Evelia R Garc

Í A Barroso

Pedro D Gonz Ález
email: pgonzalez@mat.ucm.es

P Érez

Patrick Popescu-Pampu
email: patrick.popescu-pampu@univ-lille.fr

THE COMBINATORICS OF PLANE CURVE SINGULARITIES HOW NEWTON POLYGONS BLOSSOM INTO LOTUSES

Keywords: Mathematics Subject Classification. 14H20 (primary), 14B05, 32S05 Blow ups, Branch, Characteristic exponents, Dual graph, Eggers-Wall tree, Embedded resolution, Enriques diagram, Intersection numbers, Newton polygon, Newton-Puiseux series, Plane curve singularity, Proximity relation, Resolution of singularities, Toric geometry, Toroidal embedding, Tropicalization, Valuation, Valuative tree

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs pC, oq of complex analytic curves contained in a smooth complex analytic surface S. The embedded topological type of such a pair pS, Cq is usually defined to be that of the oriented link obtained by intersecting C with a sufficiently small oriented Euclidean sphere centered at the point o, defined once a system of local coordinates px, yq was chosen on the germ pS, oq. If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of pS, Cq. One may define it by looking either at the Newton-Puiseux series associated to C relative to a generic local coordinate system px, yq, or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ pC, oq or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of pC, oq by successive toric modifications.

Introduction 2. Basic notions and examples

2.1. Basic facts about plane curve singularities 2.2. Basic facts about normalizations 2.3. Newton-Puiseux series and the Newton-Puiseux theorem 2.4. Blow ups and embedded resolutions of singularities 2.5. The minimal embedded resolution of the semicubical parabola 2.6. A Newton non-degenerate reducible example 3. Toric and toroidal surfaces and their morphisms 3.1. Two-dimensional fans and their regularizations 3.2. Toric varieties and their orbits 3.3. Toric morphisms and toric modifications 3.4. Toroidal varieties and modifications in the toroidal category 3.5. Historical comments 4. Toroidal pseudo-resolutions of plane curve singularities 4.1. Newton polygons, their tropicalizations, fans and modifications 4.2. An algorithm of toroidal pseudo-resolution 4.3. From toroidal pseudo-resolutions to embedded resolutions 4.4. The fan tree of a toroidal pseudo-resolution process their Zeta functions. The readers interested in such invariants may consult the textbooks [START_REF] Brieskorn | Plane algebraic curves. Translated from the German by John Stillwell[END_REF] of Brieskorn & Knörrer and [START_REF] Wall | Singular points of plane curves[END_REF] of Wall. There are algorithms allowing to pass between the Eggers-Wall tree, the dual graph and the Enriques diagram of C. However, they do not allow geometric representations of those passages. Our aim is to represent all these relationships using a single geometric object, called a lotus, which is a special type of simplicial complex of dimension at most two.

Our approach for associating lotuses to plane curve singularities is done in the spirit of the papers of Lê & Oka [START_REF] Lê | On resolution complexity of plane curves[END_REF], A'Campo & Oka [START_REF] A'campo | Geometry of plane curves via Tschirnhausen resolution tower[END_REF], Oka [START_REF] Oka | Geometry of plane curves via toroidal resolution[END_REF], González Pérez [52, Section 3.4], and Cassou Noguès & Libgober [START_REF] Cassou-Noguès | Multivariable Hodge theoretical invariants of germs of plane curves. II. In Valuation theory in interaction[END_REF]. Namely, we use the fact that one may obtain an embedded resolution of C by composing a sequence of toric modifications determined by the successive Newton polygons of C or of strict transforms of it, relative to suitable local coordinate systems.

One may construct a lotus using the previous Newton polygons (see Definition 5.26). Its one dimensional skeleton may be seen as a dual complex representing the space-time of the evolution of the dual graph during the process of blow ups of points which leads to the embedded resolution. Besides the irreducible components of C and the components of the exceptional divisor, one takes also into account the curves defined by the chosen local coordinate systems. If A and B are two such exceptional or coordinate curves, and them or their strict transforms intersect transversally at a point p which is blown up at some moment of the process, then a two dimensional simplex with vertices labeled by A, B and the exceptional divisor of the blow up of p belongs to the lotus. These simplices are called the petals of the lotus (see an example of a lotus with 18 petals in Figure 1). The Eggers-Wall tree, the Enriques diagram and the weighted dual graph embed simultaneously inside the lotus, and the geometry of the lotus also captures the numerical decorations of the weighted dual graph and the Eggers-Wall tree (see Theorem 5.29). For instance, the self-intersection number of a component of the final exceptional divisor is the opposite of the number of petals containing the associated vertex of the lotus. The previous lotuses associated to C have also valuative interpretations: they embed canonically in the space of semivaluations of the completed local ring of the germ pS, oq (see Remark 5.34). Let us describe the structure of the paper.

In Section 2 we introduce basic notions about complex analytic varieties, plane curve singularities, their multiplicities and intersection numbers, normalizations, Newton-Puiseux series, blow ups, embedded resolutions of plane curve singularities and the associated weighted dual graphs. The notions of Newton polygon, dual Newton fan and lotus are first presented here on a Newton non-degenerate example.

Section 3 begins with an explanation of basic notions of toric geometry: fans and their subdivisions, the associated toric varieties and toric modifications (see Subsections 3.1, 3.2 and 3.3). In particular, we describe the toric boundary of a toric variety -the reduced divisor obtained as the complement of its dense torus -in terms of the associated fan. Then we pass to toroidal geometry: we introduce toroidal varieties, which are pairs pΣ, BΣq consisting of a normal complex analytic variety Σ and a reduced divisor BΣ on it, which are locally analytically isomorphic to a germ of a pair formed by a toric variety and its boundary divisor. A basic example of toroidal surface is that of a germ pS, oq of smooth surface, endowed with the divisor L `L1 , where pL, L 1 q is a cross, that is, a pair of smooth transversal germs of curves. A morphism φ : pΣ 2 , BΣ 2 q Ñ pΣ 1 , BΣ 1 q of toroidal varieties is a complex analytic morphism such that φ ´1pBΣ 1 q Ď BΣ 2 (see Subsection 3.4).

In Section 4 we explain in which way one may associate various morphisms of toroidal surfaces to the plane curve singularity C ãÑ S. First, choose a cross pL, L 1 q on pS, oq, defined by a local coordinate system px, yq. The Newton polygon N pf q of a defining function f P Crrx, yss of the curve singularity C depends only on C and on the cross pL, L 1 q. Its associated Newton fan is obtained by subdividing the first quadrant along the rays orthogonal to the compact edges of the Newton polygon. This fan defines a toric modification of S, the Newton modification of S defined by C relative to the cross pL, L 1 q (see Subsection 4.1). The Newton modification becomes a toroidal morphism when we endow its target S with the boundary divisor BS :" L `L1 and we define the boundary divisor of its source to be the preimage of L `L1 . We emphasize the fact that those notions depend only on the objects pS, C, pL, L 1 qq, in order to insist on the underlying geometric structures. The strict transform of C by the previous Newton modification intersects the boundary divisor only at smooth points of it, which belong to the exceptional divisor and are smooth points of the ambient surface. If one completes the germ of exceptional divisor into a cross at each such point o i , then one gets again a triple of the form (surface, curve, cross), where this time the curve is the germ at o i of the strict transform of C. Therefore one may perform again a Newton modification at each such point, and continue in this way until the strict transform of C defines everywhere crosses with the exceptional divisor. The total transform of C and of all coordinate curves introduced during previous steps define the toroidal boundary BΣ on the final surface Σ. This non-deterministic algorithm produces morphisms π : pΣ, BΣq Ñ pS, BSq of toroidal surfaces, which are toroidal pseudo-resolutions of the plane curve singularity C (see Subsection 4.2). The surface Σ has a finite number of singular points, at which it is locally analytically isomorphic to normal toric surfaces. In Subsection 4.3 we show how to pass from the toroidal pseudo-resolution π to a toroidal embedded resolution by composing π with the minimal resolution of these toric singularities. Finally, we encode the process of successive Newton modifications in a fan tree, in terms of the Newton fans produced by the pseudo-resolution process (see Subsection 4.4).

In Section 5 we explain the notion of lotus. A Newton lotus associated to a fan encodes geometrically the continued fraction expansions of the slopes of the rays of the fan, as well as their common parts (see Subsection 5.2). It is composed of petals, and each petal corresponds to the blow up of the base point of a cross. One may clarify the subtitle of the paper by saying that the collection of Newton polygons appearing during the toroidal pseudo-resolution process blossomed into the associated lotus, each petal corresponding to a blow up operation. We explain how to associate to the fan tree of the toroidal pseudo-resolution a lotus, which is a 2-dimensional simplicial complex obtained by gluing the Newton lotuses associated to the Newton fans of the process (see Subsections 5.1 and 5.3). The lotus of a toroidal pseudo-resolution depends on the choices of crosses made during the process of pseudo-resolution (see Subsection 5.4). We explain then how to embed in the lotus the Enriques diagram and the dual graph of the embedded resolution. We conclude the section by defining a truncation operation on lotuses, and we explain how it may be used to understand the part of the embedded resolution which does not depend on the supplementary curves introduced during the pseudo-resolution process (see Subsection 5.5).

We begin Section 6 by introducing the notion of Eggers-Wall tree of the curve C relative to the smooth germ L (see Subsection 6.1) and by expressing the Newton polygon of C relative to a cross pL, L 1 q in terms of the Eggers-Wall tree of C `L1 relative to L (see Subsection 6.2). Then we explain that the fan tree of the previous toroidal pseudo-resolution process is canonically isomorphic to the Eggers-Wall tree relative to L of the curve obtained by adding to C the projections to S of all the crosses built during the process and how to pass from the numerical decorations of the fan tree to those of the Eggers-Wall tree (see Subsection 6.5). As preliminary results, we prove renormalization formulae which describe the Eggers-Wall tree of the strict transform of C by a Newton modification, relative to the exceptional divisor, in terms of the Eggers-Wall tree of C relative to L (see Subsections 6.3 and 6.4).

The final Section 7 begins by an overview of the construction of a fan tree and of the associated lotus from the Newton fans of a toroidal pseudo-resolution process (see Subsection 7.1). Subsection 7.2 describes perspectives on possible applications of lotuses to problems of singularity theory. The final Subsection 7.3 contains a list of the main notations used in the article.

Starting from Section 3, each section ends with a subsection of historical comments. We apologize for any omission, which may result from our limited knowledge. One may also find historical information about various tools used to study plane curve singularities in Enriques and Chisini's book [START_REF] Enriques | Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche II[END_REF], in the first chapter of Zariski's book [START_REF] Zariski | Algebraic surfaces[END_REF] and in the final sections of the chapters of Wall's book [START_REF] Wall | Singular points of plane curves[END_REF].

We tried to make this paper understandable to PhD students who have only a basic knowledge about singularities. Even if everything in this paper holds over an arbitrary algebraically closed field of characteristic zero, we will stick to the complex setting, in order to make things more concrete for the beginner. We accompany the definitions with examples and many figures. Indeed, one of our objectives is to show that lotuses may be a great visual tool for relating the combinatorial objects used to study plane curve singularities. There is a main example, developed throughout the paper starting from Section 4 (see Examples 4. 28, 4.34, 4.36, 5.28, 5.31, 5.36, 6.29 and the overview Figure 58). We recommend to study it carefully in order to get a concrete feeling of the various objects manipulated in this paper. We also recommend to those readers who are learning the subject to refer to the Section 7.1 from time to time, in order to measure their understanding of the geometrical objects presented here.

Basic notions and examples

In this section we recall basic notions about complex varieties and plane curve singularities (see Subsection 2.1), normalization morphisms (see Subsection 2.2), the relation between Newton-Puiseux series and plane curve singularities (see Subsection 2.3) and resolution of such singularties by iteration of blow ups of points (see Subsection 2.4). We describe such a resolution for the semi-cubical parabola (see Subsection 2.5). We give a flavor of the main construction of this paper in Subsection 2.6. We show there how to transform the Newton polygon of a certain Newton non-degenerate plane curve singularity with two branches into a lotus, and how this lotus contains the dual graph of a resolution by blow ups of points.

From now on, N denotes the set of non-negative integers and N ˚the set of positive integers.

Basic facts about plane curve singularities.

In this subsection we recall basic vocabulary about complex analytic spaces (see Definition 2.1) and we explain the notions of plane curve singularity (see Definition 2.5), of multiplicity and of intersection number (see Definition 2.7) for such singularities. Finally, we recall an important way of computing such intersection numbers (see Proposition 2.8).

Briefly speaking, a complex analytic space X is obtained by gluing model spaces, which are zero-loci of systems of analytic equations in some complex affine space C n . One has to prescribe also the analytic "functions" living on the underlying topological space. Those "functions" are elements of a so-called "structure sheaf" O X , which may contain nilpotent elements. For this reason, they are not classical functions, as they are not determined by their values. For instance, one may endow the origin of C with the structure sheaves whose rings of sections are the various rings Crxs{px m q, with m P N ˚. They are pairwise non-isomorphic and they contain nilpotent elements whenever m ě 2. Let us state now the formal definitions of complex analytic spaces and of some special types of complex analytic spaces. Definition 2.1. ' A model complex analytic space is a ringed space pX, O X q, where X is the zero locus of I and O X " O U {I. Here I is a finitely generated ideal of the ring of holomorphic functions on an open set U of C n , for some n P N ˚, O U is the sheaf of holomorphic functions on U and I is the sheaf of ideals of O U generated by I.

' A complex analytic space is a ringed space locally isomorphic to a model complex analytic space. ' A complex analytic space is reduced if its structure sheaf O X is reduced, that is, without nilpotent elements. In this case, one speaks also about a complex variety. ' A complex manifold is a complex variety X such that any point x P X has a neighborhood isomorphic to an open set of C n , for some n P N. If the non-negative integer n is independent of x, then the complex manifold X is called equidimensional and n is its complex dimension. ' The smooth locus of a complex variety X is its open subspace whose points have neighborhoods which are complex manifolds. Its singular locus SingpXq is the complement of its smooth locus. ' A smooth complex curve is an equidimensional complex manifold of complex dimension one and a smooth complex surface is an equidimensional complex manifold of complex dimension two. ' A complex curve is a complex variety whose smooth locus is a smooth complex curve and a complex surface is a complex variety whose smooth locus is a smooth complex surface.

By construction, the singular locus SingpXq of X is a closed subset of X. It is a deep theorem that this subset is in fact a complex subvariety of X (see [START_REF] De Jong | Local analytic geometry[END_REF]Corollary 6.3.4]).

Let S be a smooth complex surface. If o is a point of S and φ : U Ñ V is an isomorphism from an open neighborhood U of o in S to an open neighborhood V of the origin in C 2

x,y , then the coordinate holomorphic functions x, y : C 2

x,y Ñ C may be lifted by φ to two holomorphic functions on U , vanishing at o. They form a local coordinate system on the germ pS, oq of S at o. By abuse of notations, we still denote this local coordinate system by px, yq, and we see it as a couple of elements of O S,o , the local ring of S at o, equal by definition to the C-algebra of germs of holomorphic functions defined on some neighborhood of o in S. The local coordinate system px, yq establishes an isomorphism O S,o » Ctx, yu, where Ctx, yu denotes the C-algebra of convergent power series in the variables x, y. Denote by Crrx, yss the C-algebra of formal power series in the same variables. It is the completion of Ctx, yu relative to its maximal ideal px, yqCtx, yu. One has the following fundamental theorem, valid in fact for any finite number of variables (see [START_REF] De Jong | Local analytic geometry[END_REF]Corollary 3.3.17]):

Theorem 2.2. The local rings Ctx, yu and Crrx, yss are factorial.

In addition to Definition 2.1, we use also the following meaning of the term curve: Definition 2.3. A curve C on a smooth complex surface S is an effective Cartier divisor of S, that is, a complex subspace of S locally definable by the vanishing of a non-zero holomorphic function. This means that for every point o P C, there exists an open neighborhood U of o in S and a holomorphic function f : U Ñ C such that C Ă U is the vanishing locus Zpf q of f and such that the structure sheaf O C|U of C Ă U is the quotient sheaf O U {pf qO U . In this case, once U is fixed, the defining function f is unique up to multiplication by a holomorphic function on U which vanishes nowhere.

The curve C is called reduced if it is a reduced complex analytic space in the sense of Definition 2.1. This means that any defining function f : U Ñ C as above is square-free in all local rings O S,o , where o P U . For instance, the union C of coordinate axes of C 2 is a reduced curve, being definable by the function xy, which is square-free in all the local rings O C 2 ,o , where o P C. By contrast, the curve D defined by the function xy 2 is not reduced.

As results from Definition 2.3, a complex subspace C of S is a curve on S if and only if, for any o P C, the ideal of O S,o consisting of the germs of holomorphic functions vanishing on the germ pC, oq of C at o is principal. We would have obtained a more general notion of curve if we would have asked C to be a 1-dimensional complex subspace of S in the neighborhood of any of its points. For instance, if S " C 2

x,y , and C is defined by the ideal px 2 , xyq of Crx, ys, then set-theoretically C coincides with the y-axis Zpxq. But the associated structure sheaf O C 2 {px 2 , xyqO C 2 is not the structure sheaf of an effective Cartier divisor. In fact the germ of C at the origin cannot be defined by only one holomorphic function f px, yq P Ctx, yu. Otherwise, we would get that both x 2 and xy are divisible by f px, yq in the local ring Ctx, yu. As this ring is factorial by Theorem 2.2, we see that f divides x inside this ring, which implies that pf qCtx, yu " pxqCtx, yu. Therefore, px 2 , xyqCtx, yu " pxqCtx, yu which is a contradiction, as x is of order 1 and each element of the ideal px 2 , xyqCtx, yu is of order at least 2. The notion of order used in the previous sentence is defined by: Definition 2.4. Let f P Crrx, yss. Its order is the smallest degree of its terms.

For instance, the maximal ideal of Crrx, yss consists precisely of the power series of order at least 1. It is a basic exercise to show that the order is invariant by the automorphisms of the C-algebra Crrx, yss and by multiplication by the elements of order 0, which are the units of this algebra. Therefore, one gets a well-defined notion of multiplicity of a germ of formal curve on S: Definition 2.5. A plane curve singularity is a germ C of formal curve on a germ of smooth complex surface pS, oq, that is, a principal ideal in the completion ÔS,o of the local ring O S,o . It is called a branch if it is irreducible, that is, if its defining functions are irreducible elements of the factorial local ring ÔS,o . The multiplicity m o pCq of C at o is the order of a defining function f P ÔS,o of C, seen as an element of Crrx, yss using any local coordinate system px, yq of the germ pS, oq.

Example 2.6. Let α, β P N ˚and f :" x α ´yβ P Crx, ys. Denote by C the curve on C 2 defined by f . Its multiplicity at the origin O of C 2 is the minimum of α and β. The curve singularity pC, Oq is a branch if and only if α and β are coprime. One implication is easy: if α and β have a common factor ρ ą 1, then x α ´yβ " ś ω: ω ρ "1

`xα{ρ ´ωy β{ρ ˘, the product being taken over all the complex ρ-th roots ω of 1, which shows that pC, Oq is not a branch. The reverse implication results from the fact that, whenever α and β are coprime, C is the image of the parametrization N ptq :" pt β , t α q. The inclusion N pCq Ď C being obvious, let us prove the reverse inclusion. Let px, yq P C. As N p0q " O, it is enough to consider the case where xy ‰ 0. We want to show that there exists t P C ˚such that x " t β , y " t α . Assume the problem solved and consider also a pair pa, bq P Z 2 such that aα `bβ " 1, which exists by Bezout's theorem. One gets t " t aα`bβ " y a x b . Define therefore t :" y a x b . Then: t β " py a x b q β " py β q a x bβ " px α q a x bβ " x aα`bβ " x, and similarly one shows that t α " y. This proves that C is indeed included in the image of N .

Let C be a plane curve singularity on the germ of smooth surface pS, oq. If f P ÔS,o is a defining function of C, it may be decomposed as a product: [START_REF] Abhyankar | Concepts of order and rank on a complex space, and a condition for normality[END_REF] f "

ź iPI f pi i ,
in which the functions f i are pairwise non-associated prime elements of the local ring ÔS,o and p i P N ˚for every i P I. Such a decomposition is unique up to permutation of the factors f pi i and up to a replacement of each function f i by an associated one (recall that two such functions are associated if one is the product of another one by a unit of the local ring). If C i Ď S is the plane curve singularity defined by f i , then the decomposition [START_REF] Abhyankar | Concepts of order and rank on a complex space, and a condition for normality[END_REF] gives a decomposition of C seen as a germ of effective divisor C " ř iPI p i C i , where each curve singularity C i is a branch. The plane curve singularity C is reduced if and only if p i " 1 for every i P I.

The intersection number is the simplest measure of complexity of the way two plane curve singularities interact at a given point: Definition 2.7. Let C and D be two curve singularities on the germ of smooth surface pS, oq defined by functions f and g P ÔS,o respectively. Their intersection number pC ¨Dq o , also denoted C ¨D if the base point o of the germ is clear from the context, is defined by:

C ¨D :" dim C ÔS,o pf, gq P N Y t8u,
where pf, gq denotes the ideal of ÔS,o generated by f and g.

If C and D are two curve singularities, then one has that pC ¨Dq o ě m o pCqm o pDq, with equality if and only if the curves C and D are transversal (see [START_REF] Wall | Singular points of plane curves[END_REF]Lemma 4.4.1]), that is, the tangent plane of pS, oq does not contain lines which are tangent to both C and D.

Seen as a function of two variables, the intersection number is symmetric. It is moreover bilinear, in the sense that if C " ř iPI p i C i , then C ¨D " ř iPI p i pC i ¨Dq. Therefore, in order to compute C ¨D, it is enough to find C i ¨D for all the branches C i of C.

One has the following useful property (see [66, Lemma 5.1.5]):

Proposition 2.8. Let C be a branch and D be an arbitrary curve singularity on the smooth germ of smooth surface pS, oq. Denote by N : pC t , 0q Ñ pS, oq a formal parametrization of degree one of C and g P ÔS,o be a defining function of D. Then C ¨D " ν t pgpN ptqqq, where ν t phq denotes the order of a power series h P Crrtss.

Example 2.9. Let us consider two curves C, D Ď C 2 x,y , defined by polynomials f :" x α ´yβ and g :" x γ ´yδ of the type already considered in Example 2.6. Assume that α and β are coprime. This implies, as shown in Example 2.6, that the plane curve singularity pC, Oq is a branch and that N ptq :" pt β , t α q is a parametrization of degree one of it. By Proposition 2.8, if C is not a branch of D, we get:

C ¨D " ν t `pt β q γ ´pt α q δ ˘" ν t `tβγ ´tαδ ˘" mintβγ, αδu.

For more details about intersection numbers of plane curve singularities, one may consult [START_REF] Brieskorn | Plane algebraic curves. Translated from the German by John Stillwell[END_REF]Sect. 6], [113, Vol. 1, Chap. IV.1] and [START_REF] Fischer | Plane algebraic curves[END_REF]Chap. 8].

The formal parametrizations N : pC t , 0q Ñ pS, oq of degree one of a branch appearing in the statement of Proposition 2.8 are exactly the normalization morphisms of C whose sources are identified with pC, 0q. Next subsection is dedicated to the general definition of normal complex variety and of normalization morphism in arbitrary dimension, as we will need them later also for surfaces.

Basic facts about normalizations.

In this subsection we explain basic facts about normal rings (see Definition 2.10), normal complex varieties (see Definition 2.11) and normalization morphisms (see Definition 2.16) of arbitrary complex varieties. For more details and proofs one may consult [START_REF] De Jong | Local analytic geometry[END_REF]Sections 1.5,4.4] and [START_REF] Greco | Normal varieties[END_REF].

The following definition contains algebraic notions, concerning extensions of rings: Definition 2.10. Let R be a commutative ring and let R Ď T be an extension of R.

(1) An element of T is called integral over R if it satisfies a monic polynomial relation with coefficients in R. [START_REF] Abhyankar | Newton-Puiseux Expansion and Generalized Tschirnhausen Transformation[END_REF] The extension R Ď T of R is called integral if each element of T is integral over R.

(3) The integral closure of R is the set of integral elements over R of the total ring of fractions of R. (4) R is called normal if it is reduced (without nonzero nilpotent elements) and integrally closed in its total ring of fractions, that is, if it coincides with its integral closure.

The arithmetical notion of normal ring allows to define the geometrical notion of normal variety:

Definition 2.11. Let X be a complex variety in the sense of Definition 2.1.

(1) If x P X, then the germ pX, xq of X at x is called normal if its local ring O X,x is normal.

(2) The complex variety X is normal if all its germs are normal.

Normal varieties may be characterized from a more function-theoretical viewpoint as those complex varieties on which holds the following "Riemann extension property": every bounded holomorphic function defined on the smooth part of an open set extends to a holomorphic function on the whole open set (see [START_REF] De Jong | Local analytic geometry[END_REF]Theorem 4.4.15]).

Recall now the following algebraic regularity condition (see [START_REF] De Jong | Local analytic geometry[END_REF]Sect. 4.3]):

Definition 2.12. Let O be a Noetherian local ring, with maximal ideal m.

(1) The Krull dimension of O is the maximal length of its chains of prime ideals.

(2) The embedding dimension of O is the dimension of the O{m-vector space m{m 2 .

(3) The local ring O is called regular if its Krull dimension is equal to its embedding dimension.

The Krull dimension of O is always less or equal to the embedding dimension. The name embedding dimension may be understood by restricting to the case where O is the local ring of a complex space (see [START_REF] De Jong | Local analytic geometry[END_REF]Lemma 4.3.5]): Proposition 2.13. Let pX, xq be a germ of complex space. Then the embedding dimension of its local ring O X,x is equal to the smallest n P N such that there exists an embedding of germs pX, xq ãÑ pC n , 0q. In particular, O X,x is regular if and only if pX, xq is smooth, that is, a germ of complex manifold.

The normal varieties of dimension one are exactly the smooth complex curves because, more generally (see [START_REF] De Jong | Local analytic geometry[END_REF]Thm. 4 There is a canonical way to construct a normal variety X starting from any complex variety X (see [START_REF] De Jong | Local analytic geometry[END_REF]Sect. 4.4]):

Theorem 2.15. Let X be a complex variety. Then there exists a finite and generically 1 to 1 morphism N : X Ñ X such that X is normal. Moreover, such a morphism is unique up to a unique isomorphism over X.

Recall that a morphism between complex varieties is finite if it is proper with finite fibers and that it is generically 1 to 1 if it is an isomorphism above the complement of a nowhere dense closed subvariety of its target space. The existence of a morphism with the properties stated in Theorem 2.15 may be proven algebraically by considering the integral closures of the rings of holomorphic functions on the open sets of X, and showing that they are again rings of holomorphic functions on complex varieties which admit finite and generically 1 to 1 morphisms to the starting open sets. This algebraic proof extends to formal germs, by showing that the integral closure in its total ring of fractions of a complete ring of the form Crrx 1 , . . . , x n ss{I, where n P N ˚and I is an ideal of Crrx 1 , . . . , x n ss, is a direct sum of rings of the same form.

The canonical morphisms characterized in Theorem 2.15 received a special name:

Definition 2.16. Let X be a complex variety. Then a morphism N : X Ñ X is called a normalization morphism of X if it is finite, generically 1 to 1 and X is a normal complex variety.

Let now pC, oq be a germ of complex variety of Krull dimension one, that is, an abstract curve singularity. Its normalization morphisms are of the form: N : Ů iPI p Ci , o i q Ñ pC, oq, where pC i , oq iPI is the finite collection of irreducible components of pC, oq, and the restriction N i : p Ci , o i q Ñ pC, oq of N to Ci is a normalization of pC i , oq. By Theorem 2.14 and Proposition 2.13, we see that each germ p Ci , o i q is smooth, that is, isomorphic to pC, 0q. After precomposing N with such isomorphisms, we see that pC, oq admits a normalization morphism of the form Ů iPI pC, 0q Ñ pC, oq. In particular, if pC, oq is irreducible, its normalization morphism is of the form N : pC, 0q Ñ pC, oq. The same construction yields a formal parametrization when the starting germ pC, oq is formal. This is precisely a formal parametrization of degree one as used in the statement of Proposition 2.8.

Newton-Puiseux series and the Newton-Puiseux theorem.

At the end of the previous subsection we explained that normalizations of irreducible germs of complex analytic or formal curves C are holomorphic or formal parametrizations pC, 0q Ñ C of degree one. In this subsection we introduce especially nice parametrizations in the case of plane branches, which lead to the notion of Newton-Puiseux series (see Definition 2.18). The Newton-Puiseux theorem (see Theorem 2.20) implies that the field of Newton-Puiseux series is algebraically closed. Another consequence of it is stated in Theorem 6.1 below.

Let C be a branch on the smooth germ of surface pS, oq. Choose an arbitrary system of local coordinates on pS, oq. If the branch C is smooth, assume moreover that the germ at o of the y-axis Zpxq is different from C. This means that for any normalization morphism N : pC t , 0q Ñ pC, oq of C, described in this coordinate system as t Ñ pξptq, ηptqq, where ξ, η P ptqCrrtss, the power series ξptq is not identically zero. We have ξptq " t n ¨ ptq, where n P N ˚is the order of the power series ξptq and ptq is a unit in the ring Crrtss. The series ptq has exactly n different n-th roots in Crrtss, whose constant terms are the n-th roots of p0q. Pick one of them, denote it by 1{n ptq, and set λptq :" t 1{n ptq. Therefore ξptq " λptq n and ν t pλptqq " 1.

Remark 2.17. More generally, if K is an algebraically closed field of characteristic zero, then any unit of Krrtss has all its n-th roots in Krrtss. This fact is no longer true if K has positive characteristic. For instance, as a direct consequence of the binomial formula, there is no series ptq P Krrtss such that ptq p " 1 `t when K is of characteristic p. For this reason, the Newton-Puiseux Theorem 2.20 below does not always hold in positive characteristic. For more details about the situation in positive characteristic, one may consult [START_REF] Veys | Introduction to the local theory of plane algebraic curves[END_REF].

As ν t pλptqq " 1, we see that the morphism pC t , 0q Ñ pC u , 0q, which maps t Ñ λptq is an isomorphism of germs of smooth curves. By composing the morphism N : pC t , 0q Ñ pC, oq with its inverse, one gets a new normalization morphism of the form:

pC u , 0q Ñ pC, oq u Ñ pu n , ζpuqq
where ζpuq P Crruss. Therefore, if f px, yq P Crrx, yss is a defining function of C in the local coordinate system px, yq, we have:

(2) f pu n , ζpuqq " 0.

From the equations x " u n , y " ζpuq, one may deduce formally that u " x 1{n , y " ζpx 1{n q. Equation (2) becomes:

(3) f px, ζpx 1{n qq " 0.

The composition ζpx 1{n q is a Newton-Puiseux series in the following sense:

Definition 2.18. The C-algebra Crrx 1{N ss of Newton-Puiseux series consists of all the formal power series of the form ηpx 1{n q, where η P Crrtss and n P N ˚, that is, Crrx 1{N ss " Ť nPN ˚Crrx 1{n ss. Denote by ν x : Crrx 1{N ss Ñ r0, 8s the order function, which associates to every Newton-Puiseux series the smallest exponent of its terms, where ν x p0q :" 8.

The function ν x is a valuation of the C-algebra of Newton-Puiseux series, in the following sense: Definition 2.19. A valuation on an integral C-algebra A is a function ν : A Ñ R `Y t8u which satisfies the following conditions:

(1) νpf gq " νpf q `νpgq, for all f, g P A.

(2) νpf `gq ě mintνpf q, νpgqu, for all f, g P A.

(3) νpλq " 0, for all λ P C ˚.

(4) νpf q " 8 if and only if f " 0.

The basic importance of the ring of Newton-Puiseux series comes from the following Newton-Puiseux theorem (see Fischer [39, Proof. It is immediate to reduce the proof of the first sentence of the theorem to the case where f is irreducible. Assume that this is the case. By equation ( 3), there exists a Newton-Puiseux series ζpx 1{n q which is a root of f . One has necessarily n " d. Indeed, by the proof of equation ( 3), u Ñ pu n , ζpuqq is a normalization of the formal branch Zpf q. Therefore, Proposition 2.8 shows that:

n " ν u pu n q " Zpf q ¨Zpxq " Zpxq ¨Zpf q " ν y pf p0, yqq " d.

Consider now the product:

F px, yq :"

ź ω:ω n "1 ´y ´ζpωx 1{n q ¯P Crrx 1{n ssrys.
It is invariant by the changes of variables px 1{n , yq Ñ pωx 1{n , yq, where ω varies among the complex n-th roots of 1, which shows that F px, yq P Crrxssrys. As ζpx 1{n q is a root of both f px, yq and F px, yq and that f px, yq is irreducible, we see that f divides F in the ring Crrxssrys. Both being monic and of the same degree, we get the equality f " F . Therefore, all the roots of f belong to Crrx 1{n ss.

The second statement of the theorem results from the first statement and from Hensel's lemma (see [START_REF] De Jong | Local analytic geometry[END_REF]Corollary 3.3.21]), which ensures that a factorisation of f p0, yq P Crys in pairwise coprime factors lifts to an analogous decomposition of f px, yq P Crrxssrys.

The proof of Theorem 2.20 which we have sketched here also shows that the Galois group of the field extension associated to the ring extension Crrxss Ă Crrx 1{n ss is isomorphic to the cyclic group of n-th roots of 1, an element ω of this group acting on ζpx 1{n q P Crrx 1{n ss replacing it by ζpωx 1{n q.

Remark 2.21. The proof of Theorem 2.20 which we have sketched here also shows that the Galois group of the field extension associated to the ring extension Crrxss Ă Crrx 1{n ss is isomorphic to the cyclic group of n-th roots of 1, an element ω of this group acting on ζpx 1{n q P Crrx 1{n ss replacing it by ζpωx 1{n q.

Remark 2.22. Most proofs of Theorem 2.20 use the Newton polygon N pf q of f (see Definition 4.2 below). As explained in Subsection 2.5, the restrictions of f to the compact edges of N pf q allow to find the possible initial terms of the candidate roots ηpxq of the equation f px, yq " 0 seen as an equation in the single unknown y. Such proofs proceed then by showing that all those terms may be extended to true roots inside Crrx 1{N ss.

Example 2.23. Consider coprime integers α, β P N ˚and f px, yq :" x α ´yβ P Crrxssrys, as in Example 2.6. Then the Newton-Puiseux roots of f are the β series ω x α{β , where ω varies among the complex β-th roots of 1. If ω 1 is another such root of 1, it acts on ω x α{β by sending it to pω 1 q α ω x α{β .

Blow ups and embedded resolutions of singularities.

In this subsection we explain the notion of blow up of C 2 at the origin (see Definition 2.24) and more generally of a smooth complex surface at an arbitrary point of it (see Definition 2.29), the notion of embedded resolution of a curve in a smooth surface (see Definition 2.33) and the fact that an embedded resolution may be achieved after a finite number of blow ups of points (see Theorem 2.35). We conclude by recalling the notion of weighted dual graph of an embedded resolution (see Definition 2.36) and the way to compute its weights when this resolution is constructed iteratively by blowing up points.

Look at the complex affine plane C 2

x,y as a complex vector space. Denote by PpC 2 q ru:vs its projectivisation, consisting of its vector subspaces of dimension one, endowed with the projective coordinates ru : vs associated to the cartesian coordinates px, yq on C 2 . Definition 2.24. Consider the projectivisation map λ :

C 2 PpC 2 q px, yq rx : ys.

associating to each point of C 2 ztOu the line joining it to the origin O of C 2 . Let Σ be the closure of its graph in the product algebraic variety C 2 ˆPpC 2 q. Then the restriction π : Σ Ñ C 2 of the first projection C 2 ˆPpC 2 q Ñ C 2 is called the blow up of C 2 at the origin. By abuse of language, the surface Σ is also called in this way. The preimage π ´1pOq of O in Σ is called the exceptional divisor of the blow up. The restriction λ : Σ Ñ PpC 2 q to Σ of the second projection C 2 ˆPpC 2 q Ñ PpC 2 q is called the Hopf morphism.

The name "Hopf morphism" is motivated by the fact that in restriction to the preimage π ´1pS 3 q of the unit 3-dimensional sphere S 3 Ă C 2 , the morphism λ becomes the "Hopf fibration" S 3 Ñ S 2 , introduced by Hopf in [64, Section 5] (see also [START_REF] Samelson | π 3 pS 2 q[END_REF] for historical details).

