ 2 (h; u 1 , u 2 ; w; Zφ) are prepared. This ends the proof of Proposition 9.13. Definition 9.15. (Invariants). Suppose κ(x) = 3, suppose that (Z, u 1 , u 2 , v) is totally prepared. In the case where E = div(u 1 u 2 ) and ω(x) < p (so ǫ(x) < p by Definition 8.1 and d 1 + d 2 + ǫ(x) ≥ p) , we choose u 1 so that d 1 > 0 and let (i

.

For sake of simplicity, most of the time, we will skip (Z, u 1 , u 2 , v) and write A 1 (x), A 2 (x), B(x), C(x), β(x), γ(x). Proposition 9.16. Suppose x satisfies conditions (**) and (E)' with κ(x) = 3 and (Z, u 1 , u 2 , v) is totally prepared. The following holds:

(i) V ∈ Vdir(x) or x is resolved;

(ii) if B(x) = 1 and E = div(u 1 ), x is resolved or

is the unique closed point x 1 ∈ π -1 (x) in the blowing up π : X ′ → X along x such that ι(x 1 ) ≥ ι(x), and x ′ then satisfies conditions (**) and (E)';

(iii) if B(x) = 1 and ω(x) < p, x is resolved.

Proof. When B(x) > 1, clearly V ∈ Vdir(x). When B(x) = 1, then

Suppose V ∈ Vdir(x), then

, so τ ′ (x) ≥ 2 by total preparedness. By Lemma 8.5, x is resolved except possibly

As a = 0, it would mean that x = (0, 1) is a vertex of ∆ 2 . This implies that x) with notations as in Definition 9.12: a contradiction with total preparedness and (i) is proved.

Assume that E = div(u 1 ), so we have

by (i). Apply now Lemma 8.5 [START_REF] Abad | p-bases and differential operators on varieties defined over a non-perfect field[END_REF] and note that the form (8.13) is automatically achieved when (Z, u 1 , u 2 , v) is totally prepared. As B(x) = 1, in (8.3), there exists i, 0 ≤ i ≤ ω(x) with Φ i+1 = 0. If Vdir(x) =< V >, as div(u 2 ) ⊂ E, λ 2 = 0 in 8.5(1), we have

by (8.13); if Vdir(x) =< V, U 1 >, we have

by (8.13). Therefore (ii) follows from Lemma 8.5(1) and Proposition 9.8.

To prove (iii), it can be assumed that Vdir(x) =< V > by (i) and Corollary 9.2. In particular, we have

). We blow up along x and let x ′ := (Z/u 2 , u 1 /u 2 , u 2 , v/u 2 ).

Assume that E = div(u 1 ). By (ii) and (9.20), the only point to consider is x ′ . By Corollary 9.2, we are done unless ι(x ′ ) = ι(x), so x ′ satisfies again the assumption in (iii) of the proposition with E ′ = div(u ′ 1 u ′ 2 ). Note that we have A 1 (x ′ ) > 0 by (**).

Assume that E = div(u 1 u 2 ) and let x 1 ∈ π -1 (x) with ι(x 1 ) ≥ ι(x). By Corollary 9.2, we are done unless ι(x 1 ) = ι(x). If E ′ = div(u ′ 1 ), we have

satisfies the assumptions of Lemma 7.3(ii). This holds only if

Then x 1 is resolved for m(x) = p by blowing up d ′ 1 times along codimension two centers of the form (Z ′ , u ′ 1 ). Otherwise, we have < Q >=< U 1+ω(x) 1

>, x 1 = x ′ up to renumbering u 1 , u 2 , so B(x ′ ) = 1 and x ′ satisfies again the assumption in (iii) of the proposition. Note that no renumbering is necessary if A 1 (x) > 0.

Summing up, x is resolved or we construct a sequence of infinitely near points lying on the successive strict transforms of a formal curve

By Proposition 3.17 we may assume that Y is permissible of the first kind, so x is resolved by blowing up along Y.