The projectivisation map restricts to a morphism λ : C 2 ztOu Ñ PpC 2 q. This morphism cannot be extended even by continuity to the origin O, because O belongs to the closures of all its level sets, which are the complex lines of C 2 passing through O. Taking the closure of the graph of λ replaces O by the space PpC 2 q of lines passing through O. This allows the lift of λ to Σ to extend by continuity, and even algebraically, to the whole surface Σ, becoming the Hopf morphism λ. This morphism is in fact the projection morphism of the total space of a line bundle, as will be shown in Proposition 2.25 below. Before proving it, let us explain how to describe using a simple atlas of two charts the blow up surface Σ.

The projective line PpC 2 q ru:vs is covered by the two affine lines C u1 and C v2 , where:

u 1 :" u v , v 2 :" v u .
Therefore, the product C 2 ˆPpC 2 q is covered by the two affine 3-folds C 3 x,y,u1 and C 3 x,y,v2 . The surface Σ contained in C 2 ˆPpC 2 q is the zero locus Zpxv ´yuq of a homogeneous polynomial of degree one in the variables u, v. Its intersections with the two previous 3-folds are therefore:

Σ X C 3
x,y,u1 " Zpx ´yu 1 q, and Σ X C 3 x,y,v2 " Zpxv 2 ´yq. One recognizes in each case the equation of the graph of a function of two variables, those pairs of variables being pu 1 , yq and px, v 2 q respectively. Therefore, by projecting on the planes of those two pairs of variables, one gets isomorphisms:

Σ X C 3 x,y,u1 » C 2 u1,y , and Σ X C 3 x,y,v2 » C 2
x,v2 , which may be thought as the charts of an algebraic atlas of Σ. Let us replace y by u 2 in the first chart

C 2
u1,y and x by v 1 in the second chart C 2 x,v2 . The blow up morphism π : Σ Ñ C 2 gets expressed in the following way in the two charts: (4)

"

x " u 1 u 2 y " u 2 , and

" x " v 1 y " v 1 v 2 .
The previous formulae show that the exceptional divisor π ´1pOq becomes the u 1 -axis in the chart C 2 u1,u2

and the v 2 -axis in the chart C 2 v1,v2 . By composing one such morphism with the inverse of the second one, we see that Σ may be obtained from the two copies C 2 u1,u2 and C 2 v1,v2 of C 2 by gluing their open subsets C ů1 ˆCu2 and C v1 ˆCv 2 respectively using the following inverse changes of variables:

(5)

" v 1 " u 1 u 2 v 2 " u ´1 1 and " u 1 " v ´1 2 u 2 " v 1 v 2 .
The Hopf morphism λ : Σ Ñ PpC 2 q becomes the morphisms

C 2 u1,u2 Ñ C 1 u1 and C 2 v1,v2 Ñ C 1 v2 if one uses the charts C 2 u1,u2 , C 2 v1,v2 for Σ and C 1 u1 , C 1 v2
for PpC 2 q. The fibers of these two morphisms have natural structures of complex lines if one identifies them with the standard complex line C using the parameters u 2 and v 1 respectively. As the gluing maps (5) respect those structures, we get: Proposition 2.25. The Hopf morphism λ : Σ Ñ PpC 2 q is the projection morphism from the total space of a line bundle to its base PpC 2 q. Its zero-section is the exceptional divisor π ´1pOq of the blow up morphism

π : Σ Ñ C 2 .
The fundamental numerical invariant of a complex line bundle over a projective curve, which characterises it up to topological isomorphims in general and up to algebraic isomorphisms if the curve is rational, is its degree, defined by: Definition 2.26. The degree of a line bundle over a smooth connected projective curve C is the degree of the divisor on C defined by any meromorphic section of the line bundle which is neither constantly 0 nor constantly 8.

In our case, we have: Proposition 2.27. The degree of the Hopf line bundle λ : Σ Ñ PpC 2 q is equal to ´1.

Proof. Let us consider the meromorphic section s of λ which appears as the constant function 1 in the charts C 2 u1,u2 Ñ C 1 u1 . The equation of its graph is u 2 " 1. The change of variables (5) transform it into v 1 v 2 " 1. Therefore, s appears as the rational function v ´1 2 in the charts C 2 v1,v2 Ñ C 1 v2 . This shows that the section s has no zeros and a unique pole of multiplicity one. As a consequence, the degree of the divisor defined by s is equal to ´1.

On any smooth complex algebraic or analytic surface S, one may define a notion of intersection number of two divisors whenever at least one of them has compact support. This may be done algebraically, by considering first the case when one divisor is a reduced compact curve C on S, the intersection number being then the degree of the pullback of the line bundle defined by the second divisor to the normalization of C. Then, one extends this definition by linearity to arbitrary not necessarily reduced or effective divisors. There is also a topological definition, obtained by associating a homology class to one divisor, a cohomology class to the second one and then evaluating the cohomology class on the homology class. One may consult [61, Sect. V.1] for the case of algebraic surfaces and [START_REF] Laufer | Normal two-dimensional singularities[END_REF] for the case of analytic surfaces. It turns out that, either by definition or as a theorem, the self-intersection number of the zero-section of a line bundle over a smooth compact complex curve is equal to the degree of the line bundle. Therefore, Proposition 2.27 may be also stated as:

Corollary 2.28. The self-intersection number of the zero-section of the Hopf line bundle λ : Σ Ñ PpC 2 q is equal to ´1.

Till now, we have discussed in this subsection only the blow up of the origin O of C 2 . This operation may be extended to any point o of a smooth complex surface S, by choosing first local coordinates px, yq in a neighborhood U of that point. This allows to identify U with an open neighborhood of O in C 2 . Denote by π U : Σ U Ñ U the restriction to U of the blow up morphism of O in C 2 . This complex analytic morphism is an isomorphism over U ztOu. Therefore, it allows to glue Σ U and S along U ztOu, getting a surface S endowed with a morphism π : S Ñ S.

Definition 2.29. The morphism π : S Ñ S constructed above is called a blow up morphism of S at the point o.

It may be shown by a direct computation that the blow up morphism of S at o is independent of the choices of local coordinates and open set U . More precisely, given any two morphisms constructed in this way, there exists a unique isomorphism between their sources above S (see [START_REF] Wall | Singular points of plane curves[END_REF]Lemma 3.2.1]). Another way to prove this uniqueness is to characterize such morphisms by a universal property (see [61, Chap. II, Prop. 7.14]): Proposition 2.30. Let S be a smooth complex surface and π : S Ñ S a blow up morphism of S at its point o. Then for any morphism f : Y Ñ S such that the ideal sheaf defining o on S lifts to a principal ideal sheaf on Y , there exists a unique morphism g : Y Ñ S such that f " π ˝g.

One may define more generally the blow up of any complex space along a closed subspace, and again this morphism may be characterized using an analogous universal property (see [START_REF] Hartshorne | Algebraic geometry[END_REF] for a similar study in the case of schemes).

Returning to the model case of the blow up of C 2 at the origin O, relations (4) show that the lift by π to Σ of the maximal ideal px, yq of Crx, ys defining O is the principal ideal sheaf defining the exceptional divisor of π. This fact is an algebraic manifestation of the fact that on Σ all the lines of C 2 passing through O get separated: they are simply the fibers of the Hopf morphism λ. Note that in order to separate them indeed, one does not have to lift them by taking their full preimages by π (called their total transforms by π), but only by taking their strict transforms. Let us define these notions in greater generality: Definition 2.31. Let π : Y Ñ X be a morphism of complex varieties and Z Ď X a closed complex subvariety of X.

(1) The morphism π is a modification of X if it is proper and bimeromorphic, that is, if it is proper and if there exists a closed nowhere dense subvariety X 1 of X such that π ´1pX 1 q is a nowhere dense subvariety of Y and the restriction π : Y zπ ´1pX 1 q Ñ XzX 1 is an isomorphism.

(2) If X 1 is minimal with the previous properties, then X 1 is called the indeterminacy locus of π ´1 and π ´1pX 1 q is called the exceptional locus of π.

(3) The total transform π ˚pZ q of Z by π is the complex subspace of Y defined by the preimage by π of the ideal sheaf defining Z in X. (4) Assume that no irreducible component of Z is included in the indeterminacy locus X 1 of π ´1.

Then the strict transform of Z by π is the closure inside Y of π ´1pZ zX 1 q.

The blow up morphisms of surfaces at smooth points are examples of modifications. In the case of the blow up π : Σ Ñ C 2

x,y at the origin, the equations (4) show that the total transform of a line Zpy ´axq Ď C 2

x,y , for a P C ˚, may be described as Zpu 2 p1 ´au 1 qq Ď C 2 u1,u2 and Zpv

1 pv 2 ´aqq Ď C 2 u1,u2
in the two charts covering Σ. As Zpu 2 q and Zpv 1 q describe the exceptional divisor π ´1pOq in those two charts, we see that the strict transform of Zpy ´axq is the fiber of λ whose equations are u 1 " a ´1 and v 2 " a in those two charts. Assume now that C is a finite sum ř iPI L i of such lines L i passing through the origin in C 2 . The strict transform of C by π is the sum of the strict transforms Li of those lines and the total transform π ˚pC q is the sum π ´1pOq `řiPI Li of the exceptional divisor of π and of the strict transform of C. Therefore, π ˚pC q is a normal crossings divisor in the following sense: Definition 2.32. Let S be a smooth complex surface and D a divisor on it. This divisor is said to have normal crossings or to be a normal crossings divisor if its support is locally either a smooth curve or the union of two transversal smooth curves.

Coming back to the curve C " ř iPI L i in C 2 , the fact that its total transform π ˚pC q is a normal crossings divisor shows that the blow up morphism π : Σ Ñ C 2 is an embedded resolution of C, in the following sense: Definition 2.33. Let C be a curve on the smooth complex surface S, in the sense of Definition 2.3. An embedded resolution of C is a modification π : S Ñ S such that:

(1) S is smooth;

(2) the total transform π˚p Cq is a normal crossings divisor;

(3) the strict transform C of C by π is smooth.

The restriction πC : C Ñ C of an embedded resolution π of C to the strict transform C of C is a resolution of C in the following sense: Definition 2.34. Let X be a complex variety. A resolution of X is a modification π : X Ñ X such that X is smooth and the indeterminacy locus of π ´1 is equal to the singular locus of X.

If X is a complex curve, then a resolution of it is the same as a normalization morphism. This is no longer true in higher dimensions, as in each dimension at least 2, there are normal non-smooth complex varieties. For instance, a hypersurface X of C n whose singular locus has codimension at least 2 in X is normal (see [START_REF] Oka | Sur les fonctions analytiques de plusieurs variables. VIII. Lemme fondamental[END_REF], [START_REF] Abhyankar | Concepts of order and rank on a complex space, and a condition for normality[END_REF]).

Note that the second condition in Definition 2.33 does not imply the third one. For instance, if one takes the folium of Descartes C Ă C 2

x,y defined by the equation x 3 `y3 " 3xy, then C is a normal crossings divisor in C 2 (with a single singular point at the origin), therefore the identity morphism from C 2 to itself satisfies the first two conditions of Definition 2.33 but not the last one, because the strict transform of C by it is not smooth, being the curve C itself.

In order to get an embedded resolution of the folium of Descartes, it is enough to blow up C 2 at the origin O. More generally, if C is a curve in a smooth complex surface S such that at each point o of C, the branches of C at o are smooth and pairwise transversal, then the morphism obtained by blowing up S at all the singular points of C is an embedded resolution of C. Conversely, as may be seen by working with the description (4) of the blow up morphism at a point in terms of local coordinates, this property of achieving an embedded resolution by blowing up distinct points of S characterizes the previous kind of curves. What about curves with more complicated singularities? It turns out that they also have embedded resolutions, which may be obtained by blowing up points iteratively (see [START_REF] Hartshorne | Algebraic geometry[END_REF]Thm. 3.9], [START_REF] Brieskorn | Plane algebraic curves. Translated from the German by John Stillwell[END_REF], [START_REF] De Jong | Local analytic geometry[END_REF]Thm. 5.4.2], [START_REF] Casas-Alvero | Singularities of plane curves[END_REF]Section 3.7] and [START_REF] Wall | Singular points of plane curves[END_REF]Thm. 3.4.4]): Theorem 2.35. Let C be a curve on the smooth complex surface S. Define S 0 :" S and π 0 : S 0 Ñ S to be the identity. Assume that for some k ě 0 one has defined a modification π k : S k Ñ S which is not an embedded resolution of C. Denote by B k Ă S k the set of points at which either the strict transform of C is not smooth or π k pCq is not a normal crossings divisor. Define ψ k : S k`1 Ñ S k to be the blow up of S k at the points of B k and π k`1 :" π k ˝ψk : S k`1 Ñ S. Then there exists k P N such that π k is an embedded resolution of C.

If k is chosen minimal such that π k is an embedded resolution of C, then π k is called the minimal embedded resolution of C. It may be shown that any other embedded resolution of C factors through it.

The combinatorial structure of the total transform of C on a given embedded resolution π : S Ñ S of C is encoded usually by drawing its weighted dual graph: Definition 2.36. Let C be a curve on the smooth complex surface S and π : S Ñ S be an embedded resolution of C. Its weighted dual graph is a finite connected graph whose vertices are labeled by the irreducible components of the total transform π˚p Cq, two vertices being connected by an edge whenever their associated curves intersect on Σ. The vertices corresponding to the components of the strict transform of C are drawn arrowheaded. The remaining vertices are weighted by the self-intersection numbers on Σ of the associated irreducible components of the exceptional locus of π.

How to compute the weights of the dual graph of the embedded resolution π : S Ñ S? If this resolution is obtained iteratively by the process described in Theorem 2.35, then one may compute recursively the self-intersection numbers of the components of the exceptional loci of the modifications π k using Corollary 2.28 and (see [START_REF] Wall | Singular points of plane curves[END_REF]Lemma 8.1.6]): Proposition 2.37. Let C be a compact curve in the smooth complex surface S. Let o be a point of C of multiplicity m P N. If π : Σ Ñ S is the blow up of S at o, then the self-intersection C2 in Σ of the strict transform C of C by π is related to the self-intersection C 2 of C in S by the formula C2 " C 2 ´m.

The minimal embedded resolution of the semicubical parabola.

In this subsection we show how to achieve the minimal embedded resolution of the semicubical parabola using the algorithm described in Theorem 2.35 and how to compute its weighted dual graph using Proposition 2.37. It is an expansion of [61, Example V.3.9.1].

The semicubical parabola is the curve P ãÑ C 2

x,y defined as the vanishing locus of the polynomial ppx, yq :" y 2 ´x3 . The germ of P at the origin O is a branch called sometimes the standard cusp. Due to the following Jacobian criterion (see [START_REF] De Jong | Local analytic geometry[END_REF]Theorem 4.3.6] for a generalization in arbitrary dimension and codimension), the origin is the only singular point of P . x,y , defined by a holomorphic function f : U Ñ C. Then the singular locus SingpCq is the zero locus Zpf, B x f, B y f q.

We want to construct a sequence of blow ups which leads to an embedded resolution of P by following the algorithm described in Theorem 2.35, whose notations we use. Therefore, denote by π 1 :

S 1 Ñ C 2 the blow up of the origin O 0 :" O of C 2
x,y , instead of π : Σ Ñ C 2 as in Definition 2.24. We use the standard charts C 2 u1,u2 and C 2 v1,v2 for computations on S 1 , the blow up morphism π 1 being then described by the changes of variables (4). The total transform π 1 pP q of P by π 1 is defined by the composition p ˝π1 , which is expressed as follows in the two charts:

(6) ppu 1 u 2 , u 2 q " u 2 2 p1 ´u3 1 u 2 q, ppv 1 , v 1 v 2 q " v 2 1 pv 2 2 ´v1 q.
As the curve P is smooth outside the origin, its strict transform P 1 by π 1 is also smooth outside the exceptional divisor. This strict transform intersects the exceptional divisor π ´1 1 pOq only in the chart C 2 v1,v2 , because its equations in the two charts are 1 ´u3 1 u 2 " 0 and v 2 2 ´v1 " 0. The second equation is that of a parabola, therefore it defines a smooth curve. This shows that the strict transform P 1 is everywhere smooth. Therefore, the restriction of the morphism π 1 to the curve P 1 is a resolution of P , in the sense of Definition 2.34. But it is not an embedded resolution in the sense of Definition 2.33, because the total transform π 1 pP q is not a normal crossings divisor at the origin O 1 of the chart C 2 v1,v2 . Indeed, the strict transform P 1 X C 2 v1,v2 " Zpv 2 2 ´v1 q and the exceptional divisor π ´1 1 pOq X C 2 v1,v2 " Zpv 1 q are tangent at O 1 .

Blow up now the point O 1 , getting a new surface S 2 . Let ψ 1 : S 2 Ñ S 1 be this blow up morphism.

The preimage ψ ´1 1 pC 2 v1,v2 q of the chart C 2 v1,v2 of S 1 may be covered by two charts C 2 w1,w2 and C 2 z1,z2 , in which the morphism ψ 1 is described by the following analogs of equations (4): ( 7)

" v 1 " w 1 w 2 v 2 " w 2 , and
" v 1 " z 1 v 2 " z 1 z 2 .
In order to cover completely the surface S 2 , one needs also the chart C 2 u1,u2 of S 1 , which is left unchanged by the blow up morphism ψ 1 because O 1 does not appear in it.

Denote π 2 :" π 1 ˝ψ1 : S 2 Ñ C 2 . Using equations (6) we see that:

(8) p ˝π2 pw 1 , w 2 q " w 2 1 w 3 2 pw 2 ´w1 q, and p ˝π2 pz 1 , z 2 q " z 3 1 pz 1 z 2 2 ´1q. Therefore, the strict transform P 2 of P 1 by π 2 intersects again the exceptional divisor only in one of those charts, namely C 2 w1,w2 . The total transform π 2 pP q ãÑ S 2 is still not a normal crossings divisor, because its germ at the origin O 2 of C 2 w1,w2 has three branches: Zpw 1 q, Zpw 2 q, Zpw 2 ´w1 q, as shown by equation [START_REF] A'campo | Geometry of plane curves via Tschirnhausen resolution tower[END_REF]. One needs to blow up also this point, getting the morphisms ψ 2 : S 3 Ñ S 2 and π 3 :" π 2 ˝ψ2 : S 3 Ñ C 2 . The blow up ψ 2 may be described using the following analogs of equations (4) above the chart C 2 w1,w2 :

" w 1 " s 1 s 2 w 2 " s 2 , and

" w 1 " t 1 w 2 " t 1 t 2 .
Composing these changes of variables with the second equation ( 8), we get: p ˝π3 ps 1 , s 2 q " s 2 1 s 6 2 p1 ´s1 q, p ˝π3 pt 1 , t 2 q " t 6 1 t 3 2 pt 2 ´1q. In both charts of S 3 the total transform π 3 pP q is a normal crossings divisor. This being the case also in the remaining charts C 2 u1,u2 and C 2 z1,z2 , we see that π 3 is an embedded resolution of singularities of the semicubical parabola P . By Theorem 2.35, it is the minimal such resolution.

We illustrated the previous sequence of blow ups in Figure 2. We drew whenever possible the support of the total transform of P in the chart whose origin is contained in the strict transform of P . In the four charts the strict transforms of P are drawn in orange and the defining polynomial is written near it. We have used systematically the same color for a point O i which is blown up by a morphism ψ i , for the exceptional divisor E i created by this blow up and for its strict transforms E i,j by the next blow ups. Notice that the component E 0,2 appears on the chart C 2 t1,t2 , but it does not appear on the chart C 2 s1,s2 , represented on the right of Figure 2. v1,v2 one has E 0,1 " Zpv 1 q, O 1 " p0, 0q and in the chart C 2 w1,w2 one has E 1 " Zpw 2 q, E 0,1 " Zpw 1 q, O 2 " p0, 0q. Using Theorem 2.37, we get E 2 0,2 " E 2 0 ´2 " ´3 and E 2 1,2 " E 2 1 ´1 " ´2. Therefore, the weighted dual graph of the minimal embedded resolution π 3 : S 3 Ñ C 2 of the semicubical parabola P is as shown in Figure 3. Near the arrowhead vertex corresponding to the strict transform of P , we have written the defining function of the semicubical parabola.

x y Zpy 2 ´x3 q O 0 v 1 v 2 Zpv 2 2 ´v1 q O 1 E 0 E 0,1 E 1 w 1 w 2 Zpw 2 ´w1 q O 2 E 2 E 1,2 s 1 s 2 Zp1 ´s1 q π 1 ψ 1 ψ 2 Figure 2.

´3

´1 ´2

E 0,2 E 2 E 1,2
Zpy 2 ´x3 q The previous computations involve many charts, therefore many variables and changes of variables. It is easy to get lost in them. One feels the need of being able to arrive at the final result, the weighted dual graph, without such manipulations. In the next subsection we show how to achieve this goal by a simpler method, without working with charts. We will explain the method using an apparently more complicated example, with two branches. After reading it, we suggest the reader to verify that in the case of the semicubical parabola, the method leads again to the weighted tree of Figure 3.

A Newton non-degenerate reducible example.

In this subsection we present on a simple example of Newton non-degenerate plane curve singularity the notions of Newton polygon and Newton fan of a non-zero function f px, yq P Crrx, yss. Then we introduce the associated lotus and we show how to construct from it the weighted dual graph of the minimal embedded resolution of the given singularity. These notions are briefly introduced in this section to illustrate our second elementary example and will be revisited formally in Sections 4 and 5.

Let pC, Oq ãÑ pC 2

x,y , Oq be the plane curve singularity defined by the function: [START_REF] Artal Bartolo | Quasi-ordinary singularities and their zeta functions[END_REF] f px, yq :" py 2 ´4x 3 qpy 3 ´x7 q.

It is the sum of two branches, defined by the equations y 2 ´4x 3 " 0 and y 3 ´x7 " 0 respectively. Thinking of them as polynomial equations in the unknown y, as explained in Subsection 2.3, they have degrees 2 and 3. Their respective sets of roots are t˘2x 3{2 u and tωx 7{3 u, where ω varies among the complex cubic roots of 1. We could express readily in terms of x the roots of the equation f px, yq " 0 seen as a quintic polynomial equation in the variable y, because we knew a factorization of f px, yq into binomial factors. Is it possible to reach the same objective if one starts instead from the following expanded expression of f ? [START_REF] Artal Bartolo | On quasi-ordinary singularities and Newton trees[END_REF] f px, yq " y 5 ´4x 3 y 3 ´x7 y 2 `4x 10 .

By the Newton-Puiseux Theorem 2.20, we know a priori that the roots of f px, yq may be expressed as Newton-Puiseux series. Newton's fundamental insight was that one may always compute the leading terms of such series only by looking at special terms of f (see the beginning of Subsection 4.5). Let us explain this insight in the case of the polynomial [START_REF] Artal Bartolo | On quasi-ordinary singularities and Newton trees[END_REF], forgetting its factorization [START_REF] Artal Bartolo | Quasi-ordinary singularities and their zeta functions[END_REF]. Denote by cx ν the leading term (that is, the term of least degree) of such a series, where c P C ˚and ν ą 0. We have the equality: [START_REF] Barber | Reducible exceptional curves of the first kind[END_REF] f px, cx ν `opx ν qq " 0.

Using formula [START_REF] Artal Bartolo | On quasi-ordinary singularities and Newton trees[END_REF], this equality may be rewritten as:

pcx ν `opx ν qq 5 ´4x 3 pcx ν `opx ν qq 3 ´x7 pcx ν `opx ν qq 2 `4x 10 " 0, that is, as:

(13) `c5 x 5ν `opx 5ν q ˘``´4 c 3 x 3`3ν `opx 3`3ν q ˘``´c 2 x 7`2ν `opx 7`2ν q ˘`4x 10 " 0.

The left-hand side of this equation is a sum of four series, whose leading exponents are 5ν, 3 `3ν, 7 `2ν, 10, since c ‰ 0. The fundamental observation of Newton was that if the sum (13) vanishes, then the minimal value of those four exponents is reached at least twice. Now, these four exponents may be expressed as the products p1, νq ¨pa, bq :" a `bν, where pa, bq varies among the exponents pa, bq P N 2 of the monomials x a y b appearing in the expanded form [START_REF] Artal Bartolo | On quasi-ordinary singularities and Newton trees[END_REF] of f px, yq, that is, as the evaluations of the linear form l ν pa, bq :" a `bν on the support Spf q of the series f px, yq. In our example the support is finite, but it may be infinite if one allows f to be a power series in the variables x, y. It is at this point that convex geometry enters into the game, through the following property (which is a consequence of [94, Assertion III.1.5.2]): Proposition 2.39. Let S be a subset of N 2 . If l is a linear form with non-negative coefficients on R 2 , then its restriction to S achieves its minimum precisely on the subset of S lying on a face of the convex hull ConvpS `R2 `q.

Coming back to equation ( 13), we see that the linear form l ν pa, bq " a `bν, which computes the leading exponents of the terms appearing in the left-hand side of ( 13), indeed has non-negative coefficients. Therefore, the hypotheses of Proposition 2.39 are satisfied. This shows that the minimal value min t5ν, 3 `3ν, 7 `2ν, 10u is achieved on a face of the convex hull ConvpSpf q `R2 `q. This convex hull, called the Newton polygon N pf q of f P Crrx, yss (see Definition 4.2 below), is represented in Figure 4. It has three vertices, which are p0, 5q, p3, 3q, p10, 0q, corresponding to the terms y 5 , ´4x 3 y 3 and 4x 10 of the expansion [START_REF] Artal Bartolo | On quasi-ordinary singularities and Newton trees[END_REF]. It has two compact edges K 1 :" rp0, 5q, p3, 3qs and K 2 :" rp3, 3q, p10, 0qs. If the minimum is to be achieved at least twice on Spf q, then it must be achieved on one of those two compact edges, because ν ą 0. This means that the linear form l ν must be orthogonal to one of those compact edges. There are therefore two possibilities: ' Either l ν achieves its minimum on K 1 , which means that p1, νq is orthogonal to it. In other words p1, νq ¨p3 ´0, 3 ´5q " 0, that is, ν " 3{2. Writing that the sum of the terms of the left-hand side of Equation [START_REF] Brauner | Zur Geometrie der funktionen zweier komplexen Veränderlichen III[END_REF] whose leading exponents achieve the minimum vanishes, one gets the equation c 5 ´4c 3 " 0. As c ‰ 0, this is equivalent to the equation c 2 " 4, hence c " ˘2. ' Or l ν achieves its minimum on K 2 . In other words p1, νq ¨p10 ´3, 0 ´3q " 0, that is, ν " 7{3. One gets then the equation ´4c 3 `4 " 0. That is, c varies now among the cubic roots of 1. It follows that the possible leading terms of a Newton-Puiseux series η in the variable x such that f px, ηq " 0 belong to the union t˘2x 3{2 u Y tωx 7{3 : ω 3 " 1u. One recognizes the roots from the factorization (10). Newton's method shows that those are the leading terms of the roots ypxq of the equation gpx, yq " 0, for any g P Crrx, yss whose Newton polygon is the same as N pf q, and whose restrictions to the compact sides of the polygon coincide with the analogous restrictions for f . Any such function g defines a Newton non-degenerate singularity (see Definition 4.21 below), because both equations c 2 " 4 and ´4c 3 `4 " 0 obtained by restricting g to the compact edges of its Newton polygon have simple roots. Variants of Newton's previous line of thought will be followed again in the proofs of Propositions 4.11 and 4.18 below.

In general, for any series f px, yq, once a first term cx ν of a potential root of f px, yq " 0 is computed, one may perform a formal change of variables and compute a second term. Newton explained that one could compute as many terms as needed, but it was Puiseux who proved carefully that by pushing this iterative process to its limit, one gets true roots of the equation, which are Newton-Puiseux series. Moreover, he proved that whenever one starts from a convergent function f , one gets only roots of the form ξpx 1{p q, where ξptq P Crrtss is convergent and p P N ˚. This approach leads to a proof of the Newton-Puiseux Theorem 2.20, different from the one given above (see Remark 2.22).

Let us come back to our example. It turns out that in this Newton non-degenerate case, the weighted dual graph of the minimal embedded resolution is determined by the Newton polygon N pf q. In fact, one needs only the inclinations of its compact edges. This information is encoded in the associated Newton fan, obtained by subdividing the first quadrant along the rays orthogonal to the compact edges of N pf q (see the left side of Figure 5 and Definition 4.9 below). Consider now inside the first quadrant all the triangles with vertices f 1 , f 2 , f 1 `f2 , where pf 1 , f 2 q is a basis of the ambient lattice Z 2 . The edges of those triangles may be drawn recursively by starting from the segment re 1 , e 2 s which joins the elements of the canonical basis pe 1 , e 2 q and, each time a new segment rf 1 , f 2 s is drawn, by drawing also the segments rf 1 , f 1 `f2 s and rf 2 , f 1 `f2 s. If one performs this construction only whenever the interior of the segment rf 1 , f 2 s intersects one of the rays of the Newton fan, one gets its associated lotus, represented on the right side of Figure 5.

pp 7 3 q pp 3 2 q e1 e2
Figure 5. The Newton fan of f px, yq " py 2 ´4x 3 qpy 3 ´x7 q and its associated lotus

In fact, one needs to attach to it new arrowhead vertices corresponding to the branches of C, as shown in Figure 6. In this figure the lotus was redrawn as an abstract simplicial complex, without representing its precise embedding in the plane R 2 . This abstract simplicial structure is sufficient for seeing how it contains the weighted dual graph of the minimal embedded resolution of Zpxypy 2 ´4x 3 qpy 3 ´x7 qq as part of its boundary. The self-intersection number of an exceptional divisor is simply the opposite of the number of triangles containing the vertex representing this divisor (compare Figures 6 and7).

In the sequel we will associate lotuses to any plane curve singularity C (see Definition 5.26). The data needed to construct them will be a finite sequence of Newton polygons generated by a toroidal pseudoresolution algorithm (see Algorithm 4.22). We will embed analogously inside them the weighted dual graphs of associated embedded resolutions of completions of the curve (see Definition 4.15 and Theorem 5.29). We will also explain the notions of fan tree (see Definition 4.33), Enriques diagram (see Definition 4.31) and Eggers-Wall tree (see Definition 6.3) of C or of an associated toroidal pseudo-resolution process and we will show that they embed similarly in the corresponding lotus (see Theorem 5.29).

Zpyq Zpxq

Zpy 3 ´x7 q Zpy 2 ´4x 3 q ´1 ´2 ´2 ´5 ´1

´3

Figure 6. The lotus of f px, yq " py 2 ´4x 3 qpy 3 ´x7 q ´2 ´2 ´1 ´5 ´1 ´3

Zpy 3 ´x7 q Zpy 2 ´4x 3 q Zpyq Zpxq Figure 7. The weighted dual graph of the minimal embedded resolution of Zpxyf px, yqq

Toric and toroidal surfaces and their morphisms

In this section we explain basic definitions and intuitions about toric and toroidal varieties and their modifications, which will be used in the subsequent sections in the study of plane curve singularities. Namely, fans are introduced in Definition 3.3, affine toric varieties in Definition 3.14, their boundaries in Definition 3.18, toric morphisms in Subsection 3.3, in particular the toric description of 2-dimensional blow ups in Example 3.27 and the category of toroidal varieties in Subsection 3.4. Subsection 3.5 contains historical information about the development of toric and toroidal geometry and about its applications to the study of singularities.

3.1. Two-dimensional fans and their regularizations.

In this subsection we explain the basic notions of two-dimensional convex geometry needed to define toric varieties in Subsection 3.2 and toric morphisms in Subsection 3.3: lattices, rational cones and fans.

For more details about toric geometry one may consult the standard textbooks [START_REF] Oda | Convex bodies and algebraic geometry[END_REF], [41], [37] and [26].

A lattice is a free Z-module of finite rank. A pair pa, bq P Z 2 may be seen as an instruction to build two kinds of objects: the Laurent monomial x a y b and the parametrized monomial curve t Ñ pt a , t b q. The fact that monomials and curves are distinct geometrical objects indicates that it would be good to think also in two ways about the pairs pa, bq, that is, as coordinates of vectors relative to bases in two different lattices. Those two lattices are not to be chosen independently of each other. Indeed, given a monomial x a y b and a parametrized monomial curve t Ñ pt c , t d q, one may substitute the parametrization in the monomial, getting a new monomial, this time in the variable t alone: (14) px a y b q ˝pt c , t d q " t ac`bd .

This indicates that those two lattices should be seen as factors of the domain of definition of the unimodular Z-valued bilinear form pa, bq ¨pc, dq :" ac `bd, that is, that they should be dual lattices.

In order to distinguish clearly the roles of these two lattices, one denotes them usually by distinct letters, instead of simply writing for instance Z 2 and pZ 2 q _ . It became traditional after the appearance of Fulton's book [41] to denote by M the lattice whose elements are exponents of monomials in several variables, and by N the dual lattice, whose elements are thought as exponents of parametrized monomial curves in the space of the same variables. It is important to allow for changes of bases of those Z-modules, corresponding to monomial changes of variables of the form x " u α v γ , y " u β v δ , for which the matrix of exponents is unimodular:

(15) ˇˇˇα γ β δ ˇˇˇ" ˘1.
This means that one does not have to fix identifications M " Z 2 , N " Z 2 , but instead to allow those identifications to depend on the context. Note also that the elements of N may be seen as weights for the variables x, y. That is, if pc, dq P N , one gives the weight c to x and the weight d to y, which endows the monomial x a y b with the weight ac `bd appearing in the equality (14). For this reason, N is called sometimes the weight lattice associated to the monomial lattice M . We will call vectors the elements of a lattice. Those non-zero vectors which cannot be written as non-trivial integral multiples of other lattice vectors are called primitive. Any non-zero lattice vector w may be written uniquely in the form l Z pwq w 1 , with l Z pwq P N ˚and w 1 a primitive lattice vector. Definition 3.1. Let N be a lattice and w P N z t0u. The positive integer l Z pwq is the integral length of w. We extend this definition to the whole lattice N by setting l Z p0q :" 0. For w 1 , w 2 P N , the integral length l Z rw 1 , w 2 s P N of the segment rw 1 , w 2 s is equal to l Z pw 2 ´w1 q " l Z pw 1 ´w2 q.

If N is a lattice, denote by N R :" N b Z R the real vector space generated by N . We will say that the elements of N are the integral points of the real vector space N R . By a cone of N we will mean a convex rational polyhedral cone, that is, a subset of N R of the form: R `xw 1 , . . . , w k y :" R `w1 `¨¨¨`R `wk , where w 1 , . . . , w k P N . If the cone does not contain a positive dimensional vector subspace of N R , it is called strictly convex.

If the lattice N is of rank two, then the strictly convex cones are of three sorts, according to their dimensions: ' The 2-dimensional cones are of the form R `xw 1 , w 2 y, where w 1 , w 2 P N are non-proportional. In classical geometric terminology, they are strictly convex angles with apex at the origin of N R . ' The 1-dimensional cones are the closed half-lines emanating from the origin; we will call them rays. ' There is only one 0-dimensional cone: the origin of N .

As a particular case of a terminology used in any dimension, one speaks about the faces of a given cone σ Ď N R : those are the subsets of σ on which the restriction to σ of a linear form l P N _ R " M R reaches its minimum. The faces of a strictly convex 2-dimensional cone R `xw 1 , w 2 y are the cone itself, its edges R `w1 , R `w2 and the origin. The faces of a ray are the ray itself and the origin. Finally, the origin has only one face, which is the origin itself.

Endowing the 2-dimensional lattice N with a basis pe 1 , e 2 q allows to speak of the slope d{c P R Y t8u relative to pe 1 , e 2 q of any vector w " c e 1 `d e 2 P N R zt0u or of the associated ray R `w. In terms of the coordinates pc, dq, the integral length l Z pwq of w is equal to the greatest common divisor gcdpc, dq. Notations 3.2. If the basis pe 1 , e 2 q of N is fixed and clear from the context, we denote by:

σ 0 :" R `xe 1 , e 2 y Ď N R
the cone generated by it. If λ P Q `Y t8u, we denote by ppλq the unique primitive element of the lattice N contained in the cone σ 0 , and which has slope λ.

In the sequel it will be important to work with the following special sets of cones, which are fundamental in toric geometry: Definition 3.3. A fan of the lattice N is a finite set of strictly convex cones of N which is closed under the operation of taking faces of its cones and such that the intersection of any two of its cones is a face of each of them. The support |F| of a fan F is the union of its cones. A fan F refines (or subdivides) another fan F 1 if they have the same support and if each cone of F is contained in some cone of F 1 . A fan subdivides a cone σ if it subdivides the fan formed by its faces. We often denote again by σ the fan formed by the faces of a cone σ, by a slight abuse of notation.

Let us complete the previous definition, valid in arbitrary rank, with terminology and notations specific to rank two: Definition 3.4. Let pe 1 , e 2 q be a basis of the lattice N of rank two and σ 0 be the associated cone R `xe 1 , e 2 y. Any fan F subdividing σ 0 is determined by the finite set of slopes E Ă Q ˚of its rays contained in the interior of σ 0 . In this case we denote the fan by FpEq and we call it the fan of the set E. We extend the definition of FpEq to the case where E contains 0 or 8, by setting in this case FpEq :" FpE z t0, 8uq. If E " tλ 1 , . . . , λ p u, we write also Fpλ 1 , . . . , λ p q instead of FpEq.

Note that FpHq is simply the fan consisting of the cone σ 0 and its faces. Definition 3.5. A cone of a lattice N is called regular if it can be generated by elements which form a subset of a basis of N . A fan all of whose cones are regular is called regular.

It is convenient to set R `xHy :" t0u. This implies that t0u is also a regular cone. Assume that a basis pe 1 , e 2 q of the lattice N is fixed. If f 1 " αe 1 `βe 2 and f 2 " γe 1 `δe 2 are two primitive vectors of N , then the cone R `xf 1 , f 2 y generated by them is regular if and only if the matrix of the pair pf 1 , f 2 q in the basis pe 1 , e 2 q is unimodular, that is, the equality (15) holds.

Example 3.6. If E " t3{5, 2{1, 5{2u, then the rays of the fan FpEq are represented in Figure 8. On each ray of the fan which is distinct from the edges of the cone σ 0 , we indicated by a small red disc the unique primitive element of the lattice N lying on it. That is, on the ray of slope λ P E is indicated the point ppλq. The fan FpEq contains also 4 cones of dimension 2, which are R `xe 1 , p p3{5qy, R `xp p3{5q , p p2{1qy, R `xp p2{1q , p p5{2qy, R `xp p5{2q , e 2 y. Using the unimodularity criterion above, we see that R `xp p2{1q , p p5{2qy is the only 2-dimensional cone of the fan FpEq which is regular. The importance of the regularization operation in our context stems from the fact that it allows to describe combinatorially the minimal resolution of a toric surface (see Proposition 3.28 below). The regularization of a 2-dimensional cone may be described in the following way (see [START_REF] Oda | Convex bodies and algebraic geometry[END_REF]Proposition 1.19]): Proposition 3.9. Let N be a lattice of rank two and let σ be a 2-dimensional strictly convex cone of N . Then the regularization σ reg of the fan of its faces is obtained by subdividing σ using the rays directed by the integral points lying on the boundary of the convex hull of the set of non-zero integral points of σ. If F is a fan of a lattice of rank two, then its regularization is the union of the regularizations of its cones.

pp 5 2 q pp 2 1 q pp 3 5 q e 1 e 2
An alternative recursive description of σ reg was given by Mutsuo Oka in [94, Chap. II.2].

Example 3.10. Let us consider again the fan F p3{5, 2{1, 5{2q of Example 3.6. The rays of its regularization F reg p3{5, 2{1, 5{2q " F p1{2, 3{5, 2{3, 1{1, 2{1, 5{2, 3{1q are drawn in green in Figure 9. The thick orange polygonal line, on the right side of this figure, is the union of compact edges of the boundaries of the convex hulls of the sets of non-zero integral points of its 2-dimensional cones. In this subsection we explain in which way fans determine special kinds of complex algebraic varieties, called toric varieties. Namely, every rational polyhedral cone relative to a lattice determines a monoid algebra (see Definition 3.11), whose maximal spectrum is an affine toric variety (see Definition 3.14). More generally, every fan determines a toric variety by gluing the affine toric varieties associated to its cones (see Definition 3.15).

One associates with a lattice N of rank n the following complex algebraic torus of dimension n (that is, an algebraic group isomorphic to ppC ˚qn , ¨q):

(16) T N :" N b Z C ˚.
Here the factors are considered as abelian groups pN, `q and pC ˚, ¨q, therefore they are endowed with canonical structures of Z-modules, relative to which is taken the previous tensor product. This algebraic torus may be also described in terms of the dual lattice M of N , defined by: M :" Hom pN, Zq.

Namely, one has:

(17) T N " HompM, C ˚q.
Equations ( 16) and ( 17) allow in turn to give the following interpretations of the lattices N and M in terms of morphisms of algebraic groups:

N " HompC ˚, T N q " " the group of one parameter subgroups of T N ;

M " HompT N , C ˚q " " the group of characters of T N .

If w P N is seen as an element of the lattice N , we denote by t w the same element seen as a morphism of abelian groups from C ˚to T N .

Let us explain this notation in the case when N has rank 2. If t is viewed as the parameter on the source C ˚and one identifies T N with pC ˚q2 using the basis pe 1 , e 2 q of N , then the morphism becomes the following map from C ˚to pC ˚q2 : t Ñ pt c , t d q.

Here pc, dq denote as before the coordinates of w in the chosen basis pe 1 , e 2 q of N . One gets therefore a parametrized monomial curve as at the beginning of Subsection 3.1. The advantage of seeing it as an element of HompC ˚, T N q is that one gets a viewpoint independent of the choice of coordinates for T N , that is, of bases for M or for N .

It is customary to say that a morphism t w P Hom pC ˚, T N q is a one parameter subgroup of T N , even when this morphism is not injective. Note that t w is injective if and only if w is a primitive element of N . In general, when w P N zt0u, the map t w is a cyclic covering of its image, of degree l Z pwq (see Definition 3.1). Note also that t 0 is the constant map with image the unit element 1 of the group T N .

We introduced the notation t w in order to be able to distinguish between N seen as an abstract group, and seen as the lattice of one parameter subgroups of T N . In an analogous way, if m P M , one uses the notation χ m : T N Ñ C ˚for its associated character, in order to distinguish between M seen as an abstract group and seen as the lattice of characters of T N . If one denotes by w ¨m P Z the result of applying the canonical duality pairing N ˆM Ñ Z to pw, mq P N ˆM , then the composite morphism χ m ˝tw : C ˚Ñ C ˚is simply given by t Ñ t w¨m . This is the intrinsic description of the composition performed in formula ( 14).

Let us see more precisely how the choice of basis pe 1 , e 2 q of N determines an isomorphism T N » pC ˚q2 . To have such an isomorphism amounts to choosing a special pair px, yq of regular functions on T N , which are the pull-backs of the coordinate functions on pC ˚q2 . This isomorphism should be not only an isomorphism of algebraic surfaces, but also of groups. As the coordinate functions on pC ˚q2 are characters of `pC ˚q2 , ¨˘, that is, elements of HomppC ˚q2 , C ˚q, we deduce that x, y are also characters, this time of pT N , ¨q. It means that they are elements of the lattice M (see the equalities [START_REF] Campillo | Curve singularities. An algebraic and geometric approach[END_REF]). In which way does the basis pe 1 , e 2 q of N determine a pair of elements of M ? Well, this pair is simply the dual basis p 1 , 2 q of pe 1 , e 2 q! Therefore, one has px, yq " pχ 1 , χ 2 q in terms of the dual basis p 1 , 2 q P M 2 of pe 1 , e 2 q P N 2 .

The choice of coordinates px, yq allows to embed the torus T N into the affine plane C 2 with the same coordinates. The coordinate ring of this affine plane is of course Crx, ys. In our context it is important to interpret this ring as the C-algebra of the commutative monoid of monomials with non-negative exponents in the variables x and y. This monoid is isomorphic (using the map m Ñ χ m ) to the monoid R `x 1 , 2 y X M . In turn, the cone R `x 1 , 2 y is in the following sense the dual cone of σ 0 :" R `xe 1 , e 2 y: Definition 3.11. Let σ be a cone of N . Its dual is the cone σ _ of M defined by: σ _ :" tm P M R , w ¨m ě 0 for all w P σu, and its associated monoid algebra is the C-algebra of the abelian monoid pσ _ X M, `q:

Crσ _ X M s :" # ÿ finite c m χ m , m P σ _ X M and c m P C + .
Note that σ is strictly convex if and only if the dimension of σ _ is equal to the rank of the lattice M . The C-algebra Crσ _ X M s is finitely generated, since the monoid pσ _ X M, `q is finitely generated by Gordan's Lemma (see [41, Section 1.1, Proposition 1]).

The set C 2 with coordinates px, yq may now be interpreted in the two following ways:

(19) C 2 x,y
" the maximal spectrum of the ring Crσ _ 0 X M s " " Hompσ _ 0 X M, Cq. The last set of homomorphisms is taken in the category of abelian monoids, where C is considered as a monoid with respect to multiplication. This interpretation is obtained by looking at the evaluation of the monomials χ m , with m P σ _ 0 X M , at the points of C 2 .

The equalities [START_REF] Casas-Alvero | Singularities of plane curves[END_REF] may be turned into a general way to associate a complex affine variety to a cone σ of N , in arbitrary dimension: [START_REF] Cassou-Noguès | Invariants of plane curve singularities and Newton diagrams[END_REF] X σ :" the maximal spectrum of Crσ _ X M s " " Hompσ _ X M, Cq.

The equalities [START_REF] Casas-Alvero | Singularities of plane curves[END_REF] show that X σ0 " C 2 x,y , if x " χ 1 and y " χ 2 . Therefore, the affine variety X σ0 is smooth. The following proposition characterizes the cones for which the associated variety is smooth (see [41, Section 2.1, Proposition 1]): Proposition 3.12. Let σ be a strictly convex cone of the lattice N . Then the affine variety X σ is smooth if and only if σ is regular in the sense of Definition 3.5.

In the sequel, by a stratification of an algebraic variety we mean a finite partition of it into locally closed connected smooth subvarieties, called the strata of the stratification, such that the closure of each stratum is a union of strata.

Consider the following stratification of

X σ0 " C 2 x,y : (21) C 2 x,y " t0u \ pC x ˆt0uq \ `t0u ˆCẙ ˘\ pC ˚q2
x,y . One may interpret in the following way its strata in terms of vanishing of monomials whose exponents belong to σ _ 0 X M " Nx 1 , 2 y: ' 0 is the only point of C 2

x,y at which vanish exactly the monomials with exponents in pσ _ 0 z t0uqXM . ' C x ˆt0u is the set of points of C 2

x,y at which vanish exactly the monomials with exponents in

pσ _ 0 z R ` 1 q X M . ' t0u ˆCẙ is the set of points of C 2
x,y at which vanish exactly the monomials with exponents in

pσ _ 0 z R ` 2 q X M . ' pC ˚q2
x,y " T N is the set of points of C 2

x,y at which vanish no monomials, that is, at which vanish exactly the monomials with exponents in pσ _ 0 z σ _ 0 q X M . Note that the sets of exponents of monomials appearing in the previous list are precisely those of the form pσ _ 0 z τ q X M , where τ varies among the faces of the cone σ _ 0 . It is customary in toric geometry to express them in a dual way, using the following bijection between the faces of σ and of σ _ , valid in all dimensions for (not necessarily rational) convex polyhedral cones σ (see [26, Proposition 1.2.10]): Proposition 3.13. Let σ be a cone of N R . Then the map ρ Ñ ρ K X σ _ is an order-reversing bijection from the set of faces of σ to the set of faces of σ _ (see Figure 10).

Here ρ K :" tm P M R , w ¨m " 0 for all w P ρu denotes the orthogonal of the cone ρ of N . It is a real vector subspace of M R , which may be characterized as the maximal vector subspace of the convex cone

ρ _ . O ρ σ N R ÐÑ O σ _ ρ K X σ _ ρ _ M R Figure 10
. The bijection between the faces of σ and σ _

The stratification (21) of C 2 is a particular case of a stratification of any affine variety of the form X σ . In order to define it, one associates with each point p of X σ the subset of σ _ XM formed by the exponents of the monomials vanishing at p. This defines a function from X σ to the power set of σ _ X M , whose levels are precisely the strata of the stratification of X σ . The set of strata is in bijective correspondence with the set of faces of σ, the stratum O ρ corresponding to the face ρ of σ being: [START_REF] Cassou-Noguès | Newton trees for ideals in two variables and applications[END_REF] O ρ :" p P Hompσ _ X M, Cq, p ´1p0q " pσ _ z ρ K q X M ( .

In particular, O t0u " T N is the only stratum whose dimension is the same as the dimension of X σ . This shows that the torus T N embeds naturally as an affine open set in the affine surface X σ . For this reason, the following vocabulary was introduced:

Definition 3.14. If N is a lattice and σ is a strictly convex cone of N , then the variety X σ defined by the equalities ( 20) is called an affine toric variety.

Note that for X σ0 " C 2 x,y , the strata are:

' O σ0 " t0u; ' O R`e2 " C x ˆt0u; ' O R`e1 " t0u ˆCẙ ; ' O t0u " pC ˚q2
x,y " T N . One may feel difficult to remember the second and third equalities, a common error at the time of doing computations being to permute them. A way to remember them is the following: the orbit corresponding to an edge of a 2-dimensional regular cone is the complement of the origin in the axis of coordinates of C 2 defined by the vanishing of the dual variable. In our case, the dual variable of the edge R `e1 is x " χ 1 , whose 0-locus is the axis of the variable y, and conversely.

The notation O ρ is motivated by the fact that this subset of X σ is an orbit of a natural action of the algebraic torus T N on X σ . For X σ0 " C 2

x,y , case in which one may also identify T N with pC ˚q2 u,v , this action is given by pu, vq ¨px, yq :" pux, vyq. In general, the action of T N on X σ may be described in intrinsic terms by:

pM τ Ñ C ˚q ¨pσ _ X M p Ñ Cq :" pσ _ X M τ ¨p ÝÑ Cq.
In the previous equation we used again the interpretations of the points of T N and X σ as morphisms of monoids (see equations [START_REF] Campillo | Algebroid Curves in Positive Characteristic[END_REF] and ( 20)).

Assume now that F is a fan of N , in the sense of Definition 3.3. Each affine toric variety X σ , where σ P F, contains the torus T N as an affine open set. If σ and τ are two cones of F, then one has a natural identification of their respective tori, and also of their larger Zariski open subsets X σXτ Ă X σ and X σXτ Ă X τ . If one glues the various affine toric varieties pX σ q σPF using the previous identifications, one gets an abstract separated complex algebraic variety X F which still contains the torus T N as an affine open subset (see [START_REF] Oda | Convex bodies and algebraic geometry[END_REF]Theorem 1.4

and 1.5]).

Definition 3.15. The toric variety associated with a fan F of a lattice N is the variety X F constructed above.

Remark 3.16. All toric varieties constructed from fans are normal in the sense of Definition 2.11 (see [26,Theorem 1.3.5]). One has a more general notion of toric variety, which includes some non-normal varieties as well (see the paper [START_REF] González Pérez | Toric geometry and the Semple-Nash modification[END_REF] of Teissier and the second author). Those varieties can be described as before by gluing maximal spectra of algebras of not necessarily saturated finite type submonoids of lattices, the normal ones being precisely the toric varieties associated with a fan of Definition 3.15.

As a consequence of Proposition 3.12, one has a smoothness criterion for toric varieties: Proposition 3.17. Let F be a fan of the lattice N . Then the toric variety X F is smooth if and only if F is regular in the sense of Definition 3.5.

Let us come back to a fan F of a weight lattice N . When ρ varies among the cones of F, the actions of the torus T N on the affine toric varieties X ρ glue into an action on X F , whose orbits are still denoted by O ρ . The conservation of the notation ( 22) is motivated by the fact that in the gluing of X σ and X τ , the orbits denoted O ρ on both sides get identified, for every face ρ of σ X τ . If ρ is a cone of the fan F, we denote by O ρ the closure in X F of the orbit O ρ . The orbit closure O ρ has also a natural structure of normal toric variety (see [41, Chapter 3]).

The torus T N is identified canonically with the orbit O 0 corresponding to the origin of N R , seen as a cone of dimension 0. Its complement is the union of all the orbits of codimension at least 1. Let us introduce a special name and notation for this complement: Definition 3.18. Let X F be a toric variety defined by a fan F. Its boundary BX F is the complement of the algebraic torus T N inside X F .

The boundary BX F is a reduced Weil divisor inside X F , whose irreducible components are the orbit closures O ρ corresponding to the cones ρ of F which have dimension 1, that is, to the rays of the fan F.

Toric morphisms and toric modifications.

In this subsection we define the notion of toric morphism between toric varieties (see Definition 3.19) and we explain in which way refining a fan defines a special kind of toric morphism, called a toric modification (see Proposition 3.21). In Examples 3.26 and 3.27 we explain how to do concrete computations of toric modifications in dimension two, the second one giving a toric presentation of the blow ups of the origin. Finally, Proposition 3.28 explains the combinatorics of the minimal resolution of a normal affine toric surface.

Assume that N 1 and N 2 are two weight lattices, endowed with cones σ 1 and σ 2 . Let φ : N 1 Ñ N 2 be a morphism of lattices which sends the cone σ 1 into the cone σ 2 . Using the second interpretation in the equalities [START_REF] Cassou-Noguès | Invariants of plane curve singularities and Newton diagrams[END_REF] of the points of affine toric varieties, we see that φ induces an algebraic morphism between the associated toric varieties:

(23) ψ σ1 σ2,φ : X σ1 Ñ X σ2 p 1 Ñ p 1 ˝φ_
.

One sees immediately from the definitions that the adjoint

φ _ : M 2 Ñ M 1 of φ maps σ _ 2 into σ _ 1 , which shows that the composition p 1 ˝φ_ belongs indeed to X σ2 " Hompσ _ 2 X M 2 , Cq whenever p 1 P X σ1 " Hompσ _ 1 X M 1 , Cq.
The morphism ψ σ1 σ2,φ may be also described using the first interpretation in the equalities [START_REF] Cassou-Noguès | Invariants of plane curve singularities and Newton diagrams[END_REF], as the morphism of affine schemes induced by the morphism of C-algebras Crσ _ 2 X M 2 s Ñ Crσ _ 1 X M 1 s which sends each monomial χ m2 P σ _ 2 X M 2 to the monomial χ φ _ pm2q P σ _ 1 X M 1 . Assume now that N 1 and N 2 are endowed with fans F 1 and F 2 respectively, such that φ sends each cone of F 1 into some cone of F 2 . We say that φ is compatible with the two fans. It may be checked formally that the previous morphisms ψ σ1 σ2,φ : X σ1 Ñ X σ2 , for all the pairs pσ 1 , σ 2 q P F 1 ˆF2 which verify that φpσ 1 q Ď σ 2 , glue into an algebraic morphism: ψ F1 F2,φ : X F1 Ñ X F2 . This morphism is moreover equivariant with respect to the actions of T N1 and T N2 on X F1 and X F2 respectively. For this reason, one uses the following terminology: Definition 3.19. If the morphism of lattices φ : N 1 Ñ N 2 sends every cone of F 1 into some cone of F 2 , then the morphism of algebraic varieties ψ F1 F2,φ : X F1 Ñ X F2 described above is called the toric morphism associated with φ and the fans F 1 , F 2 .

The toric morphism ψ F1 F2,φ sends the torus

T N1 " X F1 z B X F1 into T N2 " X F2 z B X F2
. This fact implies the following property of toric morphisms relative to the boundaries of their sources and targets, in the sense of Definition 3.18: Proposition 3.20. Let ψ : X F1 Ñ X F2 be the toric morphism associated with φ and the fans F 1 and F 2 . Then ψ ´1pB X F2 q Ď B X F1 .

Toric morphisms have the following properties (see [START_REF] Oda | Convex bodies and algebraic geometry[END_REF]Theorems 1.13,1.15]): Proposition 3.21. Let N 1 , N 2 be two lattices and F 1 , F 2 be fans of N 1 and N 2 respectively. Let φ : N 1 Ñ N 2 be a lattice morphism compatible with the two fans. Then:

(1) The morphism ψ F1 F2,φ is birational if and only if φ is an isomorphism of lattices.

(2) The morphism ψ F1 F2,φ is proper if and only if the R-linear map φ R : pN 1 q R Ñ pN 2 q R sends the support of F 1 onto the support of F 2 . In particular, ψ F1 F2,φ is a modification in the sense of Definition 2.31 if and only if φ is an isomorphism and, after identifying N 1 and N 2 using it, the fan F 1 refines the fan F 2 in the sense of Definition 3.3.

We will consider most of the time the particular case in which N 1 " N 2 " N is a lattice of rank 2 and φ is the identity. Then, if σ Ă σ 0 is a subcone of σ 0 , we denote by ψ σ σ0 the birational toric morphism induced by the identity:

(24) ψ σ σ0 : X σ Ñ X σ0 " C 2 x,y .
When σ varies among all the cones of a fan F which subdivides the cone σ 0 , the morphisms ψ σ σ0 glue into a single equivariant birational morphism:

(25) ψ F σ0 : X F Ñ X σ0 " C 2 x,y .
By Proposition 3.21, this morphism is also proper, because F and σ 0 have the same support. Therefore,

ψ F σ0 is a modification of C 2 x,y . The strict transform of L :" Zpxq (resp. of L 1 :" Zpyq) by the modification ψ F σ0 is the orbit closure O R`e1 (resp. O R`e2 ) in X F . The preimage of 0 P C 2
x,y , called the exceptional divisor of ψ F σ0 , and the preimage of the sum L `L1 of the coordinate axes, which is the total transform of L `L1 are:

(26) pψ F σ0 q ´1p0q " O ρ1 `¨¨¨`O ρ k , and pψ F σ0 q ´1pL `L1 q " O R`e1 `Oρ1 `¨¨¨`O ρ k `OR`e2
, where ρ 1 , . . . , ρ k denote the rays of F contained in the interior of σ 0 , labeled as in Figure 11. Note that L `L1 " BX σ0 and pψ F σ0 q ´1pL `L1 q " BX F , which is a particular case of Proposition 3.21. Recall now the following classical notion of (unweighted) dual graph, which extends that of Definition 2.36 and whose historical evolution was sketched by the third author in [START_REF] Popescu-Pampu | From singularities to graphs[END_REF]: Definition 3.22. A simple normal crossings curve is a reduced abstract complex curve whose irreducible components are smooth and whose singularities are normal crossings, that is, analytically isomorphic to the germ at the origin of the union of coordinate axes of C 2 . The dual graph of a simple normal crossings curve D is the abstract graph whose set of vertices is associated bijectively with the set of irreducible components of D, the edges between two vertices corresponding bijectively with the intersection points of the associated components of D. Each vertex or edge is labeled by the corresponding irreducible component or point of D. Remark 3.23. Let σ " R `xf 1 , f 2 y Ă N R be a strictly convex cone of dimension two, not necessarily regular. One may check that the boundary BX σ " O R`f1 `OR`f2 of the affine toric surface X σ is an abstract simple normal crossings curve, according to Definition 3.22.

The dual graph of the total transform pψ F σ0 q ´1pL `L1 q may be embedded in the cone σ 0 Ď N R : Proposition 3.24. Let F be a fan which subdivides the regular cone σ 0 . Then the dual graph of the divisor pψ F σ0 q ´1pL `L1 q is a segment with extremities L and L 1 and with k intermediate points labeled in order by O ρ1 , . . . , O ρ k from L to L 1 . That is, it is isomorphic to the segment re 1 , e 2 s Ă N R , marked with its intersection points with the rays of F, the point re 1 , e 2 s X ρ i being labeled by the orbit closure O ρi .

Therefore, the rays of the fan F correspond bijectively to the irreducible components of the total transform pψ F σ0 q ´1pL `L1 q of L `L1 . The 2-dimensional cones of F correspond to the fixed points of the torus action, which are the only possible singular points of the surface X F . The orbit closures O ρ and O ρ 1 intersect at a point q P X F if and only if the cone ρ `ρ1 is a 2-dimensional cone of F and then q is the unique orbit O ρ`ρ 1 of dimension 0 of the affine toric surface X ρ`ρ 1 Ă X F . The point q is singular on the surface X F if and only if the cone ρ `ρ1 is not regular.

Example 3.25. For the fan F p3{5, 2{1, 5{2q of Figure 8 discussed in Example 3.6, the total transform pψ F σ0 q ´1pL `L1 q and its dual graph are represented in Figure 11. The 4 singular points of the total transform are also singular on the surface X F , with the exception of O ρ2 X O ρ3 . Indeed, the cone ρ 2 `ρ3 is the only regular 2-dimensional cone of the fan F, as may be seen in Figure 9.

e 2 e 1 ρ 3 ρ 2 ρ 1 ÝÑ L 1 O ρ3 O ρ2 O ρ1 L L 1 " O R`e2 O ρ3 O ρ1 O ρ2 L " O R`e1
Figure 11. The dual graph of the total transform pψ F σ0 q ´1pL `L1 q Example 3.26. Let us explain how to describe in coordinates the morphism ψ σ σ0 of ( 24), when σ is a regular subcone of σ 0 . Denote by f 1 , f 2 the primitive generators of the edges of σ, ordered in such a way that the bases pe 1 , e 2 q and pf 1 , f 2 q define the same orientation of N R (see Figure 12). Decompose pf 1 , f 2 q in the basis pe 1 , e 2 q, writing f 1 " αe 1 `βe 2 and f 2 " γe 1 `δe 2 . This means that the unimodular matrix of change of bases from pf 1 , f 2 q to pe 1 , e 2 q is:

(27) ˆα γ β δ ˙.
Denote by pϕ 1 , ϕ 2 q P M 2 the dual basis of pf 1 , f 2 q and by ( 28)

" u :" χ ϕ1 " x δ y ´γ v :" χ ϕ2 " x ´β y α ,
the associated coordinates. Then, in terms of the identifications X σ " C 2 u,v and X σ0 " C 2 x,y , the morphism ψ σ σ0 is given by the following monomial change of coordinates (compare the disposal of exponents with the matrix ( 27)): ( 29)

" x " u α v γ y " u β v δ .
Note that the system [START_REF] Cutkosky | Resolution of singularities[END_REF] implies that the expression of v " χ ϕ2 as a monomial in x and y is determined only by f 1 , being independent of the choice of f 2 . This may be explained geometrically. Indeed, as f 1 ¨ϕ2 " 0, we see that ϕ 2 belongs to the line f K 1 orthogonal to f 1 . As ϕ 2 may be completed into a basis of M , it is primitive, which determines it up to sign. This sign ambiguity is lifted by the constraint that the basis pf 1 , f 2 q determines the open half-plane bounded by the line Rf 1 on which ϕ 2 has to be positive. Note also that v is a coordinate on the orbit O R`f1 determined by the edge R `f1 of σ. This coordinate determines an isomorphism O R`f1 » C v of complex tori, and depends only on R `f1 , since the orbit O R`f1 can be realized as a subspace of the surface X R`f1 by formula [START_REF] Cassou-Noguès | Newton trees for ideals in two variables and applications[END_REF] above.

Example 3.27. In this example we use the explanations given in Example 3.26. Let F be the fan obtained by subdividing σ 0 " R `xe 1 , e 2 y using the half-line ρ generated by e 1 `e2 . It has two cones of dimension 2, denoted σ 1 :" R `xe 1 , e 1 `e2 y and σ 2 :" R `xe 1 `e2 , e 2 y (see Figure 13). Then the toric morphism ψ F σ0 may be described by its two restrictions ψ σ1 σ0 and ψ σ2 σ0 . The matrices of change of bases from pe 1 , e 1 `e2 q and pe 1 `e2 , e 2 q to pe 1 , e 2 q respectively are ˆ1 1 0 1 ˙and ˆ1 0 1 1 ˙. Denoting by pu 1 , u 2 q and pv 1 , v 2 q the coordinates corresponding to the dual bases of pe 1 , e 1 `e2 q and pe 1 `e2 , e 2 q, the e 1 general formulas ( 27) and [START_REF] Deligne | Intersections sur les surfaces régulières[END_REF] show that the morphisms ψ σ1 σ0 and ψ σ2 σ0 are given by the following changes of variables: (30)

e 2 f 1 " α e 1 `β e 2 f 2 " γe 1 `δe 2 N R " σ 0 " R `xe 1 , e 2 y σ " R `xf 1 , f 2 y 1 2 ϕ 1 " δ 1 ´γ 2 ϕ 2 " ´β 1 `α 2 M R v " x ´β y α O R`f1 u " x δ y ´γ O R`f2 y O R`e1 x O R`e2 ψ σ σ0

"

x " u 1 u 2 y " u 2 , and

" x " v 1 y " v 1 v 2 .
We get the same expressions as in equations (4). This shows that ψ F σ0 is a toric representative of the blow up morphism of C 2

x,y at the origin! Let σ be a non-regular cone of the weight lattice N of rank two. By Proposition 3.12, the affine toric surface X σ is not smooth. In fact, it has only one singular point, the orbit O σ of dimension 0. Being of dimension 2, X σ admits a minimal resolution, that is, a resolution through which factors any other resolution (recall that this notion was explained in Definition 2.34). It turns out that this minimal resolution may be given by a toric morphism, defined by the regularization of σ in the sense of Definition 3.8 (see [START_REF] Oda | Convex bodies and algebraic geometry[END_REF]Proposition 1.19]): Proposition 3.28. Let σ be a non-regular cone of the weight lattice N of rank two. Denote by σ reg the regularization of the fan formed by the faces of σ. Then the toric modification ψ σ reg σ : X σ reg Ñ X σ is the minimal resolution of X σ . As a consequence, for any fan F of N , the toric modification ψ F reg F : X F reg Ñ X F is the minimal resolution of X F .

Toroidal varieties and modifications in the toroidal category.

In this subsection we explain analytic generalizations of toric varieties and toric morphisms: the notions of toroidal variety and morphism of toroidal varieties (see Definition 3.29). Then we introduce the notion of cross on a smooth germ of surface (see Definition 3.31), and we explain how to attach to a cross a canonical oriented regular cone in a two-dimensional lattice (see Definition 3.32) and how each subdivision of this cone determines a canonical modification in the toroidal category (see Definition 3.33). The toroidal pseudo-resolutions of plane curve singularities introduced in Subsection 4.2 below will be constructed as compositions of such toroidal modifications.

Toric surfaces and morphisms are not sufficient for the study of plane curve singularities for the following reasons. One starts often from a germ of curve on a smooth complex surface which does not have a preferred coordinate system. It may be impossible to choose a coordinate system such that the germ of curve gets resolved by only one toric modification relative to the chosen coordinates (if the curve singularity is reduced and such a resolution is possible, then one says that the singularity is Newton nondegenerate, see Definition 4.21 below). Instead, what may be always achieved is a morphism of toroidal surfaces, in the following sense: Definition 3.29. A toroidal variety is a pair pΣ, BΣq consisting of a normal complex variety Σ and a reduced divisor BΣ on Σ such that the germ of pΣ, BΣq at any point p P Σ is analytically isomorphic to the germ of a pair pX σ , BX σ q at a point of X σ , where BX σ denotes the boundary of the affine toric variety X σ in the sense of Definition 3.18. Such an isomorphism is called a toric chart centered at p of the toroidal variety pΣ, BΣq. The divisor BΣ is the boundary of the toroidal variety.

A morphism ψ : pΣ 2 , BΣ 2 q Ñ pΣ 1 , BΣ 1 q between toroidal varieties is a complex analytic morphism ψ : Σ 2 Ñ Σ 1 such that ψ ´1pBΣ 1 q Ď BΣ 2 . The morphism ψ is a modification if the underlying morphism of complex varieties is a modification in the sense of Definition 2.31.

Toroidal varieties with their morphisms define a category, called the toroidal category. The previous definition implies that if pΣ, BΣq is toroidal, then the complement Σ z BΣ is smooth. Indeed, the point p is allowed to be taken outside the boundary BΣ, and the definition shows then that the germ of Σ at p is analytically isomorphic to the germ of a toric variety at a point of the associated torus, which is smooth.

If Σ is of dimension two and if p is a smooth point of BΣ, then p is a smooth point of Σ, since the germ of Σ at p is analytically isomorphic to the germ of a normal toric surface at a point belonging to a 1-dimensional orbit, which is necessarily smooth. Proposition 3.20 implies that a toric morphism ψ F1 F2,φ : X F1 Ñ X F2 becomes an element of the toroidal category if one looks at it as a complex analytic morphism from the pair pX F1 , BX F1 q to the pair pX F2 , BX F2 q, the boundaries being taken in the sense of Definition 3.18.

Remark 3.30. There exists also a more restrictive notion of toroidal morphism ψ : pΣ 2 , BΣ 2 q Ñ pΣ 1 , BΣ 1 q between toroidal varieties. By definition, such a morphism becomes monomial in the neighborhood of any point p of Σ 2 , after some choice of toric charts at the source and the target, centered at p and ψppq respectively. Toroidal morphisms belong to the toroidal category, but the converse is not true. For instance, take two copies C 2 u,v and C 2 x,y of the complex affine plane and the affine morphism

ψ : C 2 u,v Ñ C 2
x,y defined by x " u, y " up1 `vq. Consider the plane C 2 u,v as a toroidal surface with boundary equal to the union of its coordinate axes, and C 2

x,y as a toroidal surface with boundary equal to the y-axis. As ψ ´1pBC 2

x,y q Ď BC 2 u,v , ψ is a morphism of toroidal varieties. But it is not a toroidal morphism. Otherwise, it would become the morphism pu, vq Ñ pu, uq after analytic changes of coordinates in the neighborhoods of the origins of the two planes, which is impossible, because ψ is birational, therefore dominant.

Let us come back to the case of a smooth germ of surface pS, oq. Definition 3.31. A cross on the smooth germ of surface pS, oq is a pair pL, L 1 q of transversal smooth branches on pS, oq. A local coordinate system px, yq on pS, oq is said to define the cross pL, L 1 q if L " Zpxq and L 1 " Zpyq.

We chose the name cross by analogy with the denomination normal crossings divisor (see Definition 2.32). Note the subtle difference between the two notions: the pair pL, L 1 q is a cross if and only if L `L1 is a normal crossings divisor, but the knowledge of the divisor does not allow to remember the order of its branches. Definition 3.32. Let pL, L 1 q be a cross on pS, oq. We associate with it the two-dimensional lattice M L,L 1 of integral divisors supported by L Y L 1 , called the monomial lattice of the cross pL, L 1 q. The weight lattice of the cross pL, L 1 q is the dual lattice N L,L 1 of M L,L 1 . Denote by p L , L 1 q the basis L :" L, L 1 :" L 1 of M L,L 1 , by pe L , e L 1 q the dual basis of N L,L 1 , and by σ L,L 1 0 the cone R `xe L , e L 1 y.

When the cross pL, L 1 q is clear from the context, we often write simply p 1 , 2 q , pe 1 , e 2 q and σ 0 instead of p L , L 1 q, pe L , e L 1 q and σ L,L 1 0 respectively. Each time we choose local coordinates px, yq defining the cross pL, L 1 q, we identify M L,L 1 with the lattice of exponents of monomials in those coordinates. That is, a 1 `b 2 corresponds to x a y b . Such a choice of coordinates also identifies holomorphically a neighborhood of o in S with a neighborhood of the origin in C 2 and the cross pL, L 1 q with the coordinate cross in C 2 at the origin. Therefore, any subdivision F of σ 0 defines an analytic modification ψ F L,L 1 : S F Ñ S of S. As these modifications are isomorphisms over S z tou, it is easy to see that they are independent of the chosen coordinate system px, yq defining pL, L 1 q, up to canonical analytical isomorphisms above S. Moreover, if we define BS :" L `L1 and BS F :" pψ F L,L 1 q ´1pL `L1 q, the morphism ψ F L,L 1 becomes a morphism from the toroidal surface pS F , BS F q to the toroidal surface pS, BSq. Definition 3.33. If F is a fan subdividing the cone σ 0 Ă N L,L 1 , then the morphism of the toroidal category ψ F L,L 1 : pS F , BS F q Ñ pS, L `L1 q associated with F is the modification of S associated with F relative to the cross pL, L 1 q.

When the fan F is regular, the morphism ψ F L,L 1 between the underlying complex surfaces (forgetting the toroidal structures) is a composition of blow ups of points (see Definition 2.29). We will explain the structure of this decomposition of ψ F L,L 1 in Section 5 (see Propositions 5.10, 5.11).

Historical comments.