Remark 9.17. The case κ(x) = 3 and (**) is not at stable by permissible blowing ups of first or second kind. To avoid this problem, in the following Propositions 9.18, 9.20 and 9.21, we make some blowing ups π which are Hironaka permissible and keep the conditions (**) and (E)' at x ′ ∈ π -1 (x) with ι(x ′ ) ≥ (p, ω(x), 3). The transformation laws on ∆ 2 are the usual ones up to an horizontal translation by 1 in the case where D := V (z, u 1 , u 2 ). Proposition 9.18. Assume that x satisfies conditions (**) and (E)' with κ(x) = 4 and let (Z, u 1 , u 2 , v) be totally prepared. Let us call Y := V (Z, u 1 , v) with generic point y.

(1) if ω(x) < p, x is resolved;

(2) if ω(x) ≥ p and ǫ(y) ≥ 2, then (d 1 , d 2 , 1+ω(x) p ) is the only vertex of ∆ S (h; u 1 , u 2 , v; Z) in the region x 1 = d 1 . Furthermore Y is Hironakapermissible and x is resolved.

(3) if ω(x) ≥ p and E = div(u 1 ), let π : X ′ → X be the blowing up along x and x ′ ∈ π -1 (x) with ι(x ′ ) ≥ (p, ω(x), 3). Then x ′ is resolved or there is a Hironaka-permissible line

Let π ′ : X ′′ → X ′ be the blowing up along D ′ and x ′′ ∈ π ′ -1 (x ′ ) with ω(x ′′ ) ≥ ω(x ′ ). Then:

(i) x ′′ satisfies again (E)' and ω(x ′′ ) = ω(x);

(ii) x ′′ satisfies condition (**), 1) has been proved in Corollary 9.2. From now on, we assume that ω(x) ≥ p.

Let us prove (3). As κ(x) = 4, E = div(u 1 ), we have Vdir(x) =< U 1 >. By (**):

We blow up along x, let x ′ be a point above x: if ω(x ′ ) = ω(x), x ′ is on the strict transform of div(u 1 ). In the chart of origin (Z ′ , u ′ 1 , u ′ 2 , v) := (Z/v, u 1 /v, u 2 /v, v), so called v-chart, we get, before any preparation:

As 1 + ω(x) = 0 mod(p), the monomial u ′ 1 pd 1 v pd 1 +ω(x)-p v is not a p th -power, it cannot be spoilt by any translation on Z ′ : ω(x ′ ) = 1 < p ≤ ω(x), for any x ′ in this chart on the strict transform of div(u 1 ). The last point to look at is the point on the strict transform of div(u 1 ), not in the v-chart

21) As we are at the origin of a chart, (Z ′ , u ′ 1 , u ′ 2 , v) are well adapted: ǫ(x ′ ) ≤ ω(x). As ω(x) ≥ p, we keep condition (E)' at x ′ (Proposition 9.8). We are done unless ι(

297

In particular, we have in

There is an expansion

and we get ω(x ′′ ) ≤ 3: we are done for ω(x) ≥ 4.

When ω(x) = 3, in J(F p,Z ′′ , E ′′ , m S ′′ ), there is an homogeneous polynomial

Applying the Hasse-Schmidt derivation 1 2 × ∂ 2 P ∂V ′′2 = U ′′ 2 gives U ′′ 2 ∈ Vdir(x ′′ ). The reader ends the computation and sees that τ ′ (x ′′ ) = 3: x ′′ is is resolved.

When ω(x) = 2, ψ ′′ is invertible, we have Vdir(x ′′ ) =< U ′′ 1 , U ′′ 2 >. We blow up along x ′′ , at the only possible ω-near points, we have, with suitable variables:

A quick computation shows that τ ′ (x ′′′ ) = 3, so x ′′′ is resolved.

We may decompose in (9.21):

By condition (E)', the line

Let us blow up along D ′ . Let us begin with the point x ′ 2 at infinity, i.e.

As we are at the origin of a chart, the coordinates (Z ′′ , u ′′ 1 , u ′′ 2 , v ′′ ) are well adapted, so ǫ(x ′

2 ) ≤ ω(x) -1. For x 2 ∈ π ′ -1 (x ′ ) in the chart of origin

) (in particular (E)' holds) and

As 1 + ω(x) = 0 mod(p), the monomial H(x 2 )v ′′ 1+ω(x) cannot be spoilt by any translation on Z ′′ : we have (m

x ′′ is in case (**) with κ(x ′′ ) = 3. This proves (i) and (ii).