Toric varieties were called torus embeddings at the beginning of the development of toric geometry in the 1970s, following the terminology of Kempf, Knudsen, Mumford and Saint-Donat's 1973 book [START_REF] Kempf | Toroidal embeddings. I[END_REF], as these are varieties into which an algebraic torus embeds as an affine Zariski open subset. The introduction of the book [START_REF] Kempf | Toroidal embeddings. I[END_REF] The first applications of toric geometry to the study of singularities were done by Kouchnirenko, Varchenko and Khovanskii in their 1976-77 papers [START_REF] Kouchnirenko | Polyèdres de Newton et nombres de Milnor[END_REF], [START_REF] Varchenko | Zeta-function of monodromy and Newton's diagram[END_REF] and [START_REF] Khovanskii | Newton polyhedra, and toroidal varieties[END_REF] respectively. But one may see in retrospect toric techniques in Puiseux's 1850 paper [START_REF] Puiseux | Recherches sur les fonctions algébriques[END_REF]Sections 20,[START_REF] Cassou-Noguès | The Newton tree: geometric interpretation and applications to the motivic zeta function and the log canonical threshold[END_REF], in Jung's 1908 paper [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 Dumas' 1911-12 papers [31], [START_REF] Dumas | Sur les singularités des surfaces[END_REF], in Hodge's 1930 paper [START_REF] Hodge | The isolated singularities of an algebraic surface[END_REF], in Hirzebruch's 1953 paper [START_REF] Hirzebruch | Über vierdimensionale Riemannsche Flächen Mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen[END_REF] and in Teissier's 1973 paper [START_REF] Teissier | Le problème des modules pour les branches planes[END_REF]. Indeed, in all those papers, monomial changes of variables more general than those describing blow ups are used in an essential way. For instance, in his paper [START_REF] Hirzebruch | Über vierdimensionale Riemannsche Flächen Mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen[END_REF], Hirzebruch described the minimal resolution of an affine toric surface by gluing the toric charts of the resolved surface by explicit monomial birational maps. Toric surfaces appeared in Hirzebruch's paper as normalizations of the affine surfaces in C 3 defined by equations of the form z m " x p y q , with pm, p, qq P pN ˚q3 globally coprime. Interesting details about Hirzebruch's work [START_REF] Hirzebruch | Über vierdimensionale Riemannsche Flächen Mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen[END_REF] are contained in Brieskorn's paper [14].

The notion of toroidal variety of arbitrary dimension was introduced in a slightly different form in the same book [START_REF] Kempf | Toroidal embeddings. I[END_REF] of Kempf, Knudson, Mumford and Saint-Donat. The emphasis was put there on a given complex manifold V , and one looked for partial compactifications of it which were locally analytically isomorphic to embeddings of an algebraic torus into a toric variety. Such partial compactifications V were called toroidal embeddings of V . Therefore, a toroidal embedding was a pair pV , V q such that pV , V z V q is a toroidal variety in our sense. For more remarks about the toroidal category see [4, Section 1.5].

Toroidal pseudo-resolutions of plane curve singularities

In Subsection 4.1 we introduce the notions of Newton polygon N L,L 1 pCq, tropical function trop C L,L 1 , Newton fan F C L,L 1 and Newton modification ψ C L,L 1 (see Definition 4.14) determined by a curve singularity C on the smooth germ of surface pS, oq, relative to a cross pL, L 1 q. The strict transform of C by its Newton modification is a finite set of germs. If one completes for each one of them the corresponding germ of exceptional divisor into a cross, one gets again a Newton polygon, a fan and a modification. This produces an algorithm of toroidal pseudo-resolution of C (see Algorithm 4.22). It leads only to a pseudo-resolution morphism, because its source is a possibly singular surface (with toric singularities). In Subsection 4.3 we explain how to modify Algorithm 4.22 in order to get an algorithm of embedded resolution of C. In Subsection 4.4 we encode the combinatorics of this algorithm into a fan tree (see Definition 4.33), which is a rooted tree endowed with a slope function, constructed by gluing trunks associated with the Newton fans generated by the process. The final Subsection 4.5 contains historical information about Newton's and Puiseux's work on plane curve singularities, the resolution of such singularities by iteration of morphisms which are toric in suitable coordinates, and the relations with tropical geometry.

Newton polygons, their tropicalizations, fans and modifications.

This subsection begins with the definitions of the Newton polygon N pf q (see Definition 4.2), the tropicalization (see Definition 4.4) and the Newton fan Fpf q (see Definition 4.9) associated with a nonzero germ f P Crrx, yss. It turns out that they only depend on the germs L, L 1 , C defined by x, y and f respectively (see Proposition 4.13). Therefore, given a cross pL, L 1 q and a plane curve singularity C on the smooth germ pS, oq, one has associated Newton polygon, tropicalization and fan. This fan allows to introduce the Newton modification of the toroidal germ pS, L `L1 q determined by C (see Definition 4.14).

Assume that a cross pL, L 1 q is fixed on pS, oq (see Definition 3.31) and that px, yq is a local coordinate system defining it. This system allows to see any f P ÔS,o as a series in the variables px, yq, that is, in toric terms, as a possibly infinite sum of terms of the form c m pf q χ m , for c m pf q P C and m P σ _ 0 X M , where M :" M L,L 1 and σ 0 :" σ L,L 1 0 (see Definition 3.32). Denote also N :"

N L,L 1 . One has canonical identifications M » Z 2 , N » Z 2 , σ 0 » pR `q2 , and σ _ 0 » pR `q2 .
Definition 4.1. Let f P Crrx, yss be a nonzero series. The support Spf q Ď σ _ 0 X M » N 2 of f is the set of exponents of monomials with non-zero coefficients in f . That is, if [START_REF] Dumas | Sur la résolution des singularités de surfaces[END_REF] f "

ÿ mPσ _ 0 XM c m pf qχ m , then Spf q :" tm P σ _ 0 X M, c m pf q ‰ 0u.
If Y is a subset of a real affine space, then ConvpY q denotes its convex hull.

Definition 4.2. Let f P Crrx, yss. Its Newton polygon N pf q is the following convex subset of σ _ 0 » pR `q2 :

N pf q :" ConvpSpf q `pσ _ 0 X M qq. Its faces are its vertices, its edges and the whole polygon itself. If K is a compact edge of the boundary BN pf q of N pf q, then the restriction f K of f to K is the sum of the terms of f whose exponents belong to K. Suppose now that the variables x and y are weighted by non-negative real numbers. Denote by c P R the weight of x and by d P R `the weight of y. Therefore the pair w :" pc, dq may be seen as an element of the weight vector space N R " pN L,L 1 q R . More precisely, one has w P pR `q2 » σ 0 . Assuming that the non-zero complex constants have weight 0, we see that the weight wpc m pf qχ m q of a non-zero term of f is simply w ¨m P R `. Define then the w-weight of the series f P Crrx, yss as the minimal weight of its terms. One gets the function:

(32) ν w : Crrx, yss Ñ R `Y t8u f Ñ mintw ¨m, m P Spf qu .
It is an exercise to show that ν w is a valuation on the C-algebra Crrx, yss, in the sense of Definition 2.19. Instead of fixing w and letting f vary, let us fix now a non-zero series f P Crrx, yss. Considering the w-weight of f for every w P σ 0 leads to the following function:

Definition 4.4. The tropicalization trop f of f P Crrx, yss z t0u is the function:

(33) trop f : σ 0 Ñ R ẁ Ñ mintw ¨m, m P Spf qu .
Remark 4.5. Let us explain the name of tropicalization used in the previous definition (see also Subsection 4.5). Consider the set R :" R Y t`8u, endowed with the operations ' :" min and d :" `. Under both operations, R is a commutative monoid, the product d is distributive with respect to addition and the addition ' is idempotent, that is, a ' a " a, for all a P R. One says then that pR, ', dq is a tropical semiring. Consider now the expression defining trop f , and compare it with the expansion (31) of f as a power series. One sees that one gets formally trop f from (31) by replacing each constant or variable x, y by its weight, and by replacing the usual operations of sum and product by their tropical analogs. For further references see the textbook [START_REF] Mac Lagan | Introduction to tropical geometry[END_REF] on tropical geometry. Foundations for the tropical study of singularities were written by Stepanov and the third author in the paper [START_REF] Popescu-Pampu | Algebraic and Combinatorial aspects of Tropical Geometry[END_REF].

Remark 4.6. If A is a subset of a real vector space V , then its support function is the function defined on the dual vector space V _ and taking values in R Y t´8u, which associates to every element of V _ seen as a linear form on V , the infimum of its restriction to A. The tropicalization trop f is the restriction of the support function of the subset Spf q of the real vector space M R to the subset of M _ R » N R on which it does not take the value ´8. The notion of support function is an essential tool in the study of convex polyhedra (see for instance Ewald's book [37]).

For every ray ρ " R `w included in the cone σ 0 , consider the following closed half-plane of M R : [START_REF] Eisenbud | Three-dimensional link theory and invariants of plane curve singularities[END_REF] H f,ρ :" tm P M R , w ¨m ě trop f pwqu.

This definition is independent of the choice a generator w of the ray ρ.

The basic reason of the importance of the Newton polygon N pf q of f in our context is the following strengthening of Proposition 2.39: Proposition 4.7. Let the ray ρ Ă σ 0 be fixed. Then the closed half-plane H f,ρ of M R is a supporting half-plane of N pf q, in the sense that it contains N pf q and its boundary tm P M R , w ¨m " trop f pwqu has a non-empty intersection with the boundary BN pf q of N pf q.

Proof. Let w be a generating vector of the ray ρ. The inclusion N pf q Ď H f,ρ is equivalent to the property w ¨n ě trop f pwq, for all n P N pf q. These inequalities result from Definition 4.4 of the tropicalization function trop f pwq and from the following basic equality, implied by the hypothesis that w P σ 0 (see Proposition 2.39): mintw ¨m, m P Spf qu " mintw ¨m, m P N pf qu.

The boundary of the half-plane H f,ρ intersects N pf q at its points at which the restriction of the linear form w : M R Ñ R to N pf q achieves its minimum, that is, along its face N pf qXtm P M R , w¨m " trop f pwqu.

As every closed convex subset of a real plane is the intersection of its supporting half-planes, one deduces that the tropicalization trop f determines the Newton polygon N pf q in the following way: [START_REF] Enriques | Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche II[END_REF] N pf q " tm P M R , w ¨m ě trop f pwq, for all w P σ 0 u.

Formula (35) presents N pf q as the intersection of an infinite set of closed half-planes. In fact, as a consequence of the previous discussion, a finite number of them suffices: Proposition 4.8. Let Fpf q be the fan of N obtained by subdividing the cone σ 0 using the rays orthogonal to the compact edges of N pf q. Then:

(1) The tropicalization trop f is continuous and its restriction to any cone in Fpf q is linear.

(2) The relative interiors of the cones of Fpf q may be characterized as the levels of the following map from σ 0 to the set of faces of N pf q, in the sense of Definition 4.2:

w Ñ N pf q X tm P M R , w ¨m " trop f pwqu.

(3) This map realizes an inclusion-reversing bijection between Fpf q and the set of faces of N pf q. If K σ is the face of N pf q corresponding to the cone σ of Fpf q, then:

trop f pwq " w ¨m, for all w P σ, and for all m P K σ .

(4) The Newton polygon N pf q is the intersection of the closed half-planes H f,ρ defined by relation [START_REF] Eisenbud | Three-dimensional link theory and invariants of plane curve singularities[END_REF], where ρ varies among the rays of the fan Fpf q.

The fans Fpf q appearing in the previous proposition are particularly important for the sequel, that is why they deserve a name: Definition 4.9. The Newton fan Fpf q of f P Crrx, yss z t0u is the fan of N obtained by subdividing the cone σ 0 using the rays orthogonal to the compact edges of the Newton polygon N pf q Ď σ _ 0 of f , that is, by the interior normals of the compact edges of N pf q. A Newton fan in a weight lattice N and relative to a basis pe 1 , e 2 q is any fan subdividing the regular cone σ 0 " R `xe 1 , e 2 y.

Example 4.10. Consider the series f P Crrx, yss defined by: f px, yq :" ´x12 `x14 `x7 y 2 `2x 5 y 3 ´x10 y 3 `x3 y 4 `3x 7 y 4 `y9 . On the left side of Figure 15 is represented its Newton polygon N pf q, and on the right side are represented its tropicalization trop f and its Newton fan Fpf q. The support of the series f is: Spf q " tp12, 0q, p14, 0q, p7, 2q, p5, 3q, p10, 3q, p3, 4q, p7, 4q, p0, 9qu.

K 1 K 2 K 3 2 1 x 12 x 7 y 2 x 3 y 4 y 9 b a N pf q Ď M R 9d 3c `4d 7c `2d 12c c d trop f and Fpf q e 2 e 1
Among its elements, the vertices of N pf q are p12, 0q, p7, 2q, p3, 4q, p0, 9q. The corresponding monomials are marked on the left of the figure, near the associated vertices. The other elements of Spf q are marked as green dots. Now, each vertex pa, bq of N pf q may be seen as the linear function w " pc, dq Ñ ac `bd on N R . The tropicalization trop f computes the minimal value of those 4 linear functions at the points of σ 0 . The regular cone σ 0 gets decomposed into 4 smaller 2-dimensional subcones, according to the vertex which gives this minimum. On the right side of Figure 15 those subcones are represented in different colors. Each such subcone has the same color as the expression of the associated linear function and the vertex of N pf q defining it. Each ray separating two successive subcones is orthogonal to a compact edge of N pf q and both are drawn with the same color. Denoting the compact edges by K 1 :" rp0, 9q, p3, 4qs, K 2 :" rp3, 4q, p7, 2qs, K 3 :" rp7, 2q, p12, 0qs, the associated restrictions of f (see Definition 4.2) are:

f K1 " x 3 y 4 `y9 , f K2 " x 7 y 2 `2x 5 y 3 `x3 y 4 , and f K3 " ´x12 `x7 y 2 .
The Newton fan of f is Fpf q " F p3{5, 2{1, 5{2q (see Definition 3.4 for this last notation).

If α P Crrtsszt0u, we denote by c νtpαq pαq the coefficient of t νtpαq in the series α, and we call it the leading coefficient of α.

The following proposition shows why it is important to introduce trop f when studying the germ C defined by f : Proposition 4.11. Let f P Crrx, yss be a non-zero series. Let t Ñ pαptq, βptqq be a germ of formal morphism from pC, 0q to pC 2 , 0q, whose image is not contained in the union L Y L 1 of the coordinate axes. Then one has the inequality:

ν t pf pαptq, βptqqq ě trop f pν t pαq, ν t pβqq,
with equality if and only if f K pc νtpαq pαq, c νtpβq pβqq ‰ 0, where K is the compact face of N pf q orthogonal to pν t pαq, ν t pβqq P N , in the sense that its restriction to N pf q achieves its minimum on this face.

Proof. The basic idea of the proof goes back to Newton's method of computing the leading term of a Newton-Puiseux series ηpxq such that f px, ηpxqq " 0, which we explained on the example of Subsection 2.5, starting from equation [START_REF] Barber | Reducible exceptional curves of the first kind[END_REF].

The hypothesis that the image of t Ñ pαptq, βptqq is not contained in the union of coordinate axes means that both α and β are non-zero series. Therefore, they admit non-vanishing leading coefficients c νtpαq pαq and c νtpβq pβq (see Definition 2.18).

Using the expansion [START_REF] Dumas | Sur la résolution des singularités de surfaces[END_REF], we get that f pαptq, βptqq is equal to: [START_REF] Epple | Branch points of algebraic functions and the beginnings of modern knot theory[END_REF] ÿ pa,bqPSpf q c pa,bq pf q ´cνtpαq pαqt νtpαq `opt νtpαq q ¯a ´cνtpβq pβqt νtpβq `opt νtpβq q ¯b " " ÿ pa,bqPSpf q c pa,bq pf q `cνtpαq pαq ˘a `cνtpβq pβq ˘b ´taνtpαq`bνtpβq `opt aνtpαq`bνtpβq q ¯.

As a consequence:

ν t pf pαptq, βptqqq ě min pa,bqPSpf q taν t pαq `bν t pβqu " trop f pν t pαq, ν t pβqq,
where the last equality follows from Definition 4.4. This proves the inequality stated in the proposition. The case of equality follows from the fact, implied by the computation [START_REF] Epple | Branch points of algebraic functions and the beginnings of modern knot theory[END_REF], that the coefficient of the term with exponent trop f pν t pαq, ν t pβqq of the series f pαptq, βptqq is f K pc νtpαq pαq, c νtpβq pβqq.

In Proposition 4.11, K may be either an edge or a vertex of N pf q. Note that this statement plays with the two dual ways of defining a curve singularity on pC 2 , 0q, either as the vanishing locus of a function or by a parametrization.

Consider now the reduced image of the morphism t Ñ pαptq, βptqq. The hypothesis that it is not contained in L Y L 1 shows that it is a branch on pS, oq, different from L and L 1 . Endow it with a multiplicity equal to the degree of the morphism onto its image, seeing it therefore as a divisor A on pS, oq. By Proposition 2.8, the orders ν t pαptqq, ν t pβptqq which appear in Proposition 4.11 may be interpreted as ν t pαptqq " L ¨A, and ν t pβptqq " L 1 ¨A. We get the following corollary of Proposition 4.11: Proposition 4.12. Let pL, L 1 q be a cross on pS, oq and C be a curve singularity on pS, oq. Assume that the local coordinate system px, yq defines the cross pL, L 1 q and that f P ÔS,o defines C. Then, for every effective divisor A on pS, oq supported on a branch distinct from L and L 1 , one has the inequality:

C ¨A ě trop f ppL ¨Aqe 1 `pL 1 ¨Aqe 2 q.
Moreover, one has equality when A is generic for fixed values of L ¨A and L 1 ¨A.

One may describe the genericity condition involved in the last sentence of Proposition 4.12 as follows. As a consequence of the proof of Proposition 4.18 below, one has f K pc νtpαq pαq, c νtpβq pβqq ‰ 0 (which is equivalent to the equality C ¨A " trop f ppL ¨Aqe 1 `pL 1 ¨Aqe 2 q) if and only if the strict transforms of A and C by the Newton modification ψ C L,L 1 of S defined by C (see Definition 4.14 below) are disjoint. As a consequence of Propositions 4.8 (3) and 4.12 we have: Proposition 4.13. Let pL, L 1 q be a cross on pS, oq and C be a curve singularity on pS, oq. Assume that the local coordinate system px, yq defines the cross pL, L 1 q and that f P ÔS,o defines C. Then the Newton polygon N pf q, the tropicalization trop f and the Newton fan Fpf q do not depend on the choice of the defining functions x, y, f of the curve germs L, L 1 , C.

By contrast, the support of f depends on the choice of the local coordinate system px, yq defining a fixed cross, even if f P ÔS,o is fixed. For instance, the monomial xy becomes a series with infinite support if one replaces simply x by xp1 `x `x2 `¨¨¨q.

Proposition 4.13 implies that the following notions are well-defined:

Definition 4.14. Let pL, L 1 q be a cross on pS, oq, and let px, yq be a local coordinate system defining it.

Let C be a curve singularity on pS, oq, defined by a function f P ÔS,o , seen as a series in Crrx, yss using the coordinate system px, yq. Then:

' The Newton polygon N L,L 1 pCq Ď M L,L 1 of C relative to the cross pL, L 1 q is the Newton polygon N pf q.
' The tropical function trop C L,L 1 : σ 0 Ñ R `of C relative to the cross pL, L 1 q is the tropicalization trop f of the series f . ' The Newton fan F L,L 1 pCq of C relative to the cross pL, L 1 q is the fan Fpf q. ' The Newton modification ψ C L,L 1 : pS F L,L 1 pCq , BS F L,L 1 pCq q Ñ pS, L`L 1 q of S defined by C relative to the cross pL, L 1 q is the modification of S associated with F L,L 1 pCq relative to the cross pL, L 1 q, that is,

ψ C L,L 1 :" ψ F L,L 1 pCq L,L 1 (see Definition 3.33). The strict transform of C by ψ C L,L 1 is denoted C L,L 1 .
Note that we consider the Newton modification ψ C L,L 1 as a morphism in the toroidal category, by endowing S with the boundary L `L1 and the modified surface S F L,L 1 pCq with a boundary equal to the reduced total transform of L `L1 .

An algorithm of toroidal pseudo-resolution.

In this subsection we assume for simplicity that the plane curve singularity C is reduced (see Remark 4.27). We explain that, once a smooth branch L is fixed on the germ of smooth surface pS, oq, one may obtain a so-called toroidal pseudo-resolution of C on pS, oq (see Definition 4.15) by completing the smooth branch into a cross pL, L 1 q, by performing the associated Newton modification, and by iterating these steps at every point at which the strict transform of C intersects the exceptional divisor of the Newton modification (see Theorem 4.23). The algorithm stops after the first step if and only if C is Newton non-degenerate relative to the cross pL, L 1 q (see Definition 4.21).

The following definition formulates two notions of possibly partial resolution of C in the toroidal category, relative to the ambient smooth germ of surface S: Definition 4.15. Let pL, L 1 q be a cross in the sense of Definition 3.31 on the smooth germ of surface pS, oq and let C be a curve singularity on S. Consider a modification π : pΣ, BΣq Ñ pS, L `L1 q of pS, L `L1 q in the toroidal category, in the sense of Definition 3.29. It is called, in decreasing generality: ' A toroidal pseudo-resolution of C if the following conditions are satisfied:

(1) the boundary BΣ of Σ contains the reduction of the total transform π ˚pC q of C by π;

(2) the strict transform of C by π (see Definition 2.31) does not contain singular points of Σ.

' A toroidal embedded resolution of C if, moreover, the surface Σ is smooth. If π : pΣ, BΣq Ñ pS, L `L1 q is a toroidal pseudo-resolution of C, then the reduction of the image πpBΣq of BΣ in S is called the completion Ĉπ of C relative to π.

Remark 4.16. Note that if π : pΣ, BΣq Ñ pS, L `L1 q is a toroidal pseudo-resolution of C, then the strict transform of C by π is smooth and Ĉπ Ě C `L `L1 . If moreover π is an embedded resolution, then the total transform π ˚pC q is a normal crossings divisor in Σ (see Definition 2.32). Note also that if π : pΣ, BΣq Ñ pS, L `L1 q is a toroidal embedded resolution of C, then π : Σ Ñ S is an embedded resolution of C according to Definition 2.33. From now on, we will keep track carefully of the toroidal structures, considering only toroidal embedded resolutions in the sense of Definition 4.15.

Remark 4.17. If π : pΣ, BΣq Ñ pS, L `L1 q is a toroidal pseudo-resolution of C, then the strict transform of C is transversal to the critical locus of π. Our choice of terminology in Definition 4.15 is inspired by Goldin and Teissier's paper [START_REF] Goldin | Resolving singularities of plane analytic branches with one toric morphism[END_REF], where an analogous notion of (embedded) toric pseudo-resolution of a subvariety of the affine space is considered.

Let us look now at the strict transform C L,L 1 of C by the Newton modification ψ C L,L 1 defined by C relative to the cross pL, L 1 q (see Definition 4.14). The following proposition describes its intersection with the boundary BS F L,L 1 pCq : Proposition 4.18. Assume that neither L nor L 1 is a branch of C. Then the strict transform C L,L 1 of C by the Newton modification ψ C L,L 1 intersects the boundary BS F L,L 1 pCq of the toroidal surface pS F L,L 1 pCq , BS F L,L 1 pCq q only at smooth points of it. Moreover, if ρ is a ray of the Newton fan F L,L 1 pCq different from the edges of σ 0 , then C L,L 1 intersects the corresponding component O ρ of the exceptional divisor of ψ C L,L 1 only inside the orbit O ρ . The number of intersection points counted with multiplicities is equal to the integral length of the edge of the Newton polygon N L,L 1 pCq which is orthogonal to the ray ρ.

Proof. We give a detailed proof of this proposition in geometric language, in order to emphasize the roles played by the fundamental combinatorial objects N L,L 1 pCq, trop C L,L 1 and F L,L 1 pCq associated with C relative to the cross pL, L 1 q (see Definition 4.14).

The orbit O ρ is independent of the toric surface containing it, because any two such surfaces contain the affine toric surface X ρ Ą O ρ as Zariski open sets. Therefore, in order to compute the intersection of the strict transform of C with O ρ , we may choose another surface than X F L,L 1 pCq .

Choose local coordinates px, yq defining the cross pL, L 1 q. In this way M L,L 1 gets identified with the lattice of exponents of Laurent monomials in px, yq. Assume that f 1 :" αe 1 `βe 2 is the unique primitive generator of the ray ρ. Let us complete it in a basis pf 1 , f 2 q of the lattice N L,L 1 , such that the cone σ :" R `xf 1 , f 2 y is contained in one of the two cones of dimension 2 of F L,L 1 pCq adjacent to ρ. We are now in the setting of Example 3.26. As explained there, if pϕ 1 , ϕ 2 q is the basis of M L,L 1 dual to the basis pf 1 , f 2 q of N L,L 1 and u :" χ ϕ1 , v :" χ ϕ2 , then v " x ´β y α is a coordinate of the orbit O ρ . Moreover, it realises an isomorphism of its closure in the affine toric surface X σ " C 2 u,v with the affine line C v . Let K ρ be the edge of the Newton polygon N L,L 1 pCq which is orthogonal to the ray ρ. It is parallel to the line Rϕ 2 , because by definition f 1 ¨ϕ2 " 0. Orient K ρ by the vector ϕ 2 and denote then its vertices by m 0 and m 1 , such that K ρ is oriented from m 0 to m 1 . This means that m 1 ´m0 " L ρ ϕ 2 , where L ρ denotes the integral length of the segment K ρ , in the sense of Definition 3.1. Moreover, the points of K ρ X M are precisely those of the form:

(37) m :" m 0 `k ϕ 2 , for k P t0, 1, . . . , L ρ u.
Consider the smooth toric surface X σ " C 2 u,v . The orbit O ρ is its pointed v-axis C v . Therefore, one may compute the intersection of the strict transform of C with this orbit by taking the lift pψ σ σ0 q ˚f of a defining function f of C to C 2 u,v , by simplifying by the greatest monomial in σ _ X M which divides it, and then by setting u " 0. Let therefore f :" ÿ mPSpf q c m pf qχ m P Crrx, yss be a defining function of C. As the bases pf 1 , f 2 q and pϕ 1 , ϕ 2 q are dual of each other, we have the relation m " pf 1 ¨mqϕ 1 `pf 2 ¨mqϕ 2 . This implies that χ m " u f1¨m v f2¨m . As the ray ρ " R `f1 is orthogonal to the edge K ρ of the Newton polygon N L,L 1 pCq " N pf q, we know that:

f 1 ¨m ě h ρ for all m P Spf q,
where h ρ :" trop f pf 1 q, with equality if and only if m P K ρ . Therefore, the highest power of u which divides pψ σ σ0 q ˚f " ÿ mPSpf q c m pf q u f1¨m v f2¨m is u hρ , and it is achieved only on the edge K ρ of N pf q. Moreover, the linear form m Ñ f 2 ¨m achieves its minimum (at least) at the vertex m 0 of N pf q, by the hypothesis that σ is contained in one of the two 2-dimensional cones of Fpf q " F L,L 1 pCq which are adjacent to ρ. This shows that the maximal monomial in pu, vq which divides pψ σ σ0 q ˚f is u hρ v f2¨m0 . After simplifying by it and setting u " 0, one gets the following polynomial equation in the variable v, describing the intersection of the strict transform of C with the v-axis:

(38) ÿ mPKρXM c m pf q v f2¨pm´m0q " 0.
We recognize here the equation obtained from f Kρ " 0 after the change of variables from px, yq to pu, vq and the simplification of the highest dividing monomial. This illustrates the importance in our context of the operation of restriction of f to a compact edge of its Newton polygon, introduced in Definition 4.2. Using Equation (37), we see that Equation [START_REF] Favre | The valuative tree[END_REF] becomes:

(39) Lρ ÿ k"0 c m0`k ϕ2 pf q v k " 0.
The two extreme coefficients c m0 pf q and c m1 pf q of the previous polynomial equation being non-zero, we see that the strict transform of C does not pass through the origin of C 2 u,v and that it intersects the orbit O ρ in L ρ " l Z K ρ points, counted with multiplicities. The solutions of Equation ( 39) are the v-coordinates of the intersection points of the strict transform of C with the orbit O ρ .

By using the same kind of argument for all the cones of the regularization of F L,L 1 pCq, we may show also that the strict transform of C misses all the singular points of the boundary divisor of X F L,L 1 pCq .

Example 4. [START_REF] Casas-Alvero | Singularities of plane curves[END_REF]. Let us give an example of the objects manipulated in the proof of Proposition 4.18. Consider the function f P Crrx, yss of Example 4.10. Let ρ be the ray of slope 2{1 of Fpf q. Then K ρ is the side K 2 :" rp3, 4q, p7, 2qs of BN pf q (see Figure 15). One has f 1 " e 1 `2e 2 . A possible choice of the vector ϕ 2 is ϕ 2 " ´2 1 ` 2 . Therefore v " x ´2y. Orienting K ρ by this vector ϕ 2 one gets m 0 " p7, 2q and m 1 " p3, 4q. We saw in Example 4.10 that f Kρ " x 7 y 2 `2x 5 y 3 `x3 y 4 " x 3 y 2 px 4 `2x 2 y `y2 q. As v " x ´2y, Equation ( 39) is in this case 1`2v `v2 " 0. We see that its degree is indeed the integral length L ρ of the side K ρ . As it has a double root, the series f is not Newton non-degenerate (see Definition 4.21 below). The strict transform of C intersects O ρ at the single point v " ´1.

The proof of Proposition 4.18 yields easily also a proof of the following proposition : Proposition 4.20. Let pL, L 1 q be a cross and C a curve singularity on S. Let f P Crrx, yss be a defining function of C relative to any coordinate system px, yq defining the chosen cross. Then the following conditions are equivalent:

(1) the curve C is reduced and the Newton modification ψ C L,L 1 becomes a toroidal pseudo-resolution of C if one replaces the boundary BS F L,L 1 pCq by the total transform of the divisor pψ C L,L 1 q ˚pC `L`L 1 q; (2) for any ray ρ of the Newton fan F L,L 1 pCq which is orthogonal to a compact edge of N L,L 1 pCq, the polynomial equation ( 39) has only simple roots; (3) the defining function f of C has the property that all the restrictions f K of f to the compact edges K of the Newton polygon N pf q " N L,L 1 pCq define smooth curves in the torus pC ˚q2 x,y . The plane curve singularities which satisfy the equivalent conditions of Proposition 4.20 received a special name: Definition 4.21. Let pL, L 1 q be a cross and C a curve singularity on S. Let f P Crrx, yss be a defining function of C relative to any coordinate system associated to the chosen cross. The function f is called Newton non-degenerate and the curve C is called Newton non-degenerate relative to the cross pL, L 1 q if the equivalent conditions listed in Proposition 4.20 are satisfied.

Usually one speaks about Newton non-degenerate germs of holomorphic functions of several variables. We introduce here the notion of Newton non-degenerate plane curve singularity relative to a cross in order to emphasize the underlying geometric phenomena.

Let us come back to Proposition 4.18. At each point of intersection o i of the strict transform C L,L 1 with the exceptional divisor of ψ C L,L 1 , one has the following dichotomy: ' Either only one branch of C L,L 1 passes through o i , where it is moreover smooth and transversal to the exceptional divisor. The germ A i at o i of the exceptional divisor and this branch form a canonical cross on S F L,L 1 pCq . Then, one reaches locally a toroidal pseudo-resolution of C in the neighborhood of that point. ' Or one does not have a canonical cross, but only a canonical smooth branch: the germ A i at o i of the exceptional divisor pψ C L,L 1 q ´1poq itself. In the second case, one may complete A i into a cross pA i , L i q by the choice of a germ L i of smooth branch transversal to it. Then one is again in the presence of a germ of effective divisor (the germ of the strict transform C L,L 1 of C by ψ C L,L 1 ) on a germ of smooth surface endowed with a cross (the surface S F L,L 1 pCq endowed with the cross pA i , L i q). One gets again a Newton polygon, a tropical function, a Newton fan and a Newton modification, and the previous construction may be iterated. This iterative process may be formulated as the following algorithm of toroidal pseudo-resolution of the germ C: Algorithm 4.22. Let pS, oq be a smooth germ of surface, L a smooth branch on pS, oq and C a reduced germ of curve on pS, oq, which does not contain the branch L in its support. STEP 1. If pL, Cq is a cross, then STOP. STEP 2. Choose a smooth branch L 1 on pS, oq, possibly included in C, such that pL, L 1 q is a cross. STEP 3. Let F L,L 1 pCq be the Newton fan of C relative to the cross pL, L 1 q. Consider the associated Newton modification ψ C L,L 1 : pS F L,L 1 pCq , BS F L,L 1 pCq q Ñ pS, L `L1 q and the strict transform C L,L 1 of C by ψ C L,L 1 (see Definition 4.14). STEP 4. For each point õ belonging to C L,L 1 X BS F L,L 1 pCq , denote:

' L :" the germ of BS F L,L 1 pCq at õ; ' C :" the germ of C L,L 1 at õ; ' o :" õ; ' S :" S F L,L 1 pCq . STEP 5. GO TO STEP 1.
Note that one considers that only the smooth branch L is given at the beginning, and that the second branch L 1 of the cross pL, L 1 q is chosen when one executes STEP 2 for the first time. Note also that the algorithm is non-deterministic, as it involves choices of supplementary branches.

A variant of this algorithm, obtained by replacing Step 3 by a Step 3 reg , will be studied in Subsection 4. Proof. Assume that A is a curve singularity on the smooth germ of surface pS, oq, obtained after a finite number of steps of the algorithm, and that pL ¨Aq o " 1. Then pL, Aq is a cross and the algorithm stops. Therefore, in order to show that the algorithm stops, it is enough to show that after a finite number of steps all the local intersection numbers of the strict transform C L,L 1 of C with the exceptional divisor are equal to 1.

By the end statement of Proposition 4.18, a sequence of such intersection numbers at infinitely near points of o (see Definition 4.31) which dominate each other is necessarily decreasing: [START_REF] Fock | Cluster Poisson varieties at infinity[END_REF] pC ¨Lq o ě pC 1 ¨E1 q o1 ě ¨¨¨ě pC k ¨Ek q o k ě ¨¨¨.

At the k-th iteration of the algorithm we are considering the strict transform C k of C at a point o k , which belongs to the component E k of the exceptional divisor. The sequence [START_REF] Fock | Cluster Poisson varieties at infinity[END_REF] being composed of positive integers, it necessarily stabilizes. If the stable value is 1 for all choices of sequence o, o 1 , o 2 , . . . , then the algorithm stops after a finite number of steps.

Let us reason by contradiction, assuming the contrary. Therefore, one may find a sequence as before for which the stable intersection number is n ą 1. Let us assume without loss of generality, by starting our analysis after the stabilization took place, that:

(41) pC ¨Lq o " pC 1 ¨E1 q o1 " ¨¨¨" pC k ¨Ek q o k " ¨¨¨" n ą 1.

Therefore, for every k ě 1, pE k , C k q is not a cross at o k . By STEP 2 of the algorithm, a smooth germ L k was chosen at o k such that pE k , L k q is a cross at o k .

Let us reformulate the first equality [START_REF] García Barroso | Invariants des singularités de courbes planes et courbure des fibres de Milnor[END_REF] pC 1 ¨E1 q o1 " pC ¨Lq o of the sequence (41) in terms of Newton polygons. By applying again the end statement of Proposition 4.18, we see that pC 1 ¨E1 q o1 is less or equal to the integral length l Z K of the compact edge K of N L,L 1 pCq whose orthogonal ray corresponds to the prime exceptional curve E 1 . One has equality if and only if the strict transform of C intersects E 1 at a single point. In turn, the integral length l Z K is less or equal to the height pC ¨Lq o " n of N L,L 1 pCq (the ordinate of its lowest point on the vertical axis), with equality if and only if K is the only compact edge of N L,L 1 pCq and K " rp0, nq, pm 1 n, 0qs, with m 1 P N ˚.

As a consequence, one has the equality [START_REF] García Barroso | Invariants des singularités de courbes planes et courbure des fibres de Milnor[END_REF] if and only if N L,L 1 pCq has a single compact edge, of the form rp0, nq, pm 1 n, 0qs, with m 1 P N ˚, and the associated polynomial in one variable has only one root in C ˚. In terms of local coordinates px, yq on pS, oq defining the cross pL, L 1 q and a defining unitary polynomial f P Crrxssrys of the plane curve singularity C (see Theorem 6.1 below), equality holds in [START_REF] García Barroso | Invariants des singularités de courbes planes et courbure des fibres de Milnor[END_REF] if and only if f is of the form f " py ´c1 x m1 q n `¨¨¨, with c 1 P C ˚, m 1 P N ˚and where we wrote only the restriction f K of f to the compact edge K of the Newton polygon N L,L 1 pCq, in the sense of Definition 4.2. Then, STEP 3 is performed simply by considering the morphism: (43)

"

x " x 1 , y " x m1 1 pw 1 `c1 q, where px 1 , w 1 q are local coordinates at o 1 and Zpx 1 q " pE 1 , o 1 q. The hypothesis (41) implies that pE 1 , C 1 q is not a cross. Denote by L 1 1 the smooth branch at o 1 obtained by applying again STEP 2. Therefore, pE 1 , L 1 1 q is a cross at o 1 . By the formal version of the implicit function theorem, we can choose local coordinates px 1 , u 1 q defining the cross pE 1 , L 1 1 q in such a way that u 1 " w 1 ´φ1 px 1 q, for some φ 1 P Crrtss with φ 1 p0q " 0.

Let us define y 1 :" y ´xm1 pc 1 `φ1 pxqq and denote L 1 :" Zpy 1 q. Notice that the strict transform of L 1 by the modification ( 43) is equal to L 1 1 and that ( 43) can be rewritten ( 44)

"
x " x 1 , y 1 " x m1 1 u 1 with respect to the local coordinates px, y 1 q and px 1 , u 1 q. Let us denote by f 1 P Crrx 1 ssru 1 s the monic polynomial defining C 1 relative to the coordinates px 1 , u 1 q (see again Theorem 6.1). Reasoning as before, the hypothesis (41) implies that the polynomial f 1 is of the form f 1 " pu 1 ´c2 x m2 1 q n `¨¨¨, where c 2 P C ˚, m 2 P N ˚and the exponents of the monomials x i 1 u j 1 which were omitted verify that i `m2 j ą m 2 n and 0 ď j ă n. Notice that the order of vanishing of f along E 1 is equal to nm 1 . We recover a defining function of C with respect to the coordinates px, y 1 q by expressing, using the relation [START_REF] García Barroso | Variations on inversion theorems for Newton-Puiseux series[END_REF], the monomials appearing in the product x m1n 1 ¨f1 px 1 , u 1 q as monomials in px, y 1 q. We get a defining function of C of the form py 1 ´c2 x m1`m2 1 q n `¨¨¨, where the exponents of the monomials x i 1 y j 1 which are not written above verify that i `pm 1 `m2 qj ą pm 1 `m2 qn and 0 ď j ă n.

By induction on k ě 1, one may show similarly that:

' The branch L 1
k " Zpu k q is the strict transform of a smooth branch L k " Zpy k q at S, where px, y k q is a local coordinate system defining a cross at o and [START_REF] García Barroso | Ultrametric spaces of branches on arborescent singularities[END_REF] y k " y k´1 ´xm1`¨¨¨`m k pc k `φk pxqq, where φ k P Crrtss satisfies φ k p0q " 0. ' The composition of the maps in the algorithm expresses as ( 46)

" x " x 1 , y k " x m1`¨¨¨`m k 1 u k ,
with respect to the local coordinates px k , u k q at o k and the coordinates px, y k q at o. ' There exists a defining function of C of the form:

py k ´ck x m1`¨¨¨`m k q n `¨¨ẅ
here the exponents of monomials x i y j k which are not written above verify that i `pm 1 `¨¨¨`m k qj ą pm 1 `¨¨¨`m k qn and 0 ď j ă n.

In particular, we have shown that the Newton polygon N L,L k pCq has only one compact edge with vertices p0, nq and pm 1 `¨¨¨`m k , 0q, where L k ¨C " m 1 `¨¨¨`m k . When we look at the polygons N L,L k pCq as subsets of R 2 , we get a nested sequence:

(47) N L,L 1 pCq Ą N L,L1 pCq Ą ¨¨¨Ą N L,L k´1 pCq Ą N L,L k pCq.
By [START_REF] García Barroso | Ultrametric spaces of branches on arborescent singularities[END_REF], one has that y k " y ´ξk pxq with ξ k pxq P Crrxss. One may check, using the shape of relation [START_REF] García Barroso | Ultrametric spaces of branches on arborescent singularities[END_REF], that the sequence pξ k pxqq kě1 converges to a series ξ 8 pxq in the complete ring Crrxss. Set y 8 :" y ´ξ8 pxq and L 8 :" Zpy 8 q. Then pL, L 8 q is a cross at o. We deduce that L 8 ¨C " ν x f px, ξ 8 pxqq " `8 and by [START_REF] García Barroso | Quasi-ordinary singularities: tree model, discriminant, and irreducibility[END_REF] one gets the inclusion N L,L8 pCq Ă N L,L k pCq, for every k ě 1. These two facts together imply that the Newton polygon N L,L8 pCq has only one vertex p0, nq. Therefore, a local defining series for C is of the form py 8 q n . Since n ą 1, C would not be a reduced germ, contrary to the hypothesis.

Remark 4.24. The argument used in the proof of Theorem 4.23 coincides basically with one step of the proof of the Newton-Puiseux theorem (see Theorems 2.20 and 6.1), as presented in Teissier's survey [START_REF] Teissier | Complex curve singularities: a biased introduction[END_REF]. Unlike the rest of the proof of this theorem, this particular step holds without making any assumption on the characteristic of the base field. Algorithm 4.22 involves a finite number of choices, those of the smooth branches introduced in order to get crosses each time one executes STEP 2. Let us introduce the following notations: Notations 4.25. Assume that one executes Algorithm 4.22 on pS, oq, starting from the curve singularity C and the smooth branch L. Then:

(1) t o i , i P I u is the set of points at which one applies STEP 1 or STEP 2. One assumes that t1u Ď I and that o 1 :" o. (2) t pA i , B i q , i P Iu is the set of crosses considered each time one applies STEP 1 or STEP 2.

Therefore A 1 " L and for i P I z t1u, the branch A i is included in the exceptional divisor of the Newton modification performed at the previous iteration. (3) J Ď I consists of those j P I for which one performs STEP 2 at o j . Denote by L j the projection on S of the branch B j , for every j P J. Therefore, B i is a strict transform of a branch of C whenever i P I z J and B j is the strict transform of L j whenever j P J.

(4) S p1q :" S. For k ě 1, the surface S pk`1q is obtained from S pkq by performing simultaneously the Newton modification of STEP 3 at all the points o j of S pkq at which one executes STEP 2. At such a point, denote by F Aj ,Bj pCq the corresponding fan. It is the Newton fan of the germ of strict transform of C at o j , relative to the cross pA j , B j q.

(5) The previous simultaneous Newton modification is denoted π pkq : S pk`1q Ñ S pkq . We call it the k-th level of Newton modifications. (6) The toroidal boundary BS pkq is by definition the total transform on S pkq of all the crosses which appeared in the algorithm until performing STEP 2 at all the points of S pkq . In particular, BS " L `L1 . Each morphism π pkq : pS pk`1q , BS pk`1q q Ñ pS pkq , BS pkq q belongs to the toroidal category, as pπ pkq q ´1pBS pkq q Ď BS pk`1q . (7) π :" π p1q ˝¨¨¨˝π phq , where h is the number of modifications π pkq produced by the algorithm.

We denote by Σ the source of π. Therefore, π : Σ Ñ S is a modification of the initial germ S. (8) BΣ denotes BS phq . It is the underlying reduced divisor of the total transform π ˚p Ĉπ q of the completion Ĉπ " C `řjPJ L j , in the sense of Definition 4.15.

There are a lot of notations here! The only way to get used to them, to understand how those objects are related, and why they are important, is to look at examples. That is why we made a detailed one below (see Example 4.28). In fact, all the works which deal in a detailed way with processes of resolution of singularities introduce analogously plenty of notations (see for instance Zariski [START_REF] Zariski | The reduction of the singularities of an algebraic surface[END_REF], Zariski [START_REF] Zariski | General Theory of Saturation and of Saturated Local Rings II: Saturated Local Rings of Dimension 1[END_REF], Lejeune-Jalabert [START_REF] Lejeune-Jalabert | Linear systems with infinitely near base conditions and complete ideals in dimension two[END_REF], A'Campo and Oka [START_REF] A'campo | Geometry of plane curves via Tschirnhausen resolution tower[END_REF], Casas [START_REF] Casas-Alvero | Singularities of plane curves[END_REF], Wall [START_REF] Wall | Singular points of plane curves[END_REF] or Greuel, Lossen and Shustin [START_REF] Greuel | Introduction to singularities and deformations[END_REF]). This is one of the main advantages we see for the notion of lotus attached below to such a resolution process (see Definition 5.26): it allows to get a simultaneous global view of the previous objects.

We can state in the following way the output of Algorithm 4.22 in terms of Definition 4.15:

Proposition 4.26. The morphism π : pΣ, BΣq Ñ pS, L `L1 q is a toroidal pseudo-resolution of C.

Remark 4.27. We formulated Algorithm 4.22 only for reduced curve singularities C. It extends readily to an algorithm applicable to any C, by agreeing that one runs it on the reduction of C. One may agree also to define the fan tree of an arbitrary curve singularity C as the fan tree of its reduction (see Definition 4.33), each leaf being decorated with the multiplicity of the corresponding branch inside the divisor C.

Similar conventions may be chosen in order to associate a lotus to an arbitrary curve singularity C. As we do not use those more general notions in this text, we will not introduce them formally.

Let us give now an example of application of Algorithm 4.22. Instead of starting from a particular equation, we will assume that the algorithm involves three levels of toroidal modifications with prescribed Newton fans and we will describe from them the toroidal boundary of the final surface. We will see in Example 6.29 below how to write concrete equations for branches C i and L j appearing in a toroidal resolution process structured as in Example 4.28. The idea is to associate to the Newton polygons of the process a fan tree (see Definition 4.33), which may be transformed into an Eggers-Wall tree (see Definition 6.28), which in turn allows to write Newton-Puiseux series defining the branches C i and L j . One may take as their defining functions in Crrx, yss the minimal polynomials of those Newton-Puiseux series.

L L 1 o 1 :" o π p1q e L1 e L e E3 5
F L,L1 pCq E 1 E 2 C 2 C 3 E 3 C 1 L 1 o 3 o 2 L Figure 16
. First level of Newton modifications in Example 4. [START_REF] Cutkosky | Resolution of singularities[END_REF] Example 4.28. We will use Notations 4.25, but we will denote in the same way a branch and its various strict transforms by the modifications produced by the algorithm. In particular, we will write L j instead of B j , for any j P J.

Assume that, relative to the first cross pL, L 1 q, which lives on S p1q " S, the Newton fan F L,L1 pCq of the curve singularity C is as represented on the top of Figure 16. Therefore it is the same fan F p3{5, 2{1, 5{2q as in Figure 8. The associated Newton modification π p1q is represented on the bottom of Figure 16. We have drawn schematically the two boundaries BS p1q " L `L1 and BS p2q " L `E1 `E2 `E3 `L1 C1 `C2 `C3 . The components E i of the exceptional divisor of π p1q correspond to the rays R `eEi of the Newton fan F L,L1 pCq. We assume that there are three intersection points of the strict transform C L,L1 of C by π p1q at which the algorithm stops at STEP 1. The corresponding components of C are denoted C 1 , C 2 , C 3 . By contrast, at the points o 2 and o 3 , one has to apply STEP 2 of Algorithm 4.22 (which implies that t2, 3u Ď J).

One introduces two new smooth branches L 2 and L 3 passing through o 2 and o 3 respectively, transversally to the exceptional divisor E 1 `E2 `E3 of π p1q . Both points o 2 and o 3 belong to the component E 1 . Now one may get the second level of Newton modifications, by looking at the Newton fans F E1,L2 pCq and F E1,L3 pCq (note that we have written pE 1 , L j q instead of pA j , L j q, because for j P t2, 3u, A j is the germ of E 1 at o j ). We assume that those Newton fans are as represented on the top of Figure 17. The corresponding composition π p2q of Newton modifications at o 2 and o 3 is represented on the bottom of the figure, through a schematic drawing of BS p2q `L2 `L3 and of BS p3q " BΣ. We assume that the process stops at STEP 1 at three more points, through which pass the strict transforms of the branches C 4 , C 5 , C 6

E 1 E 2 C 2 C 3 E 3 C 1 L 1 L 3 L 2 o 3 o 2 L π p2q e L2 e E1
F E1,L3 pCq E 1 E 2 C 2 C 3 E 3 C 1 L 1 E 6 o 4 E 7 C 6 L 3 L E 4 E 5 C 5 C 4 L 2 Figure 17.
Second level of Newton modifications in Example 4. [START_REF] Cutkosky | Resolution of singularities[END_REF] of C (see the right bottom part of Figure 17). There remains one point o 4 , lying on the component E 6 of the exceptional divisor E 4 `E5 `E6 `E7 of π p2q , at which one has to perform STEP 2. One completes then the germ A 4 of E 6 at o 4 into a cross pE 6 , L 4 q, represented on the left bottom part of Figure 18. We assume now that the Newton fan F E6,L4 pCq is as drawn on the top of the figure. It has only one ray distinct from the edges of the cone R `xe E6 , e L4 y. Therefore, the corresponding Newton modification, which alone gives the third level of Newton modifications π p3q , introduces only one more irreducible component of exceptional divisor, denoted E 8 . It is cut by the strict transform of one more branch of C, denoted C 7 and represented on the bottom right part of Figure 18. The whole curve schematically represented here is the boundary BΣ. On the bottom left is represented the divisor BS p3q `L4 . The toroidal pseudo-resolution of C produced by the algorithm is the composition π p1q ˝πp2q ˝πp3q : pΣ, BΣq Ñ pS, L `L1 q. The singular points of the total surface Σ :" S p3q correspond bijectively to the non-regular 2-dimensional cones of the Newton fans F L,L1 pCq, F E1,L2 pCq, F E1,L3 pCq and F E6,L4 pCq produced by the algorithm. We represented them as small blue discs on the bottom right of Figure 18.

From toroidal pseudo-resolutions to embedded resolutions.

In this subsection, we explain how to get an embedded resolution of C ãÑ S from one of the toroidal pseudo-resolutions produced by Algorithm 4.22. Recall first from Definition 4.15 the difference between toroidal pseudo-resolutions and embedded ones: in the first ones the source of the modification may have toric singularities, while in the second ones the source is required to be smooth. Consider a toroidal pseudo-resolution morphism π : pΣ, BΣq Ñ pS, L `L1 q of C produced by Algorithm 4.22 (we speak about "a morphism" instead of "the morphism", because of the choices of smooth branches pL j q jPJ involved in its construction, see Definition 4.25). The surface Σ has a finite number of singular points. As explained in Example 4.36, they correspond to the 2-dimensional non-regular cones of the Newton fans which appeared during the process. Proposition 3.28 shows that one may resolve minimally those singular points by taking the regularization of each such cone. In fact, those regularizations glue into the regularizations of the Newton fans.

E 1 E 2 C 2 C 3 E 3 C 1 L 1 E 6 L 4 o 4 E 7 C 6 L 3 L E 4 E 5 C 5 C 4 L 2 π p3q e L4 e E6 e E8 1 2 F E6,L4 pCq E 1 E 2 C 2 C 3 E 3 C 1 L 1 E 6 E 8 C 7 L 4 E 7 C 6 L 3 L E 4 E 5 C 5 C 4 L 2
A way to regularize all the Newton fans produced by Algorithm 4.22 is to run a variant of it, obtained by always replacing STEP 3 with the following "regularized" version of it: STEP 3 reg . Let F reg L,L 1 pCq be the regularized Newton fan of C relative to the cross pL, L 1 q and let ψ C,reg L,L 1 : pS F reg L,L 1 pCq , BS F reg L,L 1 pCq q Ñ pS, L`L 1 q be the associated Newton modification. Consider the strict transform

C L,L 1 of C by ψ C,reg L,L 1 .
We did not change the notations for the successive strict transforms of C from STEP 3 to STEP 3 reg , because this variant of the algorithm does never modify the surfaces produced by the first algorithm in the neighborhood of those strict transforms. Indeed, the strict transforms never pass through the singular points of the modified surfaces S pkq (see Proposition 4.18 and Notations 4.25).

One has the following description of the result of running the "regularized" algorithm:

Proposition 4.29. Let π : pΣ, BΣq Ñ pS, L `L1 q be a toroidal pseudo-resolution obtained by running Algorithm 4.22. Assume that one replaces always STEP 3 with STEP 3 reg above, choosing the same smooth germs pL j q jPJ as in the construction of π. Then one gets a morphism in the toroidal category π reg : pΣ reg , BΣ reg q Ñ pS, L `L1 q, which is moreover an embedded resolution of C and which factors as π reg " π ˝η, where η : pΣ reg , BΣ reg q Ñ pΣ, BΣq is a modification in the toroidal category whose underlying modification of complex surfaces is the minimal resolution of the complex surface Σ.

Let us look at the underlying morphism of complex surfaces π reg : Σ reg Ñ S. Both surfaces are smooth, therefore this morphism is a composition of blow ups of points, by the following theorem of Zariski (see [START_REF] Hartshorne | Algebraic geometry[END_REF]Corollary 5.4] or [113, Vol.1, Ch. IV.3.4, Thm.5]):

Theorem 4.30. Let ψ : S 2 Ñ S 1 be a modification of a smooth complex surface S 1 , with S 2 also smooth. Then ψ may be written as a composition of blow ups of points.

In Section 5 we will describe explicitly the combinatorics of the decomposition of π reg : Σ reg Ñ S into blow ups of points.

Let us recall the following classical terminology about objects associated to a process of blow ups of points, starting from o P S (see [START_REF] Lejeune-Jalabert | Linear systems with infinitely near base conditions and complete ideals in dimension two[END_REF], [START_REF] Casas-Alvero | Singularities of plane curves[END_REF]Chap. 3], [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF] and [START_REF] Pereira | Fibonacci numbers and self-dual lattice structures for plane branches[END_REF]): Definition 4.31. Let pS, oq be a smooth germ of surface. ' An infinitely near point of o is either o or a point of the exceptional divisor of a smooth modification of pS, oq. Two such points, on two modifications, are considered to be the same, if the associated bimeromorphic map between the two modifications is an isomorphism in their neighborhoods. ' The Enriques diagram ΓpCq of the finite constellation C is the rooted tree with vertex set C, rooted at o, and such that there is an edge joining each point of C with its parent.

Note that the proximity binary relation on the set of all the infinitely near points of o is not a partial order, as it is neither reflexive, nor transitive. For instance, if o 1 belongs to the exceptional divisor E 0 of the blow up of o and o 2 belongs to the exceptional divisor of the blow up of o 1 but not to the srict transform of E 0 by this blow up, then o 2 Ñ o 1 Ñ o, but o 2 Û o. Therefore, the Enriques diagram of a finite constellation encodes only part of the proximity binary relation on it. For this reason, Enriques introduced in [START_REF] Enriques | Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche II[END_REF] supplementary rules for the drawing of his diagrams, allowing to reconstruct completely the proximity relation. Namely, the edges of the Enriques diagram are moreover either straight or curved and there are breaking points between some pairs of successive straight edges. As we do not insist on those aspects, we do not give the precise definitions, sending the interested reader to the literature cited above.

The fan tree of a toroidal pseudo-resolution process.

In this subsection we explain how to associate a fan tree to each process of toroidal pseudo-resolution of a curve singularity C on the smooth germ of surface pS, oq (see Definition 4.33). It is a couple formed by a rooted tree and a r0, 8s-valued function constructed from the Newton fans created by the process. It turns out that it is isomorphic to the dual graph of the boundary BΣ of the source surface Σ of the toroidal pseudo-resolution morphism π : pΣ, BΣq Ñ pS, BSq (see Proposition 4.35).

Fan trees are constructed from trunks associated with Newton fans. Let us define first those trunks: Definition 4.32. Let N be a 2-dimensional lattice endowed with a basis pe 1 , e 2 q and let F be a Newton fan of N relative to this basis, in the sense of Definition 4.9. Its trunk θpFq is the segment re 1 , e 2 s Ď σ 0 endowed with the slope function S F : re 1 , e 2 s Ñ r0, 8s which associates with each point w P re 1 , e 2 s the slope in the basis pe 1 , e 2 q of the ray R `w generated by it. Its marked points are the points of intersection of re 1 , e 2 s with the rays of F. If E Ď Q `Y t8u, we denote by θpEq the trunk of the fan FpEq introduced in Definition 3.4.

Note that the slope function of a trunk is a homeomorphism. Several examples of trunks are represented in Figure 19.

Assume now that we apply Algorithm 4.22 to the curve singularity C living on the smooth germ of surface pS, oq. Consider the set tpA i , B i q, i P Iu of crosses produced by the algorithm, as explained in Notations 4. [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF]. Note that we consider also the crosses at which the algorithm stops at an iteration of STEP 1. Denote by pe Ai , e Bi q the basis pe 1 , e 2 q of the weight lattice N Ai,Bi . The segment re Ai , e Bi s is the trunk θpF Ai,Bi pCqq. The following definition uses Notations 4.25:

Definition 4.33. The fan tree pθ π pCq, S π q of the toroidal pseudo-resolution π : pΣ, BΣq Ñ pS, L `L1 q of C is a pair formed by a rooted tree θ π pCq and a slope function S π : θ π pCq Ñ r0, 8s obtained by gluing the disjoint union of the trunks pθpF Ai,Bi pCqq, S F A i ,B i pCq q iPI in the following way:

(1) Label each marked point with the corresponding irreducible component E k , L j or C l of the boundary BΣ of the toroidal surface pΣ, BΣq.

(2) Identify all the points of Ů iPI θpF Ai,Bi pCqq which have the same label. The result of this identification is the fan tree θ π pCq and the images inside it of the marked points of Ů iPI θpF Ai,Bi pCqq are its marked points. We keep for each one of them the same label as in the initial trunks.

(3) The root of θ π pCq is the point labeled by the initial smooth branch L. (4) For every i P I, the restriction of S π to every half-open trunk θpF Ai,Bi pCqqzte Ai u " pe Ai , e Bi s ãÑ θ π pCq is equal to S F A i ,B i pCq . (5) At the root, S π pLq " S F L,L 1 pCq pLq " 0.

As in any rooted tree, the root L defines a partial order ĺ L on the set of vertices of the fan tree θ π pCq (that is, on its set of marked points), by declaring that P ĺ L Q if and only if the unique segment rL, P s joining L and P inside the tree is included in the analogous segment rL, Qs.

Note that the slope function S π is discontinuous at all the marked points of θ π pCq resulting from the identification of points of two different trunks, its directional limits jumping from a positive value to 0 when one passes from one trunk to another one in increasing way relative to the partial order ĺ L . It follows that the fan tree of a toroidal pseudo-resolution determines the trunks pθpF Ai,Bi pCqq, S F A i ,B i pCq q iPI . Example 4.34. Consider again the toroidal pseudo-resolution process of Example 4.28. The construction of the trunks associated to its Newton fans is represented in Figure 19 for all the crosses at which one applies STEP 2 of the algorithm, that is, for the crosses pA i , B i q with i P J. The remaining crosses are those at which the algorithm stops while executing STEP 1. The corresponding trunks are represented on the bottom line of Figure 19. Figure 20 shows the construction of the fan tree from the previous collection of trunks. In order to make clear the process of gluing of points with the same label, the upper part of the figure shows again the whole collection of trunks, as well as the labels of its marked points.

The following proposition is an easy consequence of Definition 4.33 and of Proposition 3.24 (recall that the notion of dual graph of an abstract simple normal crossings curve was explained in Definition 3.22): Proposition 4.35. The fan tree θ π pCq is isomorphic to the dual graph of the boundary BΣ of the source of the toroidal pseudo-resolution π : pΣ, BΣq Ñ pS, L `L1 q of the curve singularity C, by an isomorphism which respects the labels.

Example 4.36. Proposition 4.35 is illustrated in Figure 21 with the fan tree of the bottom of Figure 20 and the boundary BΣ of the bottom right of Figure 18. Both of them correspond to the toroidal pseudoresolution process of Example 4.28. The singular points of Σ may be found out from the knowledge of the slope function on the trunks composing the fan tree. Indeed, consider the slopes β{α and δ{γ of two consecutive vertices of the trunk of one of the Newton fans of the pseudo-resolution process. Then the matrix ˆα γ β δ ˙is of determinant ˘1 if and only if the intersection point o i of the irreducible components of BΣ which corresponds to this edge is non-singular on Σ. Moreover, the surface singularity pΣ, o i q is analytically isomorphic to the germ at its orbit of dimension 0 of the affine toric surface generated by the cone R `xα e 1 `β e 2 , γ e 1 `δ e 2 y and the lattice N " Zxe 1 , e 2 y. As in Figure 18, the singular points on BΣ are indicated by small blue discs. The corresponding edges of the fan trees are represented also in blue. Note that in the previous explanation it was important to say that one has to work with the slope function on the individual trunks, instead of the slope function of the fan tree. For instance, if one looks at the intersection point of the components E 1 and E 4 , the corresponding slopes are to be F E1,L2 pCq

E 1 0 L 2 8 E 4 2 3 E 5 3 4 θ pF E1,L2 pCqq e L3 e E1 e E7 e E6 3 1 5 3 F E1,L3 pCq E 1 L 3 8 E 6 E 7 θ pF E1,L3 pCqq e L4 e E6 e E8 1 2 F E6,L4 pCq E 6 0 L 4 8 E 8 1 2 θ pF E6,L4 pCqq E 3 0 C 1 8 θ pFE 3 ,C 1 pCqq E 2 0 C 2 8 θ pFE 2 ,C 2 pCqq E 2 0 C 3 8 θ pFE 2 ,C 3 pCqq E 5 0 C 4 8 θ pFE 5 ,C 4 pCqq E 4 0 C 5 8 θ pFE 4 ,C 5 pCqq E 7 0 C 6 8 θ pFE 7 ,C 6 pCqq E 8 0 C 7 8 θ pFE 8 ,C 7 pCqq

Historical comments.

The oldest method to study a plane curve singularity C, imagined by Newton around 1665, but published only in 1736 as [START_REF] Newton | The method of fluxions and infinite series[END_REF], is to express it first in local coordinates px, yq as the vanishing locus of a power series f px, yq satisfying f p0, 0q " 0 and f p0, yq ‰ 0, then to compute iteratively a formal power series ηpxq with rational positive exponents such that f px, ηpxqq " 0. Whenever Bf By p0, 0q ‰ 0, there is only one such series ηpxq which has moreover only integral exponents. This series is simply the Taylor expansion at the origin of the explicit function ypxq whose existence is ensured by the implicit function theorem applied to the function f px, yq in the neighborhood of p0, 0q. But, if Bf By p0, 0q " 0, then there are at least two such series, their number being equal to the order in y of the series f p0, yq. As explained on the example studied in Subsection 2.6, the first step of Newton's iterative method consists in finding the possible leading terms c x α of the series ηpxq. His main insight was that if one substitutes y :" c x α in the series f px, yq, getting a formal power series with rational exponents in the variable x, then there are at least two terms of this series with minimal exponent, and the sum of all such terms vanishes. This fact has two consequences. First, there is a finite number of possible exponents α, which are the slopes of the rays orthogonal to the compact edges of the Newton polygon of f px, yq. Secondly, for a fixed exponent α K corresponding to the compact edge K, there is a finite number of values of the leading coefficient c, given by the roots of the algebraic equation f K px, c x α K q " 0, where f K is the restriction of f to K in the sense of Definition 4.2. Newton wrote that his procedure may be performed iteratively in order to compute as many terms of the series ηpxq as desired. He also sketched in [88, Ch. I.LII] an explanation of the fact that, whenever f px, yq converges, the formal series with rational exponents ηpxq obtained by continuing forever the procedure also converge and satisfy indeed, all of them, the relation f px, ηpxqq " 0. But it was Puiseux, in his 1850 paper [START_REF] Puiseux | Recherches sur les fonctions algébriques[END_REF], who proved rigorously that one gets indeed as many series as the order in y of f p0, yq, that all of them are obtained by substituting some root x 1{n of the variable x into formal power series with integral exponents, and that those formal power series are in fact convergent in a neighborhood of the origin. In order to honor his work, the formal or convergent power series in a variable x of the form ξpx 1{n q, where ξpxq is a usual power series and n P N ˚are called nowadays Puiseux series or Newton-Puiseux series.

L 0 L 1 8 E 1 3 5 E 2 2 1 E 3 5 2 θ pF L,L1 pCqq E 1 0 L 2 8 E 4 2 3 E 5 3 4 θ pF E1,L2 pCqq E 1 0 L 3 8 E 6 5 3 E 7 3 1 θ pF E1,L3 pCqq E 6 0 L 4 8 E 8 1 2 θ pF E6,L4 pCqq E 3 0 C 1 8 θ pF E3,C1 pCqq E 2 0 C 2 8 θ pF E2,C2 pCqq E 2 0 C 3 8 θ pF E2,C3 pCqq E 5 0 C 4 8 θ pF E5,C4 pCqq E 4 0 C 5 8 θ pF E4,C5 pCqq E 7 0 C 6 8 θ pF E7,C6 pCqq E 8 0 C 7 8 θ pF E8,C7 pCqq gluing L 0 L 1 8 E 1 3 5 E 2 2 1 E 3 5 2 C 1 8 C 2 8 C 3 8 L 2 8 E 4 2 3 E 5 3 4 C 5 8 C 4 8 L 3 8 E 7 3 1 E 6 5 3 C 6 8 L 4 8 E 8 1 2 C 7 8 θ π pCq
E 1 E 2 C 2 C 3 E 3 C 1 L 1 E 6 E 8 C 7 L 4 E 7 C 6 L 3 L E 4 E 5 C 5 C 4 L 2 BΣ L 0 L 1 8 E 1 3 5 E 2 2 1 E 3 5 2 C 1 8 C 2 8 C 3 8 L 2 8 E 4 2 3 E 5 3 4 C 5 8 C 4 8 L 3 8 E 7 3 1 E 6 5 3 C 6 8 L 4 8 E 8 1 2 C 7 8 θ π pCq
Puiseux's approach to the proofs of the existence and the convergence of these series avoided the use of roots x 1{n , by performing changes of variables of the form x " x q 1 , y " c 1 x p 1 `y1 or of the form x " x q 1 , y " x p 1 pc 1 `y1 q, where c 1 is a non-zero constant and p{q is the irreducible expression of one of the exponents α K given by the Newton polygon of f . Both changes of variables are compositions of a birational change of variables and of the monomial change of variables x " u q , y " u p v. This monomial change of variables is birational only when q " 1, that is, when α K P N ˚. Therefore Puiseux's changes of variables are in general not birational. Nevertheless, by Lemma 6.24 below, such a map can be seen as the local analytical expression of a birational map, with respect to a particular choice of local coordinates.

Zariski saw this non-birationality as a drawback, and in his 1939 paper [START_REF] Zariski | The reduction of the singularities of an algebraic surface[END_REF] he introduced alternative changes of variables of the form x " x q 1 pc 1 `y1 q q1 , y " x p 1 pc 1 `y1 q p1 , where pp 1 , q 1 q P N ˚ˆN ˚and p 1 q ´q1 p " 1. This last condition means that Zariski's changes of variables are birational.

Let us discuss now the toric approach to the study of plane curve singularities. Note that the changes of variables used by Puiseux and by Zariski are compositions of affine morphisms and of toric ones. This fact became clear after the development of toric geometry (see Subsection 3.5).

The systematic study of plane curve singularities using sequences of toric modifications began with Mutsuo Oka's 1995-96 papers [START_REF] Lê | On resolution complexity of plane curves[END_REF], [START_REF] A'campo | Geometry of plane curves via Tschirnhausen resolution tower[END_REF], [START_REF] Oka | Geometry of plane curves via toroidal resolution[END_REF], the first one written in collaboration with Lê and the second one with A'Campo (see also Oka's 1997 book [94, Ch. III, Sect. 4]). Oka gave an introduction to this approach in his 2010 paper [START_REF] Oka | Introduction to Plane Curve Singularities. Toric Resolution Tower and Puiseux Pairs[END_REF], through the detailed examination of the case of one branch. The second author generalized this approach to quasi-ordinary hypersurface singularities of arbitrary dimension in his 2003 paper [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF] and applied it to the study of deformations of real plane curve singularities in the 2010 papers [START_REF] González Pérez | Approximate roots, toric resolutions and deformations of a plane branch[END_REF] and [START_REF] González Pérez | Multi-Harnack smoothings of real plane branches[END_REF], the second one written in collaboration with Risler.

Also during the 1990s, Pierrette Cassou-Noguès started studying plane curve singularities using Puiseux's non-birational toric morphisms, called Newton maps. References to her early works on the subject, done partly in collaboration, may be found in her 2011 paper [START_REF] Cassou-Noguès | Invariants of plane curve singularities and Newton diagrams[END_REF] with P loski, her 2014-15 papers [START_REF] Cassou-Noguès | Newton trees for ideals in two variables and applications[END_REF][START_REF] Cassou-Noguès | The Newton tree: geometric interpretation and applications to the motivic zeta function and the log canonical threshold[END_REF] with Veys, her 2014 paper with Libgober [START_REF] Cassou-Noguès | Multivariable Hodge theoretical invariants of germs of plane curves. II. In Valuation theory in interaction[END_REF] and her 2018 paper with Raibaut [START_REF] Cassou-Noguès | Newton Transformations and the Motivic Milnor Fiber of a Plane Curve[END_REF].

In his 1997 paper [START_REF] Veys | Zeta functions for curves and log canonical models[END_REF], Veys considered the log-canonical model of a plane curve singularity, obtained by contracting certain exceptional divisors on its minimal embedded resolution, in order to study associated zeta functions. The modification from the log-canonical model to the ambient germ of the plane curve singularity may be seen as a morphism associated with a toroidal pseudo-resolution of this singularity. A toroidal pseudo-resolution algorithm for plane curve singularities was described by the second author in [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF]Section 3.4]. A more general algorithm was given by Cassou-Noguès and Libgober in [21, Section 3]. Our Algorithm 4.22 of toroidal pseudo-resolution generalizes them, since it does not depend on the choice of special kinds of coordinates.

There are several approaches for the search of optimal choices of smooth branches in STEP 2 of Algorithm 4.22. Assume first that C is a branch, that f P Crrxssrys is the monic polynomial of degree n defining C in the local coordinate system px, yq and that the line L " Zpxq is transversal to C. Let a be a divisor of n. The a-th approximate root h P Crrxssrys of f is the unique monic polynomial of degree a such that the degree in y of f ´hn{a is smaller than n ´a. The importance of approximate roots for the study of plane curve singularities and of the algebraic embeddings of C in C 2 was emphasized by Abhyankar and Moh in their 1973-75 papers [START_REF] Abhyankar | Newton-Puiseux Expansion and Generalized Tschirnhausen Transformation[END_REF] and [START_REF] Abhyankar | Embeddings of the Line in the Plane[END_REF]. Certain approximate roots of f , called characteristic approximate roots, have the property that their strict transforms can be chosen at STEP 2 of Algorithm 4.22, providing in this way a toroidal pseudo-resolution of C with the minimal number of Newton modifications. This number is precisely the number of characteristic exponents of C with respect to x (see Section 6). This approach was explained by A'Campo and Oka in their 1996 paper [START_REF] A'campo | Geometry of plane curves via Tschirnhausen resolution tower[END_REF].

Some properties of the approximate roots may fail when working with a base field of positive characteristic. By contrast, the more general combinatorial notion of semiroot/maximal contact curve can be defined over fields of arbitrary characteristic and plays a similar role (see the papers [START_REF] Lejeune-Jalabert | Sur l'équivalence des singularités des courbes algébriques planes. Coefficients de Newton[END_REF] of Lejeune-Jalabert and [START_REF] García Barroso | An approach to plane algebroid branches[END_REF] of the first author and P loski). For details on applications of approximate roots and semiroots to the study of plane curve singularities, see the paper [START_REF] Gwoździewicz | On the approximate roots of polynomials[END_REF] of Gwoździewicz and P loski and [START_REF] Popescu-Pampu | Approximate roots[END_REF] of the third author. Proposition 4.35 above implies that if π : pΣ, BΣq Ñ pS, L `L1 q is a toroidal embedded resolution of C which defines its minimal resolution, then the irreducible components of the associated completion Ĉπ " πpBΣq may be thought as generalizations of the notion of semiroot to plane curve singularities with an arbitrary number of branches (see also the final comments in Example 6.33 below).

Assume now that C is an arbitrary plane curve singularity. The minimal number of Newton modifications involved in the construction of a toroidal pseudo-resolution C was characterized by Lê and Oka in [START_REF] Lê | On resolution complexity of plane curves[END_REF] in terms of properties of the dual graph of its minimal embedded resolution.