Let us prove assertion (iii) which is valid only for the point x ′′ of parameters

in particular

so the point

Let us note that if (a 0 , b 0 , c 0 ) = (ω(x), 0, 0), then, as Vdir( (9.22) gives the last inequality. When (a 0 , b 0 , c 0 ) = (ω(x), 0, 0), we get

As we saw above, ǫ(x ′′ ) = ω(x ′′ ) + 1, κ(x ′′ ) = 3 and we have (**). Then ( a 0 1+ω(x)-c 0 , 2a 0 +b 0 1+ω(x)-c 0 -1) is the vertex of ∆ 2 (h ′′ ; u ′′ 1 , u ′′ 2 ; v ′′ ; Z ′′ ) of smallest ordinate, both coordinates are < 1 and positive. As x ′ and x ′′ are origins of chart, (Z ′′ , u ′′ 1 , u ′′ 2 , v ′′ ) are well prepared and no translation on v ′′ can spoil this vertex. By (9.23)(9.24), we get:

Note that A 1 (x ′′ ) > 0 because of (9.25). This proves (iii). Let us prove (2). Since ǫ(y) > 0, we have

then, if we blow up along x, as ω(x) ≥ p ≥ 2, there is no ω-near point. The only case we have to look at is Vdir(x) =< U 1 >.

As ω(x) ≥ p, by condition (E)' at x:

Let us blow up along Y. In the first chart of origin

Proposition 9.19. Assume that x satisfies conditions (**) and (E)' with κ(x) = 4, E = div(u 1 ) and let (Z, u 1 , u 2 , v) be totally prepared. With the notations of Proposition 9.18, assume furthermore that ǫ(y) = 1 and β(Z, u 1 , u 2 , v) < 1.

Then x is resolved.

Proof. By Proposition 9.18(1), we may assume ω(x) ≥ p. As A 1 (x) > 0 by condition (**), ǫ(y) = 1 implies that ∆ S (h; u 1 , u 2 , v) has a vertex

.

On the other hand, since κ(x) = 4, we have b ≥ ω(x), i.e. b = ω(x).

Let us come back to the proof of Proposition 9.18 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF]. The only point to consider is the point x ′ at infinity, E ′ = div(u ′ 1 v). We get an expansion

) mod(v ′ ). (9.26)

The conclusion follows from Lemma 7.1 applied to the well prepared coordinates (v ′ , u ′ 1 , u 2 ; Z ′ ). The following proposition produces bounds identical to those occurring for embedded resolution of surfaces [START_REF] Cossart | Resolution of surface singularities[END_REF]. Proposition 9.20. Assume that x satisfies conditions (**) and (E)' with κ(x) ≥ 3. Consider Hironaka-permissible blowing ups π : X ′ → (X , x) of the following kinds:

Case 1: E = div(u 1 u 2 ) and ω(x) ≥ p; we blow-up along D :=V(Z, u 1 , u 2 ). Case 2: κ(x) = 3, E = div(u 1 ) or ω(x) < p ; we blow up along x. Let x ′ ∈ π -1 (x) with (m(x ′ ), ω(x ′ )) ≥ (p, ω(x)). Then ω(x ′ ) ≤ ω(x) and (x ′ is resolved or the following holds): (i) conditions (**) and (E)' are satisfied at x ′ and we have

))

in case 1 (resp. case 2), then A 1 (x ′ ) = B(x), (resp. A 1 (x ′ ) = B(x) -1) and,

))

in case 1 (resp. case 2), then

Proof. We first prove the proposition in case 1. Let x ′ be in the chart with origin (X

As 1 + ω(x) ≡ 0 mod p, the monomial u

will not be spoilt by any translation on Z ′ : x ′ satisfies (**) and (m(x ′ ), ω(x ′ )) ≤ (p, ω(x)). If ω(x) ≥ p, we have d 1 , d 2 ≥ 1, so x ′ satisfies condition (E)'. Statement γ(x ′ ) ≤ γ(x) follows from (ii) that we prove in the next lines.

The monomials defining B(x) in the expansion of f p,Z are minimal for the monomial valuation v α defined by the weight vector α := (a, a, aB(x)): 2 ) is such that (X ′ , u 1 , w, v) is a system of coordinates at x ′ .