Another toric approach to the study of plane curve singularities was initiated in Goldin and Teissier's 2000 paper [START_REF] Goldin | Resolving singularities of plane analytic branches with one toric morphism[END_REF], in the case of branches. They first reembedded in a special way the initial germ of surface in a higher dimensional space, then they resolved the branch by just one toric modification of that space. Their approach was done in the spirit of the philosophy of Teissier's 1973 paper [START_REF] Teissier | Le problème des modules pour les branches planes[END_REF], in which he saw all equisingular plane branches as deformations of a single branch of higher embedding dimension, the germ at the origin of their common monomial curve. A generalization of some of the results in [START_REF] Goldin | Resolving singularities of plane analytic branches with one toric morphism[END_REF] to the case of quasi-ordinary hypersurface singularities was obtained by the second author in [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF]. The theoretical possibility of studying analogously singularities of any dimension was established by Tevelev in his 2014 paper [START_REF] Tevelev | On a question of B. Teissier[END_REF]. See Teissier's comments in [START_REF] Teissier | Overweight deformations of affine toric varieties and local uniformization[END_REF]Section 11] for more details about his toric approach to the study of singularities.

The notions of Newton non-degenerate polynomials and series were introduced by Kouchnirenko in his 1976 paper [START_REF] Kouchnirenko | Polyèdres de Newton et nombres de Milnor[END_REF], using the last characterization of Proposition 4.20. A version of the first characterization was essential in Varchenko's theorem in [START_REF] Varchenko | Zeta-function of monodromy and Newton's diagram[END_REF] about the monodromy of Newton non-degenerate holomorphic series. Then Khovanskii introduced in [73] Newton non-degenerate complete intersection singularities, a notion which was much studied by Mutsuo Oka in a series of papers, which were the basis of his 1997 monograph [START_REF] Oka | Non-degenerate complete intersection singularity[END_REF]. Characterizations of Newton non-degenerate singularities, analogous to those of Proposition 4.20, are in fact true for complete intersection singularities (see Oka's book [START_REF] Oka | Non-degenerate complete intersection singularity[END_REF] or Teissier's paper [START_REF] Teissier | Monomial ideals, binomial ideals, polynomial ideals[END_REF]Section 5]). This last paper contains interesting comments about the evolution of the notion of Newton non-degeneracy, and an extension of it to arbitrary singularities, which are not necessarily complete intersections. This extension was further studied in Fuensanta Aroca, Gómez-Morales and Shabbir's paper [START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF].

Let us discuss now the notion of tropicalization trop f introduced in Definition 4.4. The union of the rays of the Newton fan Fpf q which intersect the interior of the regular cone σ 0 is the tropical zerolocus of the function trop f , as defined in tropical geometry, that is, the locus of non-differentiability of the continuous piecewise linear function trop f . It is also part of the local tropicalization of the zero locus Zpf q ãÑ pC 2 , 0q of f , as defined by Stepanov and the third author in [START_REF] Popescu-Pampu | Algebraic and Combinatorial aspects of Tropical Geometry[END_REF] for complex analytic singularities of arbitrary dimension embedded in germs of affine toric varieties. The local tropicalization contains also portions at infinity, in a partial compactification of the cone defining the ambient toric variety, in order to keep track of the intersections of the singularity with all the toric orbits.

A precursor of the notion of local tropicalisation was introduced under the name of "tropism of an ideal " by Maurer in his 1980 article [START_REF] Maurer | Puiseux expansion for space curves[END_REF], which was unknown to the authors of [START_REF] Popescu-Pampu | Algebraic and Combinatorial aspects of Tropical Geometry[END_REF] when they wrote that paper. In our case, the tropism of the ideal pf q Ď Crrx, yss is the set of lattice points lying on the rays of Fpf q which are different from the edges of the cone σ 0 . The term "tropism" had been used before by Lejeune-Jalabert and Teissier in their 1973 paper [START_REF] Lejeune-Jalabert | polygone de Newton, et installations[END_REF], in the expression "tropisme critique". They saw this notion as a measure of anisotropy, as explained by Teissier in [65, Footnote to Sect. 1]: " As far as I know the term did not exist before. We tried to convey the idea that giving different weights to some variables made the space "anisotropic", and we were intrigued by the structure, for example, of anisotropic projective spaces (which are nowadays called weighted projective spaces). From there to "tropismes critiques" was a quite natural linguistic movement. Of course there was no "tropical" idea around, but as you say, it is an amusing coincidence. The Greek "Tropos" usually designates change, so that "tropisme critique" is well adapted to denote the values where the change of weights becomes critical for the computation of the initial ideal. The term "Isotropic", apparently due to Cauchy, refers to the property of presenting the same (physical) characters in all directions. Anisotropic is, of course, its negation. The name of Tropical geometry originates, as you probably know, from tropical algebra which honours the Brazilian computer scientist Imre Simon living close to the tropics, where the course of the sun changes back to the equator. In a way the tropics of Capricorn and Cancer represent, for the sun, critical tropisms."

Lotuses

Throughout this section, we will assume that C is reduced. We explain the notion of Newton lotus (see Definition 5.4), its relation with continued fractions (see Subsection 5.2) and how to construct a more general lotus from the fan tree of a toroidal pseudo-resolution process (see Definition 5.26). It is a special type of simplicial complex of dimension 2, built from the Newton lotuses associated with the Newton fans generated by the process, by gluing them in the same way one glued the corresponding trunks into the fan tree. It allows to visualize the combinatorics of the decomposition of the embedded resolution morphism into point blow ups, as well as the associated Enriques diagram and the final dual graphs (see Theorem 5.29). We show by two examples that its structure depends on the choice of auxiliary curves introduced each time one executes STEP 2 of Algorithm 4.22, that is, on the choice of completion Ĉπ of C (see Subsection 5.4). In Subsection 5.5 we define an operation of truncation of the lotus of a toroidal pseudo-resolution and we explain some of its uses. In the final Subsection 5.6 we give historical information about other works in which appeared objects similar to the notion of lotus.

The lotus of a Newton fan.

In this subsection, whose content is very similar to that of [102, Section 5], we give a first level of explanation of the subtitle of this article, a second level being described in Subsection 5.3. Namely, we introduce the notion of lotus ΛpFq of a Newton fan F (see Definition 5.4). If the fan originates from a Newton polygon N pf q, that is, if F " Fpf q (see Definition 4.9), we imagine ΛpFq as a blossoming of N pf q. The lotus of a Newton fan F allows to understand visually the decomposition into blow ups of the toric modification defined by the regularized fan F reg . For instance, the dual graph of the final exceptional divisor, the Enriques diagram and the graph of the proximity relation of the associated constellation embed naturally in it, as subcomplexes of its 1-skeleton (see Propositions 5.11, 5.14 and 5. [START_REF] Brocot | Calcul des rouages par approximation, nouvelle méthode[END_REF]). Lotuses are built from petals, which are triangles with supplementary structure (see Figure 25): Definition 5.1. Let N be a 2-dimensional lattice and let pe 1 , e 2 q be a basis of it. Denote by δpe 1 , e 2 q the convex and compact triangle with vertices e 1 , e 2 , e 1 `e2 , contained in the real plane N R . It is the petal associated with the basis pe 1 , e 2 q. Its base is the segment re 1 , e 2 s, oriented from e 1 to e 2 . The points e 1 and e 2 are called the basic vertices of the petal. Its lateral edges are the segments re i , e 1 `e2 s, for each i P t1, 2u.

Once the petal δpe 1 , e 1 `e2 q is constructed, the construction may be repeated starting from each one of the bases pe 1 , e 1 `e2 q and pe 1 `e2 , e 2 q of N , getting two new petals δpe 1 , e 1 `e2 q and δpe 1 `e2 , e 2 q, and so on. Note that the bases produced by this process are ordered such as to define always the same orientation of the real plane N R -we say that they are positive bases. In this way, one progressively constructs an infinite simplicial complex embedded in the cone σ 0 : at the n-th step, one adds 2 n petals to those already constructed. Each petal, with the exception of the first one δpe 1 , e 2 q, has a common edge -its base -with exactly one of the petals constructed at the previous step, called its parent.

The pairs of vectors pf 1 , f 2 q P N 2 which appear as bases of petals δpf 1 , f 2 q during the previous process may be characterized in the following way (see [102, Remarque 5.1]): Lemma 5.2. A segment rf 1 , f 2 s, oriented from f 1 to f 2 , is the base of a petal δpf 1 , f 2 q constructed during the previous process if and only if pf 1 , f 2 q is a positive basis of the lattice N contained in the cone σ 0 . Said differently, if a positive basis pf 1 , f 2 q of N is contained in the cone σ 0 and is different from pe 1 , e 2 q, then there exists a unique permutation pi, jq of p1, 2q such that f j ´fi P σ 0 X N .

We are ready to define the simplest kinds of lotuses: Definition 5.3. The simplicial complex obtained as the union of all the petals constructed by the previous process starting from the basis pe 1 , e 2 q of N , is called the universal lotus Λpe 1 , e 2 q relative to pe 1 , e 2 q (see Figure 26). A lotus Λ relative to pe 1 , e 2 q is either the segment re 1 , e 2 s or the union of a non-empty set of petals of the universal lotus Λpe 1 , e 2 q, stable under the operation of taking the parent of a petal. The segment re 1 , e 2 s is called the base of Λ. If Λ is of dimension 2, then the petal δpe 1 , e 2 q is called its base petal. The point e 1 is called the first basic vertex and e 2 the second basic vertex of the lotus. The lotus is oriented by restricting to it the orientation of N R induced by the basis pe 1 , e 2 q. A lotus may be associated with any set E Ď r0, 8s or with any Newton fan: Definition 5.4. Let N be a lattice of rank 2, endowed with a basis pe 1 , e 2 q. ' If λ P p0, 8q, then its lotus, denoted Λpλq , is the union of petals of the universal lotus Λpe 1 , e 2 q whose interiors intersect the ray of slope λ. If λ P t0, 8u, then its lotus Λpλq is just re 1 , e 2 s.

' If E Ď r0, 8s, then its lotus ΛpEq is the union Ť λPE Λpλq of the lotuses of its elements. ' If F is a Newton fan and F " FpEq in the sense of Definition 3.4, we say that ΛpFq :" ΛpEq is the lotus of the fan F. ' A Newton lotus is the lotus of a Newton fan. That is, it is a lotus relative to pe 1 , e 2 q with a finite number of petals.

We could have called the lotuses relative to pe 1 , e 2 q finite lotuses instead of Newton lotuses. We chose the second terminology because in Definition 5.26 below we will introduce a more general kind of lotuses with a finite number of petals, and we want to distinguish the class of lotuses of Newton fans inside that more general class of lotuses.

A lotus ΛpEq, for E Ď r0, 8s, is a Newton lotus if and only E is a finite set of non-negative rational numbers. Note that, as illustrated for instance by Example 5.9 below, the structure of the lotus ΛpEq does not allow to reconstruct the initial set E. For this reason, we enrich ΛpEq with several marked points, whose knowledge allows to reconstruct E unambiguously: Definition 5.5. Fix a Newton lotus Λ. ' If Λ ‰ re 1 , e 2 s, we denote by B `Λ the compact and connected polygonal line defined as the complement of the open segment pe 1 , e 2 q in the boundary of the lotus Λ. If Λ " re 1 , e 2 s, we set B `Λ :" re 1 , e 2 s. The polygonal line B `Λ Ď Λ is called the lateral boundary of the lotus Λ.

' We denote by p Λ the homeomorphism p Λ : r0, 8s Ñ B `Λ which associates with any λ P r0, 8s the unique point p Λ pλq P B `Λ of slope λ. If Λ " ΛpEq where E Ď Q `Y t8u is finite and λ P E, then we call p ΛpEq pλq the marked point of λ (or of the ray of slope λ) in the lotus ΛpEq. We consider ΛpEq as a marked lotus using those marked points.

Remark 5.6. Notice that if λ P E, then p ΛpEq pλq is by construction the unique primitive element ppλq of the lattice N , which has slope λ relative to the basis pe 1 , e 2 q. Therefore, it is independent of the remaining elements of the set E.

We distinguish also by geometric properties several vertices of a Newton lotus: Definition 5.7. Assume that Λ is a Newton lotus. A vertex of Λ different from e 1 and e 2 is called a pinching point of the lotus Λ if it belongs to a unique petal of it. If the lotus Λ is two-dimensional, then the lattice point which is connected to e 2 (resp. to e 1 ) inside the lateral boundary B `Λ of Λ is called the last interior point (resp. first interior point) of the lateral boundary.

Remark 5.8. The pinching points of a Newton lotus ΛpEq are part of its marked points. Two Newton lotuses ΛpE 1 q and ΛpE 2 q coincide as unmarked simplicial complexes if and only if their sets of pinching points coincide. Example 5.9. In Figure 27 are represented the lotuses Λ p3{5q and ΛpEq, where E " t3{5, 2{1, 5{2u is the set whose fan FpEq was drawn in Figure 8. The lotus Λ p3{5q has only one pinching point, which is p p3{5q. The pinching points of ΛpEq are p p3{5q and p p5{2q. Its marked points are p p3{5q, p p2{1q and p p5{2q. This differentiates it from the lotus Λp3{5, 5{2q :" Λpt3{5, 5{2uq, which is the same simplicial complex if one forgets their respective marked points. The first interior point of ΛpEq is p p1{2q and its last interior point is p p3{1q. By comparing Figures 27 and9, which we combined in Figure 28, one sees that the lateral boundary of the lotus Λp3{5, 2{1, 5{2q is exactly the polygonal line constructed when one performed the regularization of the fan Fp3{5, 2{1, 5{2q (see Proposition 3.9). This is a general phenomenon, as shown by the following proposition.

Proposition 5.10. Let F be a fan subdividing the cone σ 0 . Then the regularization F reg of F is obtained by subdividing σ 0 using the rays generated by all the lattice points lying along the lateral boundary B `ΛpF q of the lotus ΛpFq.

Proof. Consider two successive marked points ppλq and ppµq of the lateral boundary B `ΛpF q. They are primitive elements of the ambient lattice N . Denote by ppλq `R`p pλq the closed half line originating from the point ppλq and generated by the vector ppλq. Consider analogously the half-line ppµq `R`p pµq. Let P pλ, µq be the polygonal line joining the points ppλq and ppµq inside B `ΛpF q. Consider the union of the three previous polygonal lines: Qpλ, µq :" pppλq `R`p pλqq Y P pλ, µq Y pppµq `R`p pµqq.

As the pinching points of ΛpFq belong to the marked points, this shows that there are no pinching points in the interior of the polygonal line P pλ, µq. Therefore, Qpλ, µq is the boundary of a closed convex set Qpλ, µq contained in the cone R `xppλq, ppµqy. The complement R `xppλq, ppµqy z Qpλ, µq is contained in the union of the complement ΛpFq z B `ΛpF q and the convex hull of the points 0, e 1 , e 2 deprived of the segment re 1 , e 2 s. Therefore, the origin 0 is the only point of N contained in R `xppλq, ppµqyz Qpλ, µq. As all the vertices of Qpλ, µq belong to N , this shows that Qpλ, µq is the convex hull of the set R `xppλq, ppµqy X pN z t0uq. One concludes using Proposition 3.9.

Consider again Figure 28. As shown by Proposition 3.24, the polygonal line on the left side gives a concrete embedding of the dual graph of the boundary BX F reg . But it does not show the order in which were performed the blow ups into which the associated modification ψ F σ0 : X F Ñ X σ0 decomposes (see Theorem 4.30). It turns out that this order is indicated by the lotus on the right side of Figure 28. To understand this fact, recall first the combinatorial description of the blow up of the orbit of dimension 0 of the smooth affine toric surface X σ0 , explained in Example 3.27: one gets it by subdividing the cone σ 0 using the ray generated by e 1 `e2 . In terms of the associated bases of N , one replaces the basis pe 1 , e 2 q by the pair of bases pe 1 , e 1 `e2 q and pe 1 `e2 , e 2 q. Graphically, this may be understood as the passage from the base re 1 , e 2 s of the petal δpe 1 , e 2 q seen as the simplest 2-dimensional lotus (see Definition 5.1) to its lateral boundary re 1 , e 1 `e2 s Y re 1 `e2 , e 2 s. Again by Proposition 3.24, we may see this passage as the replacement of the dual graph of BX σ0 by the dual graph of the boundary of the blown up toric surface. Now, each new petal in the lotus ΛpFq corresponds to the blow up of an orbit of dimension 0 of the previous toric surface. Its base may be seen as the dual graph of the irreducible components of the boundary meeting at that point. One gets: Proposition 5.11. Let F be a Newton fan. Then: ' The lateral boundary B `ΛpF q of the lotus ΛpFq is the dual graph of the boundary BX F reg of the smooth toric surface X F reg . Two vertices of it are joined by an edge of the lotus ΛpFq if and only if the corresponding orbits have intersecting closures at some moment of the process of creation of BX F reg by blow ups of orbits of dimension 0, which are particular infinitely near points of O σ0 P X σ0 . ' If one associates with each orbit of dimension 0 the corresponding petal of ΛpFq, then the parent map on the set of petals induces on the previous set of 0-dimensional orbits the restriction of the parent relation defined on the set of infinitely near points of O σ0 (see Definition 4.31).

Let us set a notation for the constellation created during a toric blow up process (see Definition 4.31): Definition 5.12. Let F be a Newton fan. Denote by C F the finite constellation above O σ0 consisting of the 0-dimensional orbits O σ , where σ varies among the regular 2-dimensional cones of the blow up process leading to the smooth toric surface X F reg . It is the constellation of the fan F.

Let σ be one of the cones mentioned in Definition 5.12. It is of the form R `xf 1 , f 2 y, where pf 1 , f 2 q is a positive basis of the lattice N . Proposition 5.11 shows that one may represent the 0-dimensional orbit O σ either by the edge rf 1 , f 2 s of the lotus ΛpFq or by the petal δpf 1 , f 2 q. How to understand the Enriques diagram of the constellation C F using the lotus ΛpFq? It turns out that this may be done easily using the representing edges rf 1 , f 2 s. In order to explain it, let us introduce first the following definition (see Figures 29 and30): Definition 5.13. Let δpf 1 , f 2 q be a petal of the universal lotus Λpe 1 , e 2 q. Assume that it is different from δpe 1 , e 2 q, which means that there exists a unique permutation pi, jq of p1, 2q such that f j ´fi P σ 0 X N (see Lemma 5.2). Then its Enriques edge is its lateral edge rf j , f 1 `f2 s, that is, its unique lateral edge which extends an edge of its parent petal. The Enriques tree of a lotus Λ is:

' the union of the Enriques edges of all its petals different from δpe 1 , e 2 q, rooted at its vertex e 1 `e2 , whenever Λ is of dimension 2; ' the vertex e 1 `e2 of δpe 1 , e 2 q, if Λ " re 1 , e 2 s. The extended Enriques tree of a lotus Λ is:

' the union of the Enriques subtree and of the lateral edge re 1 , e 1 `e2 s of the base petal δpe 1 , e 2 q of Λ, whenever Λ is of dimension 2; ' the lateral edge re 1 , e 1 `e2 s of δpe 1 , e 2 q, if Λ " re 1 , e 2 s. One has the following interpretation of the Enriques diagram of the constellation of the fan F using the lotus ΛpFq. It allows to understand for which reason we defined the Enriques tree of a lotus reduced to the base re 1 , e 2 s in the previous way: Proposition 5.14. Let F be a Newton fan. Then the Enriques diagram ΓpC F q of the constellation C F of F (see Definition 5.12) is isomorphic to the Enriques subtree of the lotus ΛpFq. This isomorphism sends each orbit O σ belonging to C F onto the point

f 1 `f2 , if σ " R `xf 1 , f 2 y.
Proof. The basic idea is that we have a bijection between the set of infinitely near points of O σ0 and the set of prime exceptional divisors created by blowing them up. Therefore, the parent binary relation may be thought as a binary relation on the set of those prime exceptional divisors. In this proposition, we restrict to the divisors which are the orbit closures O ρ , where ρ varies among the rays of the regularization F reg of F which are distinct from the edges of σ 0 . Each such a ray is generated by a lateral vertex of ΛpFq, therefore the parent binary relation among those orbit closures may be also seen as a binary relation among those lateral vertices. One may prove by induction on this number of rays, that is, on the number of petals of the associated lotus ΛpFq, that the pairs of related vertices are precisely those which are connected by an edge in the Enriques tree of ΛpFq.

The case F " σ 0 corresponds to a constellation formed by O σ0 alone. In this case one looks at the prime divisor created by blowing it up, that is, at O R`xe1`e2y . This explains why we defined ΓpC σ0 q as the vertex e 1 `e2 of the petal δpe 1 , e 2 q.

Remark 5.15. The reason why we introduced also the notion of extended Enriques tree in Definition 5.13, in addition to that of Enriques tree, will become clear after understanding point (8) of Theorem 5.29. Briefly speaking, the constellations associated to the toroidal pseudo-resolution processes have associated lotuses which are glued from lotuses of Newton fans. An analog of Proposition 5.14 is also true for them. The corresponding Enriques tree contains the Enriques trees of the Newton fans created by the toroidal process, but also other edges. Those supplementary edges are precisely the edges which have to be added to the Enriques tree of a Newton fan in order to get the corresponding extended Enriques tree (see Definition 5.26 below).

The lotus ΛpFq contains also the graph of the proximity binary relation on the constellation C F , whose set of vertices is the given constellation, two points being joined by an edge if and only if one of them is proximate to the other one (see Definition 4.31):

Proposition 5.16. Let F be a fan refining the regular cone σ 0 . Then the graph of the proximity binary relation on the finite constellation C F is isomorphic to the union of the edges of the lotus ΛpFq which do not contain the vertices e 1 and e 2 .

The proof of this proposition is based on the same principles as the proof of Proposition 5.14 and is left to the reader.

Lotuses and continued fractions.

In this subsection we explain a way to build, up to isomorphism, the lotus of a finite set of positive rational numbers in the sense of Definition 5.4, starting from the continued fraction expansions of its elements. Namely, given a positive rational number λ, we show how to construct an abstract lotus ∆pλq starting from the continued fraction expansion of λ (see Definition 5.18) and we explain that ∆pλq is isomorphic to the lotus Λpλq. Then we show how to glue two abstract lotuses ∆pλq and ∆pµq in order to get a simplicial complex isomorphic to the lotus Λpλ, µq (see Proposition 5.23). This extends readily to arbitrary finite sets of positive rationals.

Recall first the following classical notion:

Definition 5.17. Let k P N ˚and let a 1 , . . . , a k be natural numbers such that a 1 ě 0 and a j ą 0 if j P t2, . . . , ku. The continued fraction with terms a 1 , . . . , a k is the non-negative rational number:

ra 1 , a 2 , . . . , a k s :" a 1 `1 a 2 `1 ¨¨¨`1 a k . P 0 " A 2 P ´1 " A 1 P 1 P 2 P 3 " V ppr4, 2, 5sq A 2 A 1 Figure 31.
The construction of the abstract lotus ∆pr4, 2, 5sq

Any λ P Q ˚may be written uniquely as a continued fraction ra 1 , a 2 , . . . , a k s if one imposes the constraint that a k ą 1 whenever λ ‰ 1. One speaks then of the continued fraction expansion of λ. Note that its first term a 1 vanishes if and only if λ P p0, 1q.

Definition 5.18. Let λ P Q ˚. Consider its continued fraction expansion λ " ra 1 , a 2 , . . . , a k s. Its abstract lotus ∆pλq is the simplicial complex constructed as follows:

' Start from an affine triangle rA 1 , A 2 , V s, with vertices A 1 , A 2 , V . ' Draw a polygonal line P 0 P 1 P 2 . . . P k´1 whose vertices belong alternatively to the sides rA 1 , V s, rA 2 , V s, and such that P 0 :" A 2 and " P 1 P rA 1 , V q, with P 1 " A 1 if and only if a 1 " 0, P i P pP i´2 , V q for any i P t2, . . . , k ´1u.

By convention, we set also P ´1 :" A 1 , P k :" V . The resulting subdivision of the triangle rA 1 , A 2 , V s into k triangles is the zigzag decomposition associated with λ. ' Decompose then each segment rP i´1 , P i`1 s (for i P t0, . . . , k ´1u) into a i`1 segments, and join the interior points of rP i´1 , P i`1 s created in this way to P i . One obtains then a new triangulation of the initial triangle rA 1 , A 2 , V s, which is by definition the abstract lotus ∆pλq.

The base of the abstract lotus ∆pλq is the segment rA 1 , A 2 s, oriented from A 1 to A 2 . One orients also the other edges of ∆pλq in the following way: ' rP i´1 , P i s is oriented from P i to P i´1 , for each i P t1, . . . , k ´1u. ' An edge joining P i to a point of the open segment pP i´1 , P i`1 q is oriented towards P i . ' An edge contained in a segment rV, A j s is oriented towards A j , for each j P t1, 2u.

The abstract lotus ∆pλq of λ P Q ˚is a simplicial complex of pure dimension 2, isomorphic to a convex polygon triangulated by diagonals intersecting only at vertices and with a distinguished oriented base. It is well-defined, up to combinatorial isomorphism of polygons triangulated by diagonals intersecting only at vertices, respecting the bases and their orientations. The orientations of its other edges are in fact determined by the orientation of the base. Those orientations will not be important in the sequel, excepted in Proposition 5.21 below. For this reason we do not draw them in our examples of abstract lotuses.

Example 5.19. Figures 31 and32 represent the previous constructions applied to the numbers λ " r4, 2, 5s and µ " r3, 2, 1, 4s. On the left are shown the initial zigzag decompositions and on the right the final abstract lotuses ∆pλq and ∆pµq.

The abstract lotus of a positive rational number is isomorphic with its lotus: Proposition 5.20. There is a unique isomorphism between the lotus Λpλq and the abstract lotus ∆pλq, seen as simplicial complexes with a marked point and an oriented base.

Proof. The isomorphism sends A i to e i for i " 1, 2. The proof may be done by induction on k, the number of terms in the continued fraction expansion of λ. We leave the details to the reader.

P 0 " A 2 P ´1 " A 1 P 1 P 2 P 3 P 4 " V ppr3, 2, 1, 4sq A 2 A 1 Figure 32.
The construction of the abstract lotus ∆pr3, 2, 1, 4sq The previous isomorphism does not always send the orientations of the edges of Λpλq as chosen after Definition 5.1 onto the orientations of the edges of ∆pλq as fixed in Definition 5.18. The possibility of defining various canonical orientations on the edges of a lotus of the form Λpλq may be useful in applications.

A 2 A 1 V p 2 " 3 A 2 A 1 V p 1 " 2
The rational number λ ą 0 may be recovered in the following way from the structure of the corresponding abstract lotus: Proposition 5.21. Assume that λ " p 2 {p 1 with p 1 , p 2 P N ˚coprime. Then, for each j P t1, 2u, the positive integer p j is equal to the number of oriented paths not containing the base rA 1 , A 2 s and going from V to A j inside the 1-skeleton of ∆pλq, oriented as in Definition 5.18. This proposition may be easily proved by induction on the number of petals of ∆pλq. It shows a way in which the numbers leading to the construction of a Newton lotus may be interpreted as combinatorial invariants of the lotus, seen purely as a marked simplicial complex with oriented base.

Example 5.22. In Figure 33 is represented the case pp 1 , p 2 q " p2, 3q of Proposition 5.21. We have drawn twice the lotus ∆p3{2q " ∆pr1, 2sq. On the right are drawn the 2 oriented paths starting from V and arriving at A 1 . On the left are drawn the 3 oriented paths starting from V and arriving at A 2 . We see that the constraint not to contain the base is necessary, otherwise one would obtain 2 more paths from V to A 2 by adding the base to the paths from V to A 1 .

Suppose now that one has two numbers λ, µ P Q ˚. If λ " ra 1 , . . . , a k s and µ " rb 1 , . . . , b l s, let j P t0, . . . , mintk, luu be maximal such that a i " b i for all i P t1, . . . , ju. We may assume, up to permutation of λ and µ, that k " j or a j`1 ă b j`1 . Define then: [START_REF] García Barroso | Higher order polars of quasi-ordinary singularities[END_REF] λ ^µ " µ ^λ :"

$ & % ra 1 , . . . , a j s, if k " j, ra 1 , . . . , a j , a j`1 s , if k " j `1, ra 1 , . . . , a j , a j`1 `1s , if k ą j `1.
toroidal pseudo-resolution of C. We associated to it a fan tree pθ π pCq, S π q, as explained in Definition 4.33.

One may associate an analogous fan tree pθ π reg pCq, S π reg q to the toroidal resolution π reg : pΣ reg , BΣ reg q Ñ pS, L `L1 q defined in Subsection 4.3 (see Proposition 4.29). One sees that the trunks used in the two constructions are the same, as well as the gluing rules. What changes is that θ π reg pCq has more vertices than θ π pCq, those labeled by the irreducible components of the exceptional divisor of the modification η : Σ reg Ñ Σ which resolves the singularities of the surface Σ. Therefore:

Proposition 5.25. Seen as rooted trees endowed with r0, 8s-valued functions, the fan trees pθ π pCq, S π q and pθ π reg pCq, S π reg q coincide. The second one contains more vertices than the first one, labeled by the irreducible components of the exceptional divisor of the minimal resolution η : Σ reg Ñ Σ. The fan tree θ π reg pCq of the toroidal resolution π reg is isomorphic to the dual graph of the boundary BΣ reg by an isomorphism which respects the labels of the irreducible components.

The disadvantage of the fan tree pθ π reg pCq, S π reg q is that one cannot see on it at a glance the partial order of the blow ups leading to the resolution π reg : Σ Ñ S of C. We explained in Subsection 5.1 that this order may be visualized by using the notion of lotus, for each Newton modification of the regularized algorithm obtained by replacing STEP 3 with STEP 3 reg . In order to visualize the blow up structure of the resolution process leading to the modification π reg : pΣ reg , BΣ reg q Ñ pS, L `L1 q, we glue those lotuses using the same rules as those allowing to construct the fan tree from its trunks (see Definition 4.33): Definition 5.26. Let C be a reduced curve singularity and pL, L 1 q be a cross on the smooth germ pS, oq. The lotus Λ π pCq of the toroidal pseudo-resolution π : pΣ, BΣq Ñ pS, L `L1 q of C is a simplicial complex of dimension 2 endowed with a marked oriented edge called its base. It is obtained by gluing the disjoint union of the lotuses pΛpF Ai,Bi pCqqq iPI in the following way:

(1) Label each vertex of those lotuses with the corresponding irreducible component E k , L j or C l of the boundary BΣ reg of the smooth toroidal surface pΣ reg , BΣ reg q. (2) Identify all the vertices of Ů iPI ΛpF Ai,Bi pCqq which have the same label. The result of this identification is Λ π pCq and the images inside it of the labeled points of Ů iPI ΛpF Ai,Bi pCqq are its vertices. We keep for each one of them the same label as in the initial lotuses. Introduce the following terminology for the anatomy of Λ π pCq: ' The petals of Λ π pCq are the images by the gluing morphism of the petals of the initial lotuses pΛpF Ai,Bi pCqqq iPI . ' Its base is the edge labeled by the initial cross pL, L 1 q and its basic petal is the petal having it as base. ' Its basic vertices are the images inside it of the basic vertices of the 2-dimensional lotuses pΛpF Aj ,Bj pCqqq jPJ which were not identified with other vertices.

' Its lateral boundary B `Λπ pCq is the image by the gluing morphism of the union of the lateral boundaries pB `ΛpF Ai,Bi pCqqq iPI in the sense of Definition 5.5. ' Its lateral vertices are the vertices of Λ π pCq which are not basic. ' Its membranes are the images inside it of the lotuses ΛpF Ai,Bi pCqq used to construct it. ' Its Enriques tree is the union of the Enriques tree of ΛpF A1,B1 pCqq (remember that pA 1 , B 1 q " pL, L 1 q) and of the extended Enriques trees of the other Newton fans ΛpF Ai,Bi pCqq (see Definition 5.13).

We introduce the notion of Enriques tree of a lotus in order to be able to state point (8) of Theorem 5.29 below. See also Remark 5.15.

Remark 5.27. The lateral boundary B `Λπ pCq is a covering subtree of the 1-skeleton of the lotus Λ π pCq, that is, a subtree containing all of its vertices. The membranes of Λ π pCq may be obtained by removing all the vertices of Λ π pCq and by taking the closures inside Λ π pCq of the connected components of the resulting topological space. The lotus Λ π pCq is a flag complex, that is, it may be reconstructed from its 1-skeleton by filling each complete subgraph with k vertices by a pk ´1q-dimensional simplex. It turns out that there are such complete subgraphs only for k P t1, 2u, for which values of k the filling process adds nothing new, and for k " 3, for which one gets all the petals of the lotus.

F L,L1 pCq L 0 L 1 8 E 1 3 5 E 2 2 1 E 3 5 2 θ pF L,L1 pCqq E 3 E 2 E 1 L L 1 Λ ˆ3 5 , 2 1 , 5 2 ėL2 e E1 e E5 e E4 3 4 2 3
F E1,L2 pCq

E 1 0 L 2 8 E 4 2 3 E 5 3 4 θ pF E1,L2 pCqq E 4 E 5 E 1 L 2 Λ ˆ2 3 , 3 4 ėL3 e E1 e E7 e E6 3 1 5 3
F E1,L3 pCq

E 1 0 L 3 8 E 6 5 3 E 7 3 1 θ pF E1,L3 pCqq E 6 E 7 E 1 L 3 Λ ˆ5 3 , 3 1
ėL4 e E6 e E8 1 2
F E6,L4 pCq The associated lotus Λ π pCq is represented in Figure 36. It has 4 membranes of dimension 2 and 7 membranes of dimension 1. The oriented base of each lotus ΛpF Ai,Bi pCqq used to construct it is indicated in red. The base of Λ π pCq is the oriented edge whose vertices are labeled by L and L 1 . The basic vertices of Λ π pCq are those labeled by L, L 1 , L 2 , L 3 , L 4 . The part of the lateral boundary B `Λπ pCq contained in the 2-dimensional lotuses pΛpF Aj ,Bj pCqqq jPJ is represented in orange. In order to get the whole lateral

E 6 0 L 4 8 E 8 1 2 θ pF E6,L4 pCqq E 8 L 4 E 6 Λ ˆ1 2
C 4 C 5 E 1 C 1 C 3 C 2 C 6 E 7 L 4 C 7 E 4 E 5 L 2 L 0 " L L 1 E 3 E 2 L 3 E 8 E 6 Figure 36.
The lotus of the toroidal pseudo-resolution of Example 5.28 boundary, one has to add the 1-dimensional lotuses of the fans associated to the crosses at which one stops at STEP 1, that is, the segments rE 3 , C 1 s, rE 2 , C 2 s, rE 2 , C 3 s, rE 5 , C 4 s, rE 4 , C 5 s, rE 7 , C 6 s and rE 8 , C 7 s.

L 0 L 1 8 E 1 3 5 E 2 2 1 E 3 5 2 C 1 8 C 2 8 C 3 8 L 2 8 E 4 2 3 E 5 3 4 C 5 8 C 4 8 L 3 8 E 7 3 1 E 6 5 3 C 6 8 L 4 8 E 8 1 2 C 7 8 θ π pCq C 4 C 5 C 1 C 3 C 2 C 6 L 4 C 7 E 4 E 5 L 2 L 0 " L L 1 E 3 E 2 L 3 E 8 E 6 E 7 E 1 Λ π pCq
In Figure 37 are represented side by side the fan tree θ π pCq and the lotus Λ π pCq. Note that the fan tree is homeomorphic (forgetting the values of the slope function at its vertices) with the lateral boundary B `Λπ pCq, by a homeomorphism which preserves the labels. This is a general fact, as formulated in point (4) of Theorem 5.29 below. This homeomorphism is not an isomorphism of trees because some of the edges of the fan tree -the blue ones -get subdivided in the lateral boundary of the lotus. Those are precisely the edges which correspond to the singular points of the surface Σ. One may see on the lateral boundary the structure of the exceptional divisor of the minimal resolution of each such point.

For instance, the intersection point of the curves E 1 and E 6 on Σ gets resolved by replacing that point with an exceptional divisor with two components. Their self-intersection numbers in the smooth surface Σ reg are ´4 and ´3, as results from point (5) of Theorem 5.29.

Here comes the announced visualization of the structure of the decomposition of the modification π reg : Σ reg Ñ S into blow ups of points in terms of the anatomy of the lotus Λ π pCq (see Definition 5.26):

Theorem 5.29. Let C be a reduced curve singularity on the smooth germ of surface pS, oq. Consider a toroidal pseudo-resolution π : pΣ, BΣq Ñ pS, L `L1 q of C produced by Algorithm 4.22. Its lotus Λ π pCq represents the following aspects of the associated embedded resolution π reg : pΣ reg , BΣ reg q Ñ pS, L `L1 q:

(1) Its basic edges represent the crosses with respect to which STEP 2 of Algorithm 4.22 was applied.

(2) Its basic vertices represent the branches pL j q jPJ of the crosses used during the process, which were introduced each time one executed STEP 2. (3) Its lateral vertices represent the irreducible components E k of the exceptional divisor pπ reg q ´1poq of the smooth modification π reg : Σ reg Ñ S. (4) Its lateral boundary B `Λπ pCq is the dual graph of the boundary divisor BΣ reg and is homeomorphic with the fan tree θ π reg pCq, by a homeomorphism which respects the labels. (5) The opposite of the number of petals of Λ π pCq containing a given lateral vertex is the selfintersection number of the irreducible component of pπ reg q ´1poq represented by that lateral vertex. (6) The edges of Λ π pCq represent the affine charts used in the decomposition of π into a composition of blow ups of points, and the pairs of irreducible components of pπ reg q ´1p ř jPJ L j q which are strict transforms of crosses used at some stage of the composition of blow ups. [START_REF] A'campo | Le groupe de monodromie du déploiement des singularités isolées de courbes planes I[END_REF] The graph of the proximity binary relation on the constellation which is blown up is the full subgraph of the 1-skeleton of the lotus Λ π pCq on its set of non-basic vertices. Example 5.30. Assume that C is a branch. Its fan tree θ π pCq is a segment rL, Cs. Denote its interior vertices by P 1 ă L ¨¨¨ă L P k " P , with k ě 1. Here ĺ L denotes the total order on θ π pCq induced by the root L. Consider the continued fraction expansions of their slopes S π pP j q " rp j , q j , . . . s, for all j P t1, . . . , ku. Then the lotus Λ π pCq is represented in Figure 38. We explain in Examples 6.32 and 6.33 below how to give examples of branches which admit a pseudo-resolution process with such a lotus.

Example 5.31. Let us consider again our recurrent example of toroidal pseudo-resolution. Its associated lotus was represented in Figure 36. In Figure 39 are represented the Enriques trees and extended Enriques trees of its membranes of dimension 2. Finally, in Figure 40 is represented its full Enriques tree. In this figure we have also represented the petals associated to the pairs pE i , C j q, in order to draw the end edges of the Enriques tree.

The dependence of the lotus on the choice of completion.

In this subsection we show using two examples that the lotus Λ π pCq of a toroidal pseudo-resolution process π of a plane curve singularity C ãÑ S depends on the choice of auxiliary curves added each time one executes STEP 2 of Algorithm 4.22, that is, on the choice of completion Ĉπ of C (see Definition 4.15).

In the following two examples 5.32 and 5.33, we build the lotuses Λ π pCq associated with two distinct embedded resolutions π : pΣ, BΣq Ñ pS, BSq of the curve singularity C " Zpf q, defined by the power series f :" y 2 ´2xy `x2 ´x3 P Crrx, yss, relative to local coordinates px, yq on the germ pS, oq. These examples illustrate the fact that the associated lotus Λ π pCq (see Definition 5.26), which is based on the toroidal structure of Σ, depends on the choices of auxiliary curves done at STEP 2 of the Algorithm 4.22, Example 5.32. We start the algorithm by choosing L 1 :" Zpy ´xq. The cross pL, L 1 q at o is defined by the local coordinate system px, y 1 :" y ´xq. Relative to these coordinates, C has local equation y 2 1 ´x3 " 0. The Newton polygon N L,L1 pCq has only one edge and its orthogonal ray has slope 3{2, hence F L,L 1 pCq » Fp3{2q. The first trunk is just the segment re L , e L1 s with its point of slope 3{2 marked.

L 1 L q 1 p 1 L 2 q 2 p 2 L k q k p k C Figure
The Newton modification π :" ψ C, reg L,L1 : pΣ, BΣq Ñ pS, BSq has three exceptional divisors E 1 , E 2 and E 3 which correspond to the rays of the regularization F reg p3{2q " Fp1, 2, 3{2q of the fan Fp3{2q of slopes 1 and 2 and 3{2 respectively. In this case, the strict transform C L,L1 of C is smooth and intersects transversally the component E 3 of the exceptional divisor, that is, the Newton modification π is an embedded resolution of C. Note that when running the algorithm 4.22, we include the cross pE 3 , C L,L1 q in the toroidal structure of the boundary of Σ.

The lotus Λ π pCq is built by gluing the lotus Λ L,L1 pCq " Λp3{2q with the lotus re E3 , Cs associated to the cross pE 3 , C L,L1 q, identifying the points labeled by E 3 (see Figure 41).

Example 5.33. We start the algorithm by choosing L 1 :" Zpyq and the cross pL, L 1 q on pS, oq. The Newton polygon N L,L1 pCq has only one edge and its orthogonal ray has slope 1, hence F L,L1 pCq » Fp1q. The first trunk is the segment re L , e L1 s with its midpoint marked. The first lotus is just the petal Λ 1 :" Λp1q " δpe L , e L1 q with base re L , e L1 s.

The Newton modification ψ C L,L1 is the usual blow up of the point o. We restrict it to the chart C 2 v1,v2 , where x " v 1 , y " v 1 v 2 . The strict transform C 1 :" C L,L1 is defined in this chart by the equation v 2 2 ´2v 2 `1 ´v1 " 0. The exceptional divisor E 1 :" Zpv 1 q intersects the strict transform C 1 at the point o 1 defined by v 2 " 1. When running the algorithm, we have to choose a smooth branch B 2 such that pE 1 , B 2 q defines a cross at o 1 . We set B 2 :" Zpv 2 ´1q and u 1 :" v 2 ´1. Then, the local coordinates pv 1 , u 1 q define the cross pE 1 , B 2 q. We denote by L 2 the projection to S of the line B 2 " Zpu 1 q, which is parametrized by v 1 " t and v 2 " 1. One gets that L 2 , which is parametrized by x " t, y " t, has local equation y ´x " 0. The strict transform C 1 has local equation u 2 1 ´v1 " 0. The Newton polygon N E1,B2 pC 1 q has only one edge and its orthogonal ray has slope 1{2, hence its associated fan is F E1,B2 pC 1 q » Fp1{2q. The second trunk is just the segment re E1 , e L2 s with a marked point of slope 1{2. The modification ψ C1,reg E1,B2 defined by the regularization of this fan has two exceptional divisors E 2 and E 3 corresponding to the rays of the regularization of the fan Fp1{2q of slopes 1 and 1{2 respectively. When we consider the regularization of the fan F E1,B2 pC 1 q, we have to mark an additional point of slope 1 in the second trunk re E1 , e L2 s. The associated lotus is Λ 2 :" Λp1{2q, with base re E1 , e L2 s. In this example, the composition π :" ψ C1,reg E1,B2 ˝ψC L,L1 : pΣ, BΣq Ñ pS, BSq is an embedded resolution of C, since the strict transform C 2 of C is smooth and intersects transversally the exceptional divisor of π at a point o 2 P E 3 . Notice that when running the algorithm, we have to consider also the cross pE 3 , C 2 q at o 2 . Its trunk coincides with its associated lotus. It is just the segment Λ 3 :" re E3 , Cs, with no marked points.

E 3 E 2 E 1 L L 1 Λ ˆ3 5 , 2 1 , 5 2
˙E3 E 2 E 1 L L 1 Λ ˆ3 5 , 2 1 , 5 2
Ė4 E 5 E 1 L 2 Λ ˆ2 3 , 3 4
˙E4 E 5 E 1 L 2 Λ ˆ2 3 , 3 4 Ė6 E 7 E 1 L 3 Λ ˆ5 3 , 3 1
˙E6 E 7 E 1 L 3 Λ ˆ5 3 , 3 1
Ė8 L 4 E 6 Λ ˆ1 2 ˙E8 L 4 E 6 Λ ˆ1 2
C 4 C 5 E 1 C 1 C 3 C 2 C 6 E 7 L 4 C 7 E 4 E 5 L 2 L 0 " L L 1 E 3 E 2 L 3 E 8 E 6
The lotus Λ π pCq is represented in Figure 43. It is obtained from Λ 1 , Λ 2 and Λ 3 (see Figure 42) by identifying the points with the same label.

Remark 5.34. The lotus Λ π pCq may be embedded canonically into the set of semivaluations of the local C-algebra ÔS,o (semi-valuations are defined similarly to valuations, but dropping the last condition from Definition 2.19). Indeed, its base membrane ΛpF L,L1 pCqq embeds into the regular cone σ L,L1 0 of Definition 3.32, which may be interpreted valuatively by associating to each w P σ L,L1 0 the valuation ν w defined by Equation [START_REF] Dumas | Sur les singularités des surfaces[END_REF]. Each other membrane may be similarly interpreted valuatively, and one may show that one gets in this way an embedding. Details may be found in [102, Section 7].

Truncated lotuses.

In this subsection we introduce an operation of truncation of the lotus of a toroidal pseudo-resolution of a plane curve singularity C, and we explain how to use it in order to visualize the dual graph of Recall first from Definition 5.26 the construction of the lotus Λ π pCq of a toroidal pseudo-resolution π : pΣ, BΣq Ñ pS, L `L1 q of the curve singularity C ãÑ S. As stated in point (4) of Theorem 5.29, its lateral boundary B `Λπ pCq is isomorphic to the dual graph of the boundary divisor BΣ reg . Here Σ reg denotes the minimal resolution of Σ, and BΣ reg is the total transform on it of the boundary divisor BΣ of the toroidal surface pΣ, BΣq. The divisor BΣ reg is also the total transform of the completion Ĉπ of C relative to π, that is, the sum of the total transform of C by the smooth modification π reg : Σ reg Ñ S and of the strict transforms of the branches L j introduced while running Algorithm 4. [START_REF] Cassou-Noguès | Newton trees for ideals in two variables and applications[END_REF].

How to get the dual graph of the total transform of C on Σ reg from the lateral boundary B `Λπ pCq? One has simply to remove the ends of B `Λπ pCq which are labeled by the branches L j , as well as the edges which connect them to other vertices of B `Λπ pCq. This truncation operation performed on the tree B `Λπ pCq may be seen as the restriction of a similar operation performed on the whole lotus Λ π pCq. Let us explain this truncation operation on Λ π pCq, as well as some of its uses.

Consider first a petal δpe 1 , e 2 q associated to a base pe 1 , e 2 q of a lattice N (see Definition 5.1). Its axis is the median rpe 1 `e2 q{2, e 1 `e2 s of the petal, joining the vertex e 1 `e2 to the midpoint of the opposite edge. This axis decomposes the petal into two semipetals.

The semipetals of a lotus are the semipetals of all its petals. Using this vocabulary, as well as that introduced in Definition 5.26 about the anatomy of lotuses of toroidal pseudo-resolutions, we may define now the operation of truncation of such a lotus: Definition 5.35. Let Λ π pCq be the lotus of a toroidal pseudo-resolution π of the plane curve singularity C ãÑ pS, oq. Its truncation Λ tr π pCq is the union of the axis of its basic petal, of all the semipetals which do not contain basic vertices and of all the membranes which are segments, that is, of the edges of Λ π pCq which have an extremity labeled by a branch of C. The lateral boundary B `Λtr π pCq of Λ tr π pCq is the part of the lateral boundary of Λ π pCq which remains in Λ tr π pCq. Truncating the lotus Λ π pCq corresponds to forgetting its points whose corresponding semivaluations depend on the choice of the branches L j . One keeps only those semivaluations determined by the given curve singularity C and by the infinitely near points through which pass its strict transforms during the blow up process (see Remark 5.34). In fact, the third author had introduced truncated lotuses in [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF] -under the name of sails -as objects which represent the combinatorial type of a blow up process of a finite constellation, without considering any supplementary branches passing through the points of the constellation.

By construction, the lateral boundary B `Λtr π pCq is isomorphic to the dual graph of the total transform pπ reg q ˚pC q. One may read again the self-intersection number of each irreducible component of the exceptional divisor of π reg as the opposite of the number of petals, semi-petals and axis containing the vertex representing it.

Note that both lotuses of Figures 41 and43 have the same truncations. The reason is that their associated toroidal pseudo-resolutions lead to the same embedded resolution of C by regularization and that the truncated lotus is a combinatorial object encoding the decomposition of this resolution into blow ups of points.

Example 5.36. For instance, in Figure 45 is shown the truncation of the lotus of Figure 36. Its lateral boundary is emphasized using thick orange segments. The component of the exceptional divisor represented by the unique vertex of the lotus contained in the axis has self-intersection number ´4, as this vertex is contained in the axis, in two semi-petals and in one petal of Λ tr π pCq. Consider now the Enriques tree of the toroidal pseudo-resolution π. Its edges are certain lateral edges of the 2-dimensional petals of Λ π pCq and of the 2-dimensional petals constructed from the 1-dimensional petals of Λ π pCq as bases (see Definition 5.26). For each edge rA, Bs of the Enriques tree, one may consider instead the homothetic segment p1{2qrA, Bs, joining the points p1{2qA and p1{2qB. This homothety is well-defined if one interprets the elements of the segment rA, Bs as valuations (see Remark 5.34). If both A and B are vertices of the same petal, then the segment p1{2qrA, Bs joins two edge midpoints of this petal. Otherwise, the interior points of the segment p1{2qrA, Bs are disjoint from the lotus Λ π pCq.

The union of such segments p1{2qrA, Bs -which were called ropes by the third author in [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF] -is isomorphic to the Enriques tree of π. Therefore it is another representation of the Enriques diagram of the constellation whose blow up creates the resolution π reg .

Figure 46. Two ways of visualizing the Enriques tree on a truncated lotus

It is convenient to draw in a same picture both the truncation Λ tr π pCq and the union of the ropes. For instance, for the lotus of Figure 36 this union is represented on the right side of Figure 46. For comparison, the Enriques tree is represented on the left side. An advantage of the right-side drawing is that the ropes whose interiors lie outside the truncation are exactly the ropes which were represented by Enriques as curved arcs. One may similarly determine from this drawing which edges go straight in Enriques' convention. For details, one may consult [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF]Thm. 6.2]. Note that the kites of the title of [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF] (in French cerf-volants) were the unions of truncated lotuses and of their ropes, as represented on the right side of Figure 46.

Assume now that the combinatorial type of a plane curve singularity is given either using the dual graph of its total transform by an embedded resolution, weighted by the self-intersection numbers of the components of its exceptional divisor, or using the Enriques diagram of the decomposition of the resolution morphism into blow ups of infinitely near points of o. How to get a series f P Crrx, yss defining a curve singularity with the given combinatorial type?

One may apply the following steps:

' Pass from the given tree to the associated truncated lotus. If the given tree is an Enriques diagram, it may be more convenient for drawing purposes to think about it as the union of ropes of the truncated lotus which is searched for. ' Complete the truncated lotus into a lotus having it as truncation. This step is not canonical, as shown by the comparison of Figures 41 and43 above. ' Proceed as in Example 6.29 below, by constructing the fan tree of the lotus, then the associated Eggers-Wall tree and writing finally a finite set of Newton-Puiseux series whose associated Eggers-Wall tree is isomorphic with this one.

5.6. Historical comments.

The study of plane curve singularities by using sequences of blow ups of points was initiated by Max Noether in his 1875 paper [START_REF] Noether | Ueber die singulären Werthsysteme einer algebraischen Function und die singulären Punkte einer algebraischen Curve[END_REF], and became common in the meantime, as shown by the works [START_REF] Noether | Les combinaisons caractéristiques dans la transformation d'un point singulier[END_REF] of Noether, [START_REF] Enriques | Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche II[END_REF] of Enriques and Chisini, [START_REF] Val | Reducible exceptional curves[END_REF] of Du Val and [134, Sections I.2, II.2], [START_REF] Zariski | Polynomial ideals defined by infinitely near base points[END_REF] of Zariski.

Nowadays, a modification of C 2 obtained as a sequence of blow ups of points is studied most of the time through the structure of its exceptional divisor. One encodes the incidences between its components, as well as their self-intersection numbers in a weighted dual graph, which is a tree (see [START_REF] Popescu-Pampu | From singularities to graphs[END_REF] for a description of the development of this idea). When one looks at an embedded resolution of the plane curve singularity C, one adds new vertices to this graph, corresponding to the strict transforms of the branches of C.

Figure 47. An Enriques diagram

The dual trees of exceptional divisors were not the first graphs associated with a process of blow ups of points. Another kind of tree, an Enriques diagram, encoding the proximity relation between the infinitely near points which are blown up in the process (see Definition 4.31), was associated with such a process in the 1917 book [START_REF] Enriques | Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche II[END_REF] of Enriques and Chisini. An example of an Enriques diagram, extracted from [35, Page 383], may be seen in Figure 47. Details about the notion of Enriques diagram may be found in Casas' book [START_REF] Casas-Alvero | Singularities of plane curves[END_REF] or in the third author's papers [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF], [START_REF] Pereira | Fibonacci numbers and self-dual lattice structures for plane branches[END_REF], the second written in collaboration with Pe Pereira. The proximity relation was extended to higher dimensions by Semple in his 1938 paper [START_REF] Semple | Singularities of space algebraic curves[END_REF]. Details about this generalization and about other approaches to the study of curve singularities of higher embedding dimension may be found in Campillo and Castellanos' 2005 book [START_REF] Campillo | Curve singularities. An algebraic and geometric approach[END_REF].

In order to understand the relation between the Enriques diagram of a finite constellation and the dual graph of the blow up of the constellation, the third author introduced the notion of kite in his 2011 paper [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF]. A kite was defined by gluing lotuses into a sail, and attaching then ropes to this sail. The ropes were lying inside each lotus as the veins in a leaf, and they allowed to visualize the Enriques diagram. In turn, the dual graph could be visualised as the lateral boundary of the sail. A sail was composed not only of petals, but also of axes and semi-petals. The lotuses were also used in Castellini's thesis [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF], written under the supervision of the third author. Castellini was able to do everything with petals, eliminating the use of axes, semi-petals and ropes, as what we call here the Enriques tree of a lotus proved to be more convenient to visualize the Enriques diagram. Also, the terminology was simplified, the gluing of lotuses resulting again in lotuses, instead of sails, as we do in the present paper.

It turns out that lotuses already appeared in disguise before the paper [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF]. Their oldest ancestor is probably the proximity relation, defined in Enriques and Chisini's book [START_REF] Enriques | Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche II[END_REF]Page 381]. Indeed (see Theorem 5.29 [START_REF] A'campo | Le groupe de monodromie du déploiement des singularités isolées de courbes planes I[END_REF]), the graph of the proximity relation among all the points whose blow up composes the embedded resolution produced by the second algorithm described in our paper may be identified with the full subgraph of the 1-skeleton of the associated lotus on the set of vertices which are not basic. The oldest drawings of such proximity graphs seem to be those of Du Val's 1944 paper [START_REF] Val | On absolute and non-absolute singularities of algebraic surfaces[END_REF] (see Figures 48 and49, in which one may also recognize what we call the "Enriques tree" of a lotus, drawn with continuous segments). Before, the proximity binary relation was related to the exceptional divisor of the associated blow up process in Barber and Zariski's 1935 paper [START_REF] Barber | Reducible exceptional curves of the first kind[END_REF] and Du Val's 1936 paper [START_REF] Val | Reducible exceptional curves[END_REF]. Du Val introduced the notion of proximity matrix, equivalent to that of proximity binary relation. In his 1939 paper [START_REF] Zariski | The reduction of the singularities of an algebraic surface[END_REF], Zariski began a new ideal-theoretical and valuation-theoretical trend in the study of infinitely near points. A geometrical presentation of the previous approaches of study of infinitely near points was given by Lejeune-Jalabert in her 1995 paper [START_REF] Lejeune-Jalabert | Linear systems with infinitely near base conditions and complete ideals in dimension two[END_REF].

The graph of the proximity relation was mentioned again by Deligne in his 1973 paper [START_REF] Deligne | Intersections sur les surfaces régulières[END_REF], by Morihiko Saito in his 2000 paper [START_REF] Saito | Exponents of an irreducible plane curve singularity[END_REF] and by Wall in his 2004 book [START_REF] Wall | Singular points of plane curves[END_REF]Sections 3.5,3.6]. One may find drawings of simple such graphs only in the first and the third reference.

Another occurrence of lotuses in disguise may be found in Schulze-Röbbecke's 1977 Diplomarbeit [START_REF] Schulze-Röbbecke | Algorithmen zur Auflösung und Deformation von Singularitäten ebener Kurven[END_REF] written under the supervision of Brieskorn. In that paper are described particular divides (generic immersions of segments in a disc) obtained by applying to branches A'Campo's method of constructing δ-constant deformations explained in the 1974-75 papers [6] and [START_REF] A'campo | Le groupe de monodromie du déploiement des singularités isolées de courbes planes I[END_REF]. The diagram of Figure 50, extracted from page 57 of [START_REF] Schulze-Röbbecke | Algorithmen zur Auflösung und Deformation von Singularitäten ebener Kurven[END_REF], indicates the general shape of the divides constructed in that paper. One may recognize inside it part of the lotus associated to a toroidal resolution process of a branch. In his already mentioned 2015 PhD thesis [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF], Castellini could extend Schulze-Röbbecke's description to arbitrary plane curve singularities, using in a crucial way the notion of lotus of a blow up process. Page 116], in reference to the 1858 paper [START_REF] Stern | Über eine zahlentheoretische Funktion[END_REF] of Stern and the 1860 paper [START_REF] Brocot | Calcul des rouages par approximation, nouvelle méthode[END_REF] of Brocot. This tree represents the successive generation of the positive rational numbers starting from the sequence p0{1, 1{0q. At each step of the generating process, one performs the Farey addition pa{b, c{dq Ñ pa `cq{pb `dq on the pairs of successive terms of the increasing sequence of rationals obtained at the previous steps. The vertices of the Stern-Brocot tree correspond bijectively with the positive rationals. For each Farey addition pa{b, c{dq Ñ pa `cq{pb `dq in which c{d was created after a{b, one joins the vertices corresponding to c{d and to pa `cq{pb `dq. The embedding of the Stern-Brocot tree represented in Figure 29 is obtained by sending each vertex corresponding to λ P Q X p0, 8q to the primitive vector ppλq P N X σ 0 (see Notations 3.2) and each edge to a Euclidean segment. Another embedding in the cone σ 0 of the same part of the Stern-Brocot tree as above was described in [START_REF] Popescu-Pampu | Le cerf-volant d'une constellation[END_REF]Rem. 5.7]. That embedding may be obtained from the embedding of Figure 29 by applying a homothety of factor 1{2.

The sequence of continued fractions [START_REF] García Barroso | An approach to plane algebroid branches[END_REF] appearing in the proof of Proposition 5.23 was called the slow approximation ("approximation lente") of ra 1 , . . . , a k s in Lê, Michel and Weber's paper [82,Appendice]. They used such sequences in order to describe the construction of the dual graph of the minimal embedded resolution of a plane curve singularity starting from the generic characteristic exponents of its branches and the orders of coincidence between such branches.

The zigzag decompositions introduced in Definition 5.18 are a variant of the zigzag diagrams of the third's author 2007 paper [START_REF] Popescu-Pampu | The geometry of continued fractions and the topology of surface singularities. Dans Singularities in Geometry and Topology[END_REF]Section 5.2]. Those diagrams allow to relate geometrically the usual continued fractions to the so-called Hirzebruch-Jung continued fractions. Those Hirzebruch-Jung continued fractions are the traditional tool, going back to Jung's 1908 paper [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] and Hirzebruch's 1953 paper [START_REF] Hirzebruch | Über vierdimensionale Riemannsche Flächen Mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen[END_REF], to describe the regularization of a 2-dimensional strictly convex cone. They are also crucial for the understanding of lens spaces, which becomes obvious once one sees that those 3-manifolds are exactly the links of toric surface singularities. See Weber's survey [START_REF] Weber | Lens spaces among 3-manifolds and quotient surface singularities[END_REF] for more details and historical explanations about the relations between lens spaces and complex surface singularities.

In [87, Section 9.1], Neumann and Wahl described a method for reconstructing the dual graph of the minimal resolution of a complex normal surface singularity whose link is an integral homology sphere from the so-called splice diagram of the link. This method is based on the construction of a finite sequence of rationals interpolating between two given positive rational numbers λ and µ. It may be described in the following way using lotuses of sequences of positive rational numbers: ' Construct by successive additions of petals the lotus Λpλ, µq as the union of Λpλq and Λpµq. ' Consider the increasing sequence of slopes of vertices of Λpλ, µq lying between λ and µ, that is, of vertices of the lateral boundary B `Λpλ, µq (see Definition 5.5) lying on the arc joining the primitive vectors ppλq and ppµq of N .

In [40, Section 2.2], Fock and Goncharov described the tropical boundary hemisphere of the Teichmüller space of the punctured torus as an infinite simplicial complex with integral vertices embedded in the real affine space associated to a two-dimensional affine lattice. This simplicial complex is a union of universal lotuses (see [START_REF] Fock | Cluster Poisson varieties at infinity[END_REF]Fig. 1]).

Relations of fan trees and lotuses with Eggers-Wall trees

In Subsection 6.1 we explain how to associate an Eggers-Wall tree Θ L pCq to a plane curve singularity C ãÑ pS, oq, relative to a smooth branch L. It is a rooted tree endowed with three structure functions, the index i L , the exponent e L and the contact complexity c L . In Subsection 6.2 we express the Newton polygon of C relative to a cross pL, L 1 q in terms of the Eggers-Wall tree Θ L pC `L1 q of C `L1 relative to L (see Corollary 6.17). In Subsection 6.5 we prove that the fan tree θ π pCq associated with a toroidal pseudoresolution process of C is canonically isomorphic with the Eggers-Wall tree Θ L p Ĉπ q of the completion of C relative to this process (see Theorem 6.27), and we explain how to compute the triple pi L , e L , c L q of functions starting from the slope function of the fan tree (see Proposition 6.28). As a prerequisite, in Subsections 6.3 and 6.4 we prove renormalization formulae, which compare the Eggers-Wall tree of C relative to L and those of its strict transform relative to the exceptional divisor of a Newton modification. 6.1. Finite Eggers-Wall trees and the universal Eggers-Wall tree.

In this subsection we define the Eggers-Wall tree Θ L pCq of a reduced plane curve singularity C ãÑ pS, oq relative to a smooth branch L (see Notations 6.7). It is constructed from the Newton-Puiseux series of C relative to a local coordinate system px, yq such that L " Zpxq (see Definition 6.3), but it is independent of this choice (see Proposition 6.6). It is a rooted tree whose root is labeled by L and whose leaves are labeled by the branches of C. It is endowed with three functions, the index i L , the exponent e L and the contact complexity c L , which allow to compute the characteristic exponents of the Newton-Puiseux series mentioned above and the intersection numbers of the branches of C (see Proposition 6.11). Finally, we introduce the universal Eggers-Wall tree of pS, oq relative to L (see Definition 6.12), as the projective limit of the Eggers-Wall trees of the plane curve singularities contained in S. For more details and proofs one may consult our papers [START_REF] García Barroso | Ultrametric spaces of branches on arborescent singularities[END_REF]Subsection 4.3] and [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]Section 3].

Let L be a smooth branch on pS, oq. Assume in the whole subsection that C is reduced. Let px, yq be a local coordinate system on pS, oq, such that L " Zpxq, and let f P Crrx, yss be a defining function of C in this coordinate system. As a consequence of the Newton-Puiseux Theorem 2.20, one has: Theorem 6.1. Assume that C does not contain L, that is, that x does not divide f px, yq. Then there exists a finite set Z x pf q of Newton-Puiseux series of Crrx 1{N ss and a unit upx, yq of the local ring Crrx, yss, such that: [START_REF] Goldin | Resolving singularities of plane analytic branches with one toric morphism[END_REF] f px, yq " upx, yq ź ηpxqPZxpf q py ´ηpxqq.

The set Z x pf q is obviously independent of the defining function f of C. For this reason, we will denote it instead Z x pCq. It is the disjoint union of the sets Z x pC l q, when C l varies among the branches of C. It allows to associate to f the following objects: Definition 6.2. Let px, yq be a local coordinate system of pS, oq such that L " Zpxq and let C be a reduced curve singularity on pS, oq not containing L.

' The finite subset Z x pCq :" Z x pf q from the statement of Theorem 6.1 is called the set of Newton-Puiseux roots of C relative to x. ' The order of coincidence k x pξ, ξ 1 q of two Newton-Puiseux series ξ, ξ 1 is equal to ν x pξ ´ξ1 q.

' The order of coincidence k x pC l , C m q of two distinct branches C l and C m of C is the maximal order of coincidence of Newton-Puiseux roots of the two branches: maxtk x pξ, ξ 1 q, ξ P Z x pC l q, ξ 1 P Z x pC m qu.

' The set of characteristic exponents Ch x pC l q of a branch C l of C relative to the variable x is the set of orders of coincidence of pairs of distinct Newton-Puiseux roots of it: tk x pξ, ξ 1 q, ξ, ξ 1 P Z x pC l q, ξ ‰ ξ 1 u. This shows that for each ξ P Z x pC l q, there exists some ξ 1 P Z x pC m q such that ν x pξ ´ξ1 q " k x pC l , C m q. Therefore, knowing a Newton-Puiseux root of C l determines some Newton-Puiseux root of C m until their order of coincidence k x pC l , C m q. This fact motivates the following construction of a rooted tree endowed with two functions: Definition 6.3. Let px, yq be a local coordinate system such that L " Zpxq and C be a reduced curve singularity on pS, oq.

' The Eggers-Wall tree Θ x pC l q of a branch C l ‰ L of C relative to x is a compact segment endowed with a homeomorphism e x : Θ x pC l q Ñ r0, 8s called the exponent function, and with marked points, which are the preimages by the exponent function of the characteristic exponents of C l relative to x. The point pe x q ´1p0q is labeled by L and pe x q ´1p8q is labeled by C l . The index function i x : Θ x pC l q Ñ N ˚whose value i x pP q on a point P P Θ x pC l q is equal to the lowest common multiple of the denominators of the exponents of the marked points belonging to the half-open segment rL, P q.

' The Eggers-Wall tree Θ x pLq is reduced to a point labeled by L, at which e x pLq " 0 and i x pLq " 1.

' The Eggers-Wall tree Θ x pCq of C relative to x is obtained from the disjoint union of the Eggers-Wall trees Θ x pC l q of its branches by identifying, for each pair of distinct branches C l and C m of C, their points with equal exponents not greater than the order of coincidence k x pC l , C m q. Its marked points are its ramification points and the images of the marked points of the trees Θ x pC l q by the identification map. Its labeled points are analogously the images of the labeled points of the trees Θ x pC l q, the identification map being label-preserving. The tree is rooted at the point labeled by L. It is endowed with an exponent function e x : Θ x pCq Ñ r0, 8s and an index function i x : Θ x pCq Ñ N ˚obtained by gluing the exponent functions and index functions on the trees Θ x pC l q respectively. Note that, by construction, the exponent function is surjective in restriction to every segment rL, C l s " Θ x pC l q of Θ x pCq such that C l ‰ L and that the ends of Θ x pCq are labeled by the branches of C and by the smooth reference branch L. The marked points of Θ x pCq which are images of marked points of the subtrees Θ x pC l q may be recovered from the knowledge of the index function, as its set of points of discontinuity. Therefore, the index function is constant on each open edge between two consecutive marked points of Θ x pCq. Moreover, it is continuous from below relative to the partial order ĺ L defined by the root L of Θ x pCq.

The Eggers-Wall tree allows to determine visually the characteristic exponents of each branch C l . One has simply to follow the segment going from the root to the leaf representing the branch and to read all the vertex weights of the discontinuity points of the index function. In particular, if an internal vertex of such a segment is not a ramification vertex of the tree, then its exponent is necessarily a characteristic exponent of C l . 0 Zpxq Zpy 3 ´x7 q Example 6.4. Consider again the plane curve singularity C " Zpf px, yqq of Subsection 2.6. That is, f px, yq " py 2 ´4x 3 qpy 3 ´x7 q. Its Eggers-Wall tree is drawn on the left side of Figure 51. On the right side is drawn the Eggers-Wall tree of the singularity Zpxypy 2 ´4x 3 qpy 3 ´x7 qq, which is the sum of C and of the coordinate axes.

Look at the segment joining the root to the branch Zpy 3 ´x7 q, on the left side of Figure 51. It contains two internal vertices, with exponents 3{2 and 7{3. The vertex of exponent 7{3 is not a ramification vertex of the tree, therefore 7{3 is a characteristic exponent of this branch. In turn, 3{2 is not a characteristic exponent of this branch, as the value of the index function does not increase when crossing the corresponding vertex. Note that, by contrast, it increases when crossing the same vertex on the segment joining the root to the leaf corresponding to the branch Zpy 2 ´4x 3 q, which shows that 3{2 is a characteristic exponent of that branch.

We have represented both the Eggers-Wall tree of C and of its union with the coordinate axes in order to show that the second one is homeomorphic to the dual graph of the total transform of the union by its minimal embedded resolution, while our example shows that this is not true if one looks at the total transform of C alone (see Figure 7). The previous homeomorphism is a general phenomenon, valid for any plane curve singularity, as seen by combining Proposition 4.35 and Theorem 6.27 below. Note that in full generality one needs to add to C more branches than simply the coordinate axes, considering a completion in the sense of Definition 4.15.

Example 6.5. Consider a plane curve singularity C whose branches C i , 1 ď i ď 3, are defined by the Newton-Puiseux series ξ i , where:

ξ 1 " x 7{2 ´x4 `2x 17{4 `x14{3 , ξ 2 " x 5{2 `x8{3 , ξ 3 " x 2 .
The sets of characteristic exponents of the branches are Ch x pC 1 q " t7{2, 17{4, 14{3u, Ch x pC 2 q " t5{2, 8{3u, Ch x pC 3 q " H. One has k x pC 1 , C 2 q " 5{2, k x pC 1 , C 3 q " k x pC 2 , C 3 q " 2. The Eggers-Wall trees Θ x pC 1 q and Θ x pCq relative to x are drawn in Figure 52. We represented the value of the corresponding exponent near each marked or labeled point, and the value of the corresponding index function near each edge.

In fact, the objects introduced in Definition 6.3 depend only on C and L, not on the coordinate system px, yq such that L " Zpxq (see [START_REF] García Barroso | Ultrametric spaces of branches on arborescent singularities[END_REF]Proposition 103]): Proposition 6.6. Let px, yq be a local coordinate system such that L " Zpxq and C be a reduced curve singularity on pS, oq. Then the tree Θ x pCq endowed with the pair of functions pi x , e x q is independent of the choice of local coordinate system such that L " Zpxq. Proposition 6.6 motivates us to introduce the following notations: Notations 6.7. Let L be a smooth branch and C be a reduced curve singularity on pS, oq. We denote pΘ L pCq, i L , e L q :" pΘ x pCq, i x , e x q, for any local coordinate system px, yq on pS, oq such that L " Zpxq. We say that this rooted tree endowed with two structure functions is the Eggers-Wall tree of C relative to L. Remark 6.8. Let L be a smooth branch and C be a reduced curve singularity on pS, oq. Then for any point Q P Θ L pCq, we have: [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF] i L pQq " minti L pAq , A is a branch on S such that Q ĺ L Au, where Q ĺ L A has a meaning in the Eggers-Wall-tree Θ L pC `Aq Ě Θ L pCq. Indeed, if Q ĺ L C l for a branch C l of C, and if B is a branch on S parametrized by the truncation of a Newton-Puiseux series ξ P Z x pC l q, obtained by keeping only the terms of ξ of exponent ă e L pQq, then Q ĺ L B and i L pQq " i L pBq.

The exponent function and the index function determine a third function on the tree Θ L pCq, the contact complexity function (see [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]Def. 3.19]): Definition 6.9. Let C be a reduced curve singularity on pS, oq. The contact complexity function c L : Θ L pCq Ñ r0, 8s is defined by the formula:

c L pP q :" ż P L de L i L .
Note that in restriction to a segment rL, C l s " Θ L pC l q of Θ L pCq, the contact complexity function is a bijection rL, C l s Ñ r0, 8s. Remark 6.10. It follows immediately from Definition 6.9 that the contact complexity function together with the index function determine the exponent function by the following formula: [START_REF] González Pérez | Approximate roots, toric resolutions and deformations of a plane branch[END_REF] e L pP q "

ż P L i L dc L .
The importance of the contact complexity function stems from the following property, which in different formulation goes back at least to Smith [START_REF] Smith | On the higher singularities of plane curves[END_REF]Section 8], Stolz [START_REF] Stolz | Die Multiplicität der Schnittpunkte zweier algebraischer Curven[END_REF]Section 9] and Max Noether [START_REF] Noether | Les combinaisons caractéristiques dans la transformation d'un point singulier[END_REF]: Proposition 6.11. Let L be a smooth branch and C be a reduced curve singularity on pS, oq, not containing L. Let A and B be two distinct branches of C. Denote by A ^L B the infimum of the points of Θ L pCq labeled by A and B, relative to the partial order ĺ L defined by the root L. Then: [START_REF] González Pérez | Multi-Harnack smoothings of real plane branches[END_REF] c L pA ^L Bq " A ¨B pL ¨Aq ¨pL ¨Bq .