Of course, we take w = u ′ 2 when x ′ is the origin of the chart. In this special case where x ′ is the origin, the argument is the same as in [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic II[END_REF] Lemma I.5.3 on page 1966. This relies on the characteristic free Proposition 2.18 which asserts that no changes in Z ′ need to be performed in order to get well adapted data: (X ′ , u 1 , w, v) is totally prepared. It is easy to see that ∆ 2 (x ′ ) is obtained from ∆ 2 (x) by applying the affine transformation: (v 1 , v 2 ) → (v 1 , v 1 + v 2 ) when D is the center of the blowing up (resp. (v 1 , v 2 ) → (v 1 , v 1 + v 2 -1) when x is the center of the blowing up) and adding quadrants. The reader verifies that all the statements of (ii) are true, despite of the fact that in case where D is the center of the blowing up, there is not the usual shift by -1.

From now on, E ′ = div(u 1 ). Monomials defining B(x) become the monomials defining A 1 (x ′ ) = B(x). The monomials defining the vertices of smaller abscissa of ∆ 2 (h ′ ; u ′ 1 , w, v; X ′ ) are those minimal for the valuation v α ′ given by v α ′ (X ′ ) = 1, v α ′ (u 1 ) = a, v α ′ (w) = 0, v α ′ (v) = aB(x).

We set

When G α ′ = 0, we have A 1 (x ′ ) = B(x), β(x ′ ) = 0, so (ii) holds. Assume now that G α ′ = 0. Subcase 1.1: when

∂F p,Z,α ∂V ∈< V ω(x) > .

We expand (9.27) with λ = 0, Q i = 0 or Q i divisible neither by U 1 , nor by U 2 . For Q i = 0:

By Proposition 3.9(v), H(x ′ ) -1 ∂F p,X ′ ,α ′ ∂V is the strict transform of H(x) -1 ∂F p,Z,α ∂V (with some abuse of notation for H(x), H(x ′ )). Then, by [START_REF] Cossart | Resolution of surface singularities[END_REF] Lemma 6.2.3 a and page 92, the lowest abscissa of the vertices of the polygon ∆(H(x ′ ) -1 ∂F p,X ′ ,α ′ ∂V ; U 1 , w; V ) is B(x). The non compact face of lowest abscissa is not solvable and, after a possible translation:

the ordinate β ′ of the vertex of lowest abscissa of ∆(H(x ′ ) -1 ∂F p,Z ′ ,α ′ ∂V ; U 1 , w; V ) satisfies

where β 2 (x) is the ordinate of the left vertex of the initial face of the polygon ∆(H(x ′ ) -1 ∂F p,Z,α ∂V ; U 1 , U 2 ; V ). Then we have .28) This implies all the assertions in subcase 1-1. Subcase 1.2: when

∂F p,Z,α ∂V ∈< V ω(x) > .

We now have an expansion (9.29) with λ = 0, Q i = 0 or Q i divisible neither by U 1 , nor by U 2 . For Q i = 0: a j (i) ≥ (1 + ω(x)pi)A j (x), deg(Q i ) ≤ (1 + ω(x)pi)C(x).

Take i 0 , 1 ≤ i 0 < (1 + ω(x))/p maximal such that U pd 1 +a 1 (i 0 ) 1 U pd 2 +a 2 (i 0 ) 2 Q i 0 is not a p th -power. This i 0 exists by total preparation. By (9.29), the transform of ∂F p,Z,α ∂V now reads x) , λ ′ a unit. (9.30) Preparation along the face of abscissa B(x) will thus be a translation Z ′ = X ′ + φ ′ on X ′ , no translation on v: this will just add a p th -power to the term

2 ) in (9.29), which will become of the form

By the usual computations ( [START_REF] Cossart | Resolution of surface singularities[END_REF] page 92 or the blowing up formula applied to U pd 1 +a 1 (i 0 ) 1 U pd 2 +a 2 (i 0 ) 2

(9.31) This implies all the assertions in subcase 1-2, x ′ not the origin and (ii) is proved. Permuting u 1 and u 2 gives (iii). We now turn to case 2. Let x ′ be in the chart of origin (X ′ := Z u 1 , u 1 , u ′ 2 , v ′ ). By Proposition 9.16(ii), we may assume that B(x) > 1, i.e. < V >= Vdir(x), so v ′ ∈ m S ′ . In the expansion of f p,Z the monomial u pd 1 1 u pd 2 2 v 1+ω(x)-i u a 1 u b 2 , 0 ≤ i ≤ 1 + ω(x)i becomes u pd 1 +pd 2 +1+ω(x)-p 1 u ′ 2 pd 2 v 1+ω(x)-i u a+b 1 u ′ 2 b in the expansion of f p,Z ′ . This leads to:

Then x ′ is resolved or x ′ satisfies conditions (**) and (E)' as in case 1. Then the proof runs along the same lines as above: equations (9.28) and (9.31) remain true. The case where x ′ is the origin of the second chart is given by a permutation of u 1 and u 2 in the computations above and the fact that the vertices of ∆ 2 (h ′ ; u 1 /u 2 , u 2 ; v/u 2 ; Z/u 2 ) are the transforms of the left vertices of ∆ 2 (h; u 1 , u 2 ; v; Z) by the affinity (x 1 , x 2 ) → (x 1 , x 1 + x 2 -1): they are totally prepared.

Proposition 9.21. Assume that x satisfies conditions (**) and (E)'. Let µ be a valuation of L = k(X ) centered at x. There exists a finite and independent composition of local Hironaka-permissible blowing ups w.r.t. E: (X , x) =: (X 0 , x 0 ) ← (X 1 , x 1 ) ← • • • ← (X r , x r ), (9.32) where x i ∈ X i is the center of µ, such that x r is resolved or (x r satisfies again conditions (**) and (E)' together with one of the following):

(i) E r = div(u 1,r ), β(x r ) < 1;

(ii) E r = div(u 1,r u 2,r ), C(x r ) = 0.

Proof. Let (Z, u 1 , u 2 , v) be totally prepared. Let Y = V (Z, u 1 , v) with generic point y. We define by induction on i ≥ 0 a sequence of local Hironakapermissible blowing ups w.r.t. E, or composition of two such local blowing ups. Take i = 0 w.l.o.g. in the following definition.

(1) if (E = div(u 1 ), κ(x) = 3), blow up along x (Proposition 9.20, case 2);

(2) if (E = div(u 1 ), κ(x) = 4, ǫ(y) ≤ 1), blow up along x, then along

2 ) (notations of Proposition 9.18(3)); (3) if (E = div(u 1 ), κ(x) = 4, ǫ(y) ≥ 2), blow up along Y (Proposition 9.18(2)); (4) if (E = div(u 1 u 2 ), ω(x) ≥ p), blow up along D = V(Z, u 1 , u 2 ) (Proposition 9.20, case 1); [START_REF] Abhyankar | Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry[END_REF] if (E = div(u 1 u 2 ), ω(x) < p), blow up along x (Proposition 9.20, case 2).

We must prove that (A) this algorithm is well defined, i.e. x 1 is resolved or satisfies again conditions (**) and (E)', so it builds up a sequence (9.32), then (B) this sequence is finite.

Note that any x fits into exactly one of (1)-( 5). To prove (A)(B), we recollect results from the previous propositions. Proposition 9.18 [START_REF] Abhyankar | Local uniformization on algebraic surfaces over ground fields of characteristic p = 0[END_REF] shows that x is resolved when x is in case (3) above. In case (2), Proposition 9.18(3) produces x 1 satisfying again the assumptions of Proposition 9.21 and fitting into (4) with κ(x 1 ) = 3, γ(x 1 ) = 1.

We now turn to Proposition 9.20 in the cases (1)(4)(5) above. Statement (i) shows that x 1 is resolved or satisfies again the assumptions of the lemma. The proof of (A) is thus complete and we turn to (B). Assume w.l.o.g. that Proof. We assume that (Z, u 1 , u 2 , v) is totally prepared. Let c(x) := (A 1 (x), β(x)) with lexicographical ordering. First suppose that A 1 (x) < 1 and (x is in case (iii) =⇒ A 2 (x) < 1).

(9.34)

If E = div(u 1 u 2 ) and κ(x) = 3, we blow up along x. Let x ′ be a point ω near to x. When x ′ is the origin of a chart, by Proposition 9.20(i)-(iii), x ′ satisfies again the assumptions of the proposition with c(x ′ ) < c(x). When x ′ is in the first chart with E ′ = div(u 1 ), Proposition 9.20(ii) gives A 1 (x ′ ) = B(x) -1 ≤ A 1 (x) + β(x) -1 < A 1 (x) and β(x ′ ) < 1.

In both cases, x ′ satisfies again the assumptions of the proposition together with (9.34) and c(x ′ ) < c(x).

If E = div(u 1 u 2 ) and κ(x) = 4, we let Y j := V (Z, v, u j ) with generic point y j , j = 1, 2. The condition ǫ(y j ) ≥ 2 is equivalent to A j (x) > 1 1+ω(x) . We apply Proposition 9.18(1)(2): then x is resolved except possibly if A j (x) ≤ 1 1+ω(x) , j = 1, 2. Then 1 -1 1 + ω(x) ≤ B(x) ≤ A 1 (x) + β(x) < 1.

We deduce that equality holds and that Vdir(x) =< U 1 , U 2 >. Since ω(x) ≥ p ≥ 2, we obtain ω(x ′ ) < ω(x) after blowing up along x, so x is resolved. If E = div(u 1 ) and κ(x) = 3, we blow up along x. Note that β(x) > 0 since A 1 (x) < 1. Let

), E ′ = div(u ′ 1 u 2 ).

If x 1 = x ′ , Proposition 9.20(iv) gives A 1 (x ′ ) = B(x) -1 ≤ A 1 (x) + β(x) -1 < A 1 (x) and β(x ′ ) ≤ β(x).

Therefore x 1 satisfies again assumption (i) of the proposition together with (9.34) and c(x ′ ) < c(x).

Clearly ι(x ′ ) ≤ (p, ω(x), 2) and x is resolved for (p, ω(x), 3). There remains to prove the Proposition in case (iii) with A i (x) ≥ 1, i = 1 or 2. See [START_REF] Cossart | Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings[END_REF] II.6.2 and II.6.3 on pp. 1950-1951. The argument is similar to the one used in the proof of Proposition 6.17(b)(c).

If (ω(x) ≥ p and A 1 (x) ≥ 1), then Y := V (Z, u 1 , v) is permissible of the first kind. Blowing up along Y, the only point which may be ω-near x is the point x ′ as in (9.36). We have

Then x ′ satisfies again conditions (**) and (E)'. A descending induction on max{A 1 (x), A 2 (x)} reduces to A 1 (x), A 2 (x) < 1 which is (9.34) and the proof is complete. If 1 + ω(x) < p, we argue by induction on c ′ (x) := (max{A 1 (x), A 2 (x)}, max{d 1 , d 2 }, n)

where n := 2 if (A 1 (x) = A 2 (x), d 1 = d 2 ), n := 1 otherwise. Suppose that A 1 (x) ≥ 1, d 1 + 1+ω(x) p ≥ 1. Up to renumbering u 1 , u 2 , it can be assumed that c ′ (x) = (A 1 (x), d i , n), i = 1, 2 or c ′ (x) = (A 2 (x), d 1 , 1) with d 2 + 1+ω(x) p < 1. Blowing up along Y := (Z, u 1 , v), the only point which may be ω-near x is the point x ′ as in (9.36). If (m(x ′ ), ω(x ′ )) = (p, ω(x)), x ′ is in case (**) and we have

It is easily seen that c ′ (x ′ ) < c ′ (x). The remaining case: up to renumbering u 1 , u 2 , we have

We then blow up along x. As case (i) is resolved, we have just to look at the origins of both charts. Let us look at the first chart, of origin the point x ′ as above. If (m(x ′ ), ω(x ′ )) = (p, ω(x)), x ′ is in case (**) and we have

311 Therefore c ′ (x ′ ) < c ′ (x). The last point to look at is the point

If (m(x ′′ ), ω(x ′′ )) = (p, ω(x)), x ′′ is in case (**), and we have A 1 (x ′′ ) = A 1 (x) and A 2 (x ′′ ) = A 1 (x) + A 2 (x) -1 < A 2 (x), C(x ′′ ) = 0.

Therefore c ′ (x ′′ ) < c ′ (x). This concludes the proof.

Index

A 1 , A 2 , B, C, β, β 2 , γ, when there is maximal contact, Notation 6.5(6.6)(6.7)(6.9), 166 A 1 , A 2 , B, C, α 2 , β 2 , γ in case κ(x) = 2(*), see also Definition 7.