Proof. One may find a proof of Proposition 6.11 in [START_REF] Wall | Singular points of plane curves[END_REF]Thm. 4 The finite multi-set of rational numbers whose elements are summed may be expressed in terms of the characteristic exponents of A and B which are not greater than the order of coincidence of A and B. A little computation finishes the proof.

If C and D are two reduced plane curve singularities on pS, oq, with C Ď D, then by construction one has a natural embedding of rooted trees Θ L pCq Ď Θ L pDq. The uniqueness of the segment joining two points of a tree allows to define a canonical retraction Θ L pDq Ñ Θ L pCq. One may consider then either the direct limit of the previous embeddings, or the projective limit of the previous retractions, for varying C and D. Both limits have natural topologies. The direct limit, which may be thought simply as the union of all Eggers-Wall trees pΘ L pCqq C , is not compact, but the projective limit is compact. It is in fact a compactification of the direct limit. For this reason, the projective limit is more suitable in many applications. Let us introduce a special notation for this notion, which will be used in Subsection 6.3 below. Definition 6.12. Let L be a smooth branch on pS, oq. The universal Eggers-Wall tree Θ L of pS, oq relative to L is the projective limit of the Eggers-Wall trees Θ L pCq of the various reduced curve singularities C on pS, oq, relative to the natural retraction maps Θ L pDq Ñ Θ L pCq associated to the inclusions C Ď D.

From Eggers-Wall trees to Newton polygons.

In this subsection we explain how the Newton polygon N L,L 1 pCq of a plane curve singularity C relative to the cross pL, L 1 q (see Definition 4.14) may be determined from the Eggers-Wall tree Θ L pC `L1 q (see Corollary 6.17).

The Minkowski sum K 1 `K2 of two subsets of a real vector space is the set of sums v 1 `v2 , where each v i varies independently among the elements of K i . It is a commutative and associative operation. When both subsets are convex, their Minkowski sum is again convex.

The following property is classical and goes back at least to Dumas' 1906 paper [30, Section 3] (where it was formulated in a slightly different, p-adic, context): Proposition 6.13. If C and D are germs of effective divisors on pS, oq, then:

N L,L 1 pC `Dq " N L,L 1 pCq `NL,L 1 pDq.
Proof. This is a direct consequence of formula [START_REF] Enriques | Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche II[END_REF] and Proposition 4.12.

One may extend the notion of Newton polygon to series in two variables with non-negative rational exponents whose denominators are bounded. They have again only a finite number of edges. The simplest Newton polygons are those with at most one compact edge: Definition 6.14. Assume that a, b P Q ˚. One associates them the following elementary Newton polygons (see Figure 53):

x a y b x a y b x a y b pa, 0q p0, bq p0, bq pa, 0q ! a b ) :" N px a `yb q ! a 8
) :" N px a q ! 8
! a b ) :" N px a `yb q, ! a
) :" N px a q, ! 8
) :" N py b q.

The quotient a{b is the inclination of the elementary Newton polygon ! a b

) .

Note that for any a P Q Proof. This is a consequence of the following property, which in turn may be proved by induction on p P N ˚: If N 1 , N 2 , . . . , N p are elementary Newton polygons with finite non-zero strictly increasing inclinations, then their Minkowski sum N has exactly p compact edges which are translations of the compact edges of N 1 , N 2 , . . . , N p . Moreover, they are met in this order when one lists them starting from the unique vertex of N lying on the vertical axis.

The next proposition explains how to compute the Newton polygon of a branch C relative to a cross pL, L 1 q, starting from the Eggers-Wall tree of C `L1 relative to L: Lemma 6.16. Let pL, L 1 q be a cross and let C ‰ L be a branch on pS, oq. Then the Newton polygon N L,L 1 pCq may be expressed as follows in terms of the Eggers-Wall tree pΘ L pC `L1 q, e L , i L q:

N L,L 1 pCq " i L pCq " e L pC ^L L 1 q 1 * .
The fan F L,L 1 pCq has a unique ray in the interior of the cone σ 0 , and its slope is equal to e L pC ^L L 1 q. That is:

F L,L 1 pCq " F `eL pC ^L L 1 q ˘.
Proof. This is a consequence of Theorem 6.1. Indeed, let f P Crrxssrys be a defining function for C relative to a local coordinate system px, yq defining the cross pL, L 1 q. We know that its set of Newton-Puiseux roots Z x pf q has C ¨L " i L pCq elements. All of them have the same support, since C is a branch, which implies that they form a single orbit under the Galois action of multiplication of x 1{i L pCq by the group of i L pCq-th roots of 1. The order of any such series is equal to k x pL 1 , Cq " e L pC ^L L 1 q. We deduce from Proposition 6.13 that the Newton polygon N L,L 1 pCq is equal to the Minkowski sum of the factors of f in formula [START_REF] Goldin | Resolving singularities of plane analytic branches with one toric morphism[END_REF]. The first assertion follows since the Newton polygon of y ´ηpxq is equal to " e L pC ^L L 1 q 1 * , for any series ηpxq P Z x pf q, and then by taking into account formula [START_REF] González Pérez | Motivic Milnor fiber of a quasi-ordinary hypersurface[END_REF]. The second assertion is an immediate consequence of the first one.

As a corollary we get the announced expression of the Newton polygon relative to pL, L 1 q of a reduced curve singularity C in terms of the Eggers-Wall tree Θ L pC `L1 q of C `L1 relative to L: Corollary 6.17. Let pL, L 1 q be a cross and let C be a reduced curve singularity on pS, oq not containing the branch L. The Newton polygon N L,L 1 pCq of the germ C with respect to the cross pL, L 1 q is equal to the Minkowski sum:

(56) ÿ l i L pC l q " e L pC l ^L L 1 q 1 * ,
where C l runs through the branches of C.

Proof. By Proposition 6.13, the Newton polygon N L,L 1 pCq is the Minkowski sum of the Newton polygons of its branches. One uses then Lemma 6.16 for each such branch.

Note that the previous result extends to not necessarily reduced curve singularities C if one defines their Eggers-Wall tree as the Eggers-Wall tree of their reduction, each leaf being endowed with the multiplicity of the corresponding branch in the divisor C. Then, in the right-hand side of Equation [START_REF] González Pérez | Toric geometry and the Semple-Nash modification[END_REF], each branch C l has to be counted with its multiplicity.

Renormalization of Eggers-Wall trees.

Let pL, L 1 q be a cross on pS, oq. In this subsection we will denote sometimes by Θ o,L pCq the Eggers-Wall tree denoted before by Θ L pCq, in order to emphasize the point at which it is based. Indeed, we want to compare the previous tree with the Eggers-Wall tree Θ ow,Ew pC w q of the germ pC w , o w q of the strict transform C w of C at a smooth point o w of the exceptional divisor E w of a Newton modification relative to the cross pL, L 1 q, with respect to the germ at o w of the exceptional divisor E w itself. Notice that if C is a reduced curve, then the strict transform C w may consist of several germs of curves, one for each point of intersection of C w with E w . We show that the universal Eggers-Wall tree Θ ow,Ew in the sense of Definition 6.12 embeds naturally in the universal Eggers-Wall tree Θ o,L and we explain how to relate their triples of structure functions (index, exponent and contact complexity). We conceive the passage from Θ o,L pCq to Θ ow,Ew pC w q as a renormalization operation, which explains the title of this subsection. We will give another proof of the renormalization Proposition 6.22 in Section 6.4, in terms of Newton-Puiseux series.

Let us fix a cross pL, L 1 q on pS, oq. Fix also a weight vector w " c w e 1 `dw e 2 P σ 0 X N L,L 1 . Denote by π w : pS w , BS w q Ñ pS, L `L1 q the modification obtained by subdividing σ 0 along the ray R `w. If A is a branch on S, we denote by A w the strict transform of A by π w . We look at the modification π w in the toroidal category, relative to the boundaries BS :" L `L1 and BS w :" L w `Ew `L1 w , where E w is the exceptional divisor of the morphism π w .

Denote by W the point of Θ L pL 1 q corresponding to w, that is, the unique point of Θ L pL 1 q whose exponent is the slope of the ray R `w in the basis pe 1 , e 2 q: [START_REF] Graham | Concrete Mathematics. A foundation for computer science[END_REF] e L pW q " d w c w .

(4) c L pA ^L Bq " 1 c 2 w c Ew pA w ^Ew B w q `dw c w .

Proof. Notice first that the hypothesis and Lemma 6.18 imply that the strict transforms A w , B w of A and B by π w intersect E w z pL w Y L 1 w q. If C is a branch on pS, oq, denote by pπ ẘC q ex the exceptional part of the total transform divisor pπ ẘC q " pπ ẘC q ex `Cw of C on S w .

Proof of (1). We have:

L

¨A piq "" pπ ẘLq ¨pπ ẘAq " piiq "" pπ ẘLq ¨Aw " piiiq "" pπ ẘLq ex ¨Aw " pivq "" ord Ew pLqpE w ¨Aw q " pvq "" ν w pχ 1 qpE w ¨Aw q " pviq "" ppc w e 1 `dw e 2 q ¨ 1 qpE w ¨Aw q " pviiq "" c w pE w ¨Aw q.

Let us explain each one of the previous equalities:

' Equality piq results from the birational invariance of the intersection product, if one works with total transforms of divisors. ' Equality piiq is a consequence of the equality pπ ẘLq ¨pπ ẘAq ex " 0, which results from the projection formula (see [START_REF] Hartshorne | Algebraic geometry[END_REF]Appendix A1]), applied to the divisors L on S, pπ ẘAq ex on S w and to the proper morphism π w . ' Equality piiiq follows from the hypothesis L w ¨Aw " 0 and the bilinearity of the intersection product. ' Equality pivq is a consequence of the equality pπ ẘLq ex " ord Ew pLqE w . ' Equality pvq results from the equalities ord Ew " ν w (see Equation 32) and x " χ 1 . ' Equality pviq results from the fact that w " c w e 1 `dw e 2 . ' Equality pviiq results from the fact that p 1 , 2 q is the dual basis of pe 1 , e 2 q.

L 1 w L w B w A w A 1 w E w L L 1 A A 1 B W Figure 54. The choice of branch A 1 in the proof of Proposition 6.20 (2)
Proof of (2). Let us choose a branch A 1 on pS, oq such that: [START_REF] Greuel | Introduction to singularities and deformations[END_REF] i L pAq " i L pA 1 q and W " A ^L L 1 " A 1 ^L L 1 .

Using Lemma 6.19, we can translate this hypothesis in terms of the total transform of the branches A, A 1 by π w . On the left side of Figure 54 is represented the total transform of L `L1 `A `A1 `B by π w and on its right side is represented the Eggers-Wall tree Θ L pL `L1 `A `A1 `Bq, for some branch B. Then:

A ¨B piq "" pπ ẘAq ¨pπ ẘBq " piiq "" pπ ẘAq ¨Bw " piiiq "" A w ¨Bw `pπ ẘAq ex ¨Bw " pivq "" A w ¨Bw `pπ ẘA 1 q ex ¨Bw " pvq "" A w ¨Bw `pπ ẘA 1 q ¨Bw " pviq "" A w ¨Bw `A1 ¨B " pviiq "" A w ¨Bw `pL ¨A1 qpL ¨Bq c L pW q " pviiiq "" A w ¨Bw `pL ¨AqpL ¨Bq d w c w .

Let us explain each one of the previous equalities:

' Equalities piq and piiq are analogs of the equalities piq and piiq in the proof of point (1) above. ' Equality piiiq results from the bilinearity of the intersection product. ' Equality pivq results from the hypothesis (59) and Lemma 6.18, which imply that ord Ew pAq " ord Ew pA 1 q. Then one concludes using the equality pπ ẘC q ex ¨Bw " ord Ew pCqpE w ¨Bw q, for each C P tA, A 1 u. ' Equality pvq results from the fact that, by construction, A 1 w and B w are disjoint. ' Equality pviq results from the projection formula. ' Equality pviiq results from Lemma 6.19. ' Equality pviiiq results from Equation ( 58) and from the equality L ¨A " L ¨A1 , which is a consequence of the hypothesis (59) and the equality L ¨C " i L pCq for each C P tA, A 1 u.

Proof of (3). By hypothesis, the strict transforms A w and B w intersect the set E w zpL w Y L 1 w q, which is equal to the torus orbit O R`w . By the proof of Proposition 4.18, this implies that w is orthogonal to the compact edges of the Newton polygons N L,L 1 pAq and N L,L 1 pBq. Lemma 6.16 implies that e L pW q " e L pA ^L L 1 q " e L pB ^L L 1 q. As the three points W, A ^L L 1 , B ^L L 1 belong to the segment rL, L 1 s and that e L is strictly increasing on it, we get the equalities W " A ^L L 1 " B ^L L 1 . This implies that W ĺ L A, W ĺ L B. The claim follows from point (2) by using Lemma 6.19.

Proof of (4). Dividing both sides of the formula of point (2) by the product pL ¨AqpL ¨Bq, we get: A ¨B pL ¨Aq ¨pL ¨Bq " A w ¨Bw pL ¨Aq ¨pL ¨Bq `dw c w .

Using point (1), we get: A w ¨Bw pL ¨Aq ¨pL ¨Bq " 1 c 2 w A w ¨Bw pE w ¨Aw q ¨pL w ¨Bw q .

By applying formula (54) twice we obtain the desired formula:

(60) c L pA ^L Bq " 1 c 2 w c Ew pA w ^Ew B w q `dw c w .
Let us define in combinatorial terms a natural embedding of the universal Eggers-Wall tree Θ ow,Ew into the universal Eggers-Wall tree Θ o,L (see 6.12): Definition 6.21. Let A w be a branch on the germ of surface pS w , o w q. Denote by A its image by the modification π w . The natural embedding of the universal Eggers-Wall tree Θ ow,Ew into the universal Eggers-Wall tree Θ o,L is defined by sending each point Q of the Eggers-Wall segment Θ ow,Ew pA w q to the unique point Q 1 of Θ o,L pAq which satisfies:

(61) c L pQ 1 q " 1 c 2 w c Ew pQq `dw c w .
If pC w , o w q is a reduced curve on pS w , o w q, then the embedding of the Eggers-Wall tree Θ ow,Ew pC w q in Θ o,L pCq is well-defined thanks to Formula (60) applied to any pair A w , B w of branches of pC w , o w q. That is, the embeddings of the Eggers-Wall segments of its branches glue into an embedding of Θ ow,Ew pC w q in Θ o,L pCq. Notice that the root E w of Θ ow,Ew pC w q corresponds to the point W P Θ o,L pL 1 q defined by relation [START_REF] Graham | Concrete Mathematics. A foundation for computer science[END_REF] and that the leaf of Θ ow,Ew labeled by A w corresponds to the leaf of Θ o,L labeled by A.

The following proposition describes how to pass from the functions pi Ew , e Ew q on the tree Θ ow,Ew pC w q to the functions pi L , e L q on Θ o,L pCq: Proposition 6.22. Let pC w , o w q be a reduced curve singularity on pS w , o w q. Identify the tree Θ ow,Ew pC w q with the subtree of Θ o,L pCq defined by the natural embedding of Definition 6.21. One has the following relations in restriction to this subtree:

(1) i L " c w i Ew .

(2) e L " 1 c w e Ew `dw c w .

Proof.

Proof of (1). We show first the assertion for an end of Θ ow,Ew pC w q corresponding to a branch B w of C w . By the definition of the index function, we have the equalities i L pBq " L ¨B and i Ew pB w q " E w ¨Bw . Combining these equalities with point (1) of Proposition 6.20, we get:

(62) i L pBq " c w i Ew pB w q.

Let Q ‰ E w be any rational point of Θ ow,Ew pC w q. By the equality ( 52), there exists a branch A w on the germ of surface pS w , o w q such that i Ew pA w q " i Ew pQq. We get:

i L pQq p52q ď i L pAq p62q " c w i Ew pA w q " c w i Ew pQq.
This implies that i L pQq ď c w i Ew pQq. Analogously, using again equality [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF], there exists a branch B on the germ pS, oq such that W ă L Q ă L B and i L pBq " i L pQq. By Definition 6.21 of the natural embedding of Θ ow,Ew in Θ o,L , this implies that Q ă Ew B. Therefore:

i L pQq " i L pBq p62q " c w i Ew pB w q p52q ě c w i Ew pQq.
It follows that i L pQq " c w i Ew pQq. We have shown that the equality in point (1) holds in restriction to the rational points of Θ ow,Ew pC w q, and by the continuity properties of the index functions, it holds for every point of Θ ow,Ew pC w q.

Proof of (2). Let P be a point of Θ ow,Ew pC w q. This implies that W ĺ L P . By the integral formula (53), we get:

e L pP q " ż P L i L dc L " ż W L i L dc L `ż P W i L dc L .
Using again equation ( 53), we have:

(63) ż W L i L dc L " e L pW q " d w c w .
We compute the second integral ş P W i L dc L by making a change of variable. Differentiating formula (61), we get dc L " p1{c 2 w qdc Ew . Using the expression for i L of point (1), we obtain:

(64)

ż P W i L dc L " 1 c w ż P W i Ew dc Ew " 1 c w e Ew pP q,
where we have used again the integral formula [START_REF] González Pérez | Approximate roots, toric resolutions and deformations of a plane branch[END_REF]. We end the proof by combining the equalities ( 63) and ( 64):

e L pP q " d w c w `1 c w e Ew pP q.

Remark 6.23. Identify the tree Θ ow,Ew with the subtree of the universal Eggers-Wall tree Θ o,L defined by the embedding of Definition 6.21. As a consequence of Proposition 6.22, the two formulae stated in it also hold on Θ ow,Ew .

6.4. Renormalization in terms of Newton-Puiseux series.

We give a different proof of Proposition 6.22 by using Newton-Puiseux series. This proof relates the Newton modifications in the toroidal category of Definition 4.14 with the Newton maps, which appear sometimes in the algorithmic construction of Newton-Puiseux series (see Subsection 6.6).

We keep the notations introduced at the beginning of Section 6.3. Let A be a branch on pS, oq such that A w intersects E w at a point o w P E w zpL w Y L 1 w q. Consider local coordinates px, yq defining the cross pL, L 1 q. Recall from Definition 6.2 that Z x pAq denotes the set of Newton-Puiseux roots of A relative to x. Let us choose η P Z x pAq. It may be expressed as:

(65) η " ÿ kěm α k x k{n ,
where n " A ¨L, m " A ¨L1 . Hence α m ‰ 0. All the series in Z x pAq have the same support, since they form a single orbit under the Galois action of multiplication of x 1{n by the complex n-th roots of 1 (see Remark 2.21). Let us denote p :" gcdpn, mq. Our hypothesis that A w meets E w zpL w Y L 1 w q implies that: [START_REF] De Jong | Local analytic geometry[END_REF] n " c w ¨p, m " d w ¨p.

The branch A is defined by f " 0, where:

(67) f " ź γ n "1
py ´pγ ¨ηqpxqq " py cw ´αcw m x dw q p `. . . .

We have only written on the right-hand side of (67) the restriction of f to the unique compact edge of the Newton polygon of f px, yq.

Lemma 6.24. There exist local coordinates px 1 , y 1 q on the germ pS w , o w q such that E w " Zpx 1 q and the map π w is defined by:

"

x " x cw 1 , y " x dw 1 pα m `y1 q. Proof. Consider a vector w 1 " a w e 1 `bw e 2 such that: [START_REF] Kapranov | Veronese curves and Grothendieck-Knudsen moduli space M 0[END_REF] b w c w ´aw d w " 1.

Therefore the cone σ " R `xw, w 1 y is regular and included in one cone of the fan F L,L 1 pAq. As explained in the proof of Proposition 4.18, we can look at the intersection of A w with the orbit O R`w " E w zpL w Y L 1 w q in the open subset corresponding to this orbit on the toric surface X σ " C 2 u,v . The toric morphism ψ σ σ0 is the monomial map defined by "

x " u cw v aw y " u dw v bw (see Example 3.26). The orbit O R`w , seen on the surface C 2 u,v , is the pointed axis C v . The maximal monomial in pu, vq which divides pψ σ σ0 q ˚f is equal to pu cwdw v awdw q p . After factoring out this monomial and setting u " 0 we get: [START_REF] Kapranov | The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation[END_REF] pv awcw´bwdw ´αcw m q p p69q " pv ´αcw m q p . This shows that the point o w has coordinates pu, vq " p0, α cw m q. The formulae

" u " x 1 py 1 `αm q ´aw , v " py 1 `αm q cw , define local coordinates px 1 , y 1 q at o w , since the jacobian determinant of pu, v ´αcw m q with respect to px 1 , y 1 q does not vanish at p0, 0q. Notice also that Zpx 1 q " Zpuq " E w . By (69) we get: "

x " x cw 1 py 1 `αm q ´awcw py 1 `αm q awcw " x cw 1 y " x dw 1 py 1 `αm q bwcw´dwaw " x dw 1 pα m `y1 q.

Proposition 6.25. With respect to the coordinates px 1 , y 1 q introduced in Lemma 6.24, the series

η w :" ÿ kąm α m x pk´mq{p 1
, is a Newton-Puiseux series parametrizing the branch A w on pS w , o w q.

Proof. By formula (70), we have that pA w ¨Ew q ow " p. It follows that the Newton-Puiseux series in Z x1 pA w q must have exponents in p1{pqN ˚. By composing [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] with the change of variable (72)

x 1 " x p 2 , we get: (73)

"

x " x n 2 , y " x dwp 2 pα m `y1 q. Apply the substitution [START_REF] Khovanskii | Newton polyhedra, and toroidal varieties[END_REF] to the factor y ´pγ ¨ηqpx 1{n q, using that x 2 " x 1{n by definition, and factor out the monomial x dwp 2 . We get the series [START_REF] Kouchnirenko | Polyèdres de Newton et nombres de Milnor[END_REF] pα m `y1 q ´αm γ m ´ÿ kąm α k γ k x k´m 2 P Crrx 2 , y 1 ss.

This series has vanishing constant term if and only if γ m " 1. Since γ n " 1 and gcdpn, mq " p, one may check that this condition holds if and only if γ p " 1, and in this case for any k ą m one has γ k " γ k´m . It follows that the series (74) which are non-units are precisely the conjugates of the series y 1 ´ηw px 1{p 1 q under the Galois action, since x 2 " x 1{p 1 by definition [START_REF] Kennedy | Monodromy of plane curves and quasi-ordinary surfaces[END_REF]. Therefore, the product of all the conjugates of y 1 ´ηw px 1{p 1 q under the Galois action defines a polynomial in Crrx 1 ssry 1 s which divides the strict transform of f by the map [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]. The remaining factor is a series with nonzero constant term, and must belong to the ring Crrx 1 , y 1 ss since it is invariant under the Galois action. Corollary 6.26. Let A, B be two branches on pS, oq such that o w P A w X B w X E w . Then:

k x pA, Bq " d w c w `cw ¨kx1 pA w , B w q.
Proof. By point (3) of Proposition 6.20, the inequality A w ¨Bw ą 0 (which results from the hypothesis that o w P A w X B w ) implies that k x pA, Bq ą d w {c w . It follows that if we fix a Newton-Puiseux series η P Z x pAq, then there exists ξ P Z x pBq with the same order and the same leading coefficient. We can apply Lemma 6.24, using this leading coefficient, to define suitable local coordinates px 1 , y 1 q at the point o 1 . The formula results from Proposition 6.25 by taking into account the facts that η w P Z x1 pA w q and ξ w P Z x1 pB w q.

Corollary 6.26 implies readily Proposition 6.22.

6.5. From fan trees to Eggers-Wall trees.

In this subsection we assume that C is reduced. We explain that there exists a canonical isomorphism from the fan tree θ π pCq of a toroidal pseudo-resolution π of C produced by running Algorithm 4.22, to the Eggers-Wall tree of the completion Ĉπ of C (see Theorem 6.27). We also explain how to compute the index, exponent and contact complexity functions on the Eggers-Wall tree from the slope function on the fan tree (see Proposition 6.28).

Let L be a smooth branch on the germ pS, oq. Assume that we run Algorithm 4.22, arriving at a toroidal pseudo-resolution π : pΣ, BΣq Ñ pS, L `L1 q. Consider the corresponding completion Ĉπ , in the sense of Definition 4.15. There are two trees associated with this setting which have their ends labeled by the branches of Ĉπ , the fan tree θ π pCq and the Eggers-Wall tree Θ L p Ĉπ q. How are they related? It turns out that they are isomorphic: Theorem 6.27. There is a unique isomorphism from the fan tree θ π pCq to the Eggers-Wall tree Θ L p Ĉπ q, which preserves the labels of the ends of both trees by the branches of Ĉπ .

Proof. At the first step of Algorithm 4.22, one chooses a smooth branch L 1 such that pL, L 1 q is a cross on pS, oq. By definition, the branch L 1 is a component of the completion Ĉπ . Let us consider the segment rL, L 1 s of Θ L p Ĉπ q and the first trunk θ F L,L 1 pCq " re L , e L 1 s. We have a homeomorphism Ψ o : re L , e L 1 s Ñ rL, L 1 s " Θ L pL 1 q sending a vector w P re L , e L 1 s to the unique point W P rL, L 1 s whose exponent e L pW q is equal to the slope of w with respect to the basis pe L , e L 1 q of N L,L 1 . By Corollary 6.17, the map Ψ o defines also a bijection between the set of marked points of the trunk, according to Definition 4.33, and the set of the marked points of the tree Θ L p Ĉπ q which belong to the segment rL, L 1 s according to Definition 6.3.

Let o i be a point of BS F L,L 1 pCq , lying on the strict transform of C. The point o i is considered at the fourth step of Algorithm 4.22. Let A i denote the germ of BS F L,L 1 pCq at o i and let pA i , B i q be the cross at o i chosen when one passes again through the first and second steps of Algorithm 4.22. By definition, L i :" π L,L 1 pB i q is a branch of Ĉπ . We denote by Ĉπ,oi (resp. C oi ) the germ of the strict transform of Ĉπ (resp. C) at the point o i . We use the Notations 4.25. Let us consider the segment rA i , L i s of the Eggers-Wall tree Θ oi,Ai p Ĉπ,oi q and the trunk θ F A i ,B i pCo i q " re Ai , e Bi s. Arguing as before, we obtain a homeomorphism Ψ oi : re Ai , e Bi s Ñ rA i , L i s which sends w P re Ai , e Bi s to the unique point W P rA i , L i s such that e Ai pW q is equal to the slope of w with respect to the basis pe Ai , e Bi q of the lattice N Ai,Bi . In addition, we get also that the homeomorphism Ψ oi defines a bijection between the marked points of the trunk θ F A i ,B i pCo i q and the marked points of Θ oi,Ai p Ĉπ,oi q on the segment rA i , L i s. By Proposition 6.22, we have an embedding of the Eggers-Wall tree Θ oi,Ai p Ĉπ,oi q such that the root A i of this tree is sent to the marked point L 1 ^L L i of Θ L p Ĉπ q. By Definition 4.33, the point e Ai of the trunk θpF Ai,Bi pCqq is identified with the marked point labeled by A i on θpF L,L 1 pCqq, during the construction of the fan tree θ π pCq.

If T is a tree and P 1 , . . . , P s P T , we denote by rP 1 , . . . , P s s the smallest subtree of T containing P 1 , . . . , P s . We apply this notation for the subtree re L , e L 1 , e Bj s of θ π pCq and the subtree rL, L 1 , L j s of Θ L p Ĉπ q. The previous discussion implies that the homeomorphisms Ψ o and Ψ oi can be glued into a homeomorphism re L , e L 1 , e Bj s Ñ rL, L 1 , L j s, which sends the ramification vertex e Ai of the tree re L , e L 1 , e Bj s to the ramification vertex L 1 ^L L j of rL, L 1 , L j s. We repeat this construction each time we pass through a cross at the first and second steps during the iterations of Algorithm 4. [START_REF] Cassou-Noguès | Newton trees for ideals in two variables and applications[END_REF]. By induction, we get a finite number of homeomorphisms Ψ oj , which glue into a homeomorphism Ψ : θ π pCq Ñ Θ L p Ĉπ q which respects the labelings of the ends of both trees by the branches of Ĉπ .

Identify the two rooted trees θ π pCq and Θ L p Ĉπ q by the isomorphism of Theorem 6.27. For every point P P θ π pCq, define the set δ P Ă rL, P q as the finite subset of discontinuity points of the restriction of the slope function S π to the segment rL, P q. If λ P Q ˚, denote by denpλq the denominator q of λ, when one writes it in the form p{q, with pp, qq P Z ˆN˚, and p, q coprime. The fan tree θ π pCq comes endowed with only one function, the slope function S π , while the Eggers-Wall tree is endowed with the index i L , the exponent e L and the contact complexity c L functions. These functions are related by: Proposition 6.28. For every P P θ π pCq, one has:

(1) i L pP q " ź QPδ P denpS π pQqq.

(2) e L pP q " ż P L 1 i L dS π .

(3) c L pP q "

ż P L 1 i 2 L dS π .
Proof. In order to follow the proof, one has to keep in mind the isomorphism of the fan tree with the Eggers-Wall tree built in Theorem 6.27. If the set δ P is empty, that is, if the slope function S π is continuous in restriction to rL, P q, then P belongs to the first trunk rL, L 1 s. By definition, for any Q P rL, L 1 s we have:

(75) i L pQq " 1, e L pQq " S π pQq.

Hence the equalities (1), ( 2) and (3) hold trivially for P . We prove the assertions ( 1) and ( 2) by induction on the number of elements of the set δ P of discontinuity points. Assume that δ P " tW " W 1 , W 2 , . . . , W k u with k ě 1, and W ă L W 2 ă L ¨¨¨ă L W k ă L P . By construction, the point W belongs to the first trunk of θ π pCq. Then, using the notation (57), we have e L pW q " d w {c w " S π pW q, with c w " denpS π pW qq. We decompose the integral of the second member of equality (2) in the form:

ż P L 1 i L dS π " ż W L 1 i L dS π `ż P W 1 i L dS π .
By [START_REF] Kuo | On analytic function germs of two complex variables[END_REF], one has:

(76)

ż W L 1 i L dS π " e L pW q " d w c w .
With the notations of Section 6.3, we consider the reduced curve C w at pS w , o w q, consisting of those branches A w which are the strict transforms of branches A of C such that W ă L P ă L A (see point (3) of Proposition 6.20). Proposition 6.22 implies that:

(77) i L pQq " c w i Ew pQq, for Q P rW, P s Ă Θ Ew pC w q.

Hence:

(78)

ż P W 1 i L dS π " 1 c w ż P W 1 i Ew dS π " 1
c w e Ew pP q.

To understand the last equality of (78), apply the induction hypothesis to the integral ş P W p1{i Ew qdS π , with respect to the set tW 2 , . . . , W k u of discontinuity points of the restriction of the slope function S π to rW, P q. The equality (2) follows from [START_REF] Laufer | Normal two-dimensional singularities[END_REF], [START_REF] Lejeune-Jalabert | Linear systems with infinitely near base conditions and complete ideals in dimension two[END_REF] and point (2) of Proposition 6.22.

The equality (1) follows similarly by [START_REF] Lejeune-Jalabert | Sur l'équivalence des singularités des courbes algébriques planes. Coefficients de Newton[END_REF] and the induction hypothesis applied to i Ew pP q.

Let us prove the equality [START_REF] Abhyankar | Embeddings of the Line in the Plane[END_REF]. By point (2) one has de L " p1{i L qdS π . Therefore:

c L pP q " ż P L 1 i L de L " ż P L 1 i 2 L dS π .
Example 6.29. Consider the toroidal pseudo-resolution process of Example 4.28. Figure 55 shows the fan tree θ π pCq and the corresponding Eggers-Wall tree Θ L p Ĉπ q, for which are indicated the values of the exponent and the index functions. We computed them using Proposition 6.28. For instance, we have i L pE 6 q " 1 ¨5 " 5, e L pE 6 q " 3 5 `1 5 ¨5 3 " 14 15 , i L pE 8 q " 1 ¨5 ¨3 " 15 and e L pE 8 q " 14 15 `1 15 ¨1 2 " 29 30 .

Proposition 6.28 allows us to define a concrete reduced curve singularity C which admits the toroidal resolution process described in Example 4.28, whose lotus was represented in Figure 36 and whose Enriques tree was represented in Figure 40. Namely, we fix local coordinates px, yq and we choose Newton-Puiseux series η 1 pxq, . . . , η 7 pxq defining branches C 1 , . . . , C 7 , then we take supplementary series L4 q is that on the right side of Figure 55. For instance, one may choose: η 1 pxq :" x 5{2 , η 2 pxq :" x 2 , η 3 pxq :" ´x2 , η 4 pxq :" x 3{5 `x3{4 , η 5 pxq :" x 3{5 `x11{15 η 6 pxq :" 2x 3{5 `x6{5 , η 7 pxq :" 2x 3{5 `x14{15 `x29{30 , λ 1 pxq :" 0, λ 2 pxq :" x 3{5 , λ 3 pxq :" 2x 3{5 , λ 4 pxq :" 2x 3{5 `x14{15 .

Remark 6.30. The right part of Figure 55 shows the Eggers-Wall tree of the completion of a plane curve singularity generated by a toroidal pseudo-resolution process. One may verify that it satisfies the following property which characterizes the Eggers-Wall trees of such completions: each vertex which is not an end of the tree is contained in the interior of a segment in restriction to which the index function is constant (in particular, such an Eggers-Wall tree has no vertices of valency 2). When one has such an Eggers-Wall tree, it originates from a fan tree as described in Proposition 6.28. But this fan tree is not unique. One has to determine first which segments of the Eggers-Wall tree are trunks of the fan tree, and there may be different choices. For instance, in Figure 55 one could decide that the segment rL, C 2 s is a trunk, instead of rL, L 1 s. Once the trunks are chosen, the sets δ P are determined for every point P of the tree. This allows to compute the slope function S π by integrating the differential relation dS π " i L de L , which is a consequence of Proposition 6.28 [START_REF] Abhyankar | Newton-Puiseux Expansion and Generalized Tschirnhausen Transformation[END_REF].

Proposition 6.28 may be written more explicitly as follows:

Corollary 6.31. Let P be a vertex of θ π pCq " Θ L p Ĉπ q, different from the root L. Assume that when one moves on the segment rL, P s from L to P , one meets successively the vertices P 1 , . . . , P k " P of δ P Y tP u. Denote S π pP j q " d j {c j with coprime c j , d j P N ˚, for all j P t1, . . . , ku (with c k " 1 and d k " 8 if P is a leaf of the tree). Then:

(1) i L pP q " c 1 ¨¨¨c k´1 .

(2) c L pP q " d 1 c 1 `d2 c for all j P t1, . . . , ku. Here the positive integers pm 1 , . . . , m k q and pn 1 , . . . , n k q are chosen such that m j and n j are coprime for all j P t1, . . . , ku. The relations (79) may be reexpressed in the following way:

(80) pc j , d j q " pn j , m j ´nj ¨mj´1 q, for all j P t1, . . . , ku (with the convention m 0 :" 0). Sometimes the couples pm j , n j q are called the Puiseux pairs and the couples pd j , c j q are called the Newton pairs of the given Newton-Puiseux series. The importance of using both sequences of pairs in the topological study of plane curve singularities was emphasized by Eisenbud and Neumann in their book [START_REF] Eisenbud | Three-dimensional link theory and invariants of plane curve singularities[END_REF]Page 6]. More details may be found in Weber's survey [132, Section 6.1].

Example 6.33. This is a continuation of Example 6.32. Consider pairs of coprime integers pn j , m j q P N ˚ˆN ˚with n j ą 1, for j " 1, . . . , k and the Newton-Puiseux series

x m1{n1 `xm2{pn1n2q `¨¨¨`x m k {pn1¨¨¨n k q , defining a branch C. We can build a toroidal pseudo-resolution π of C with respect to L " Zpxq, such that Ĉπ " L `C `řk j"1 L j and the branches L 1 , . . . , L k are defined by the Newton-Puiseux series: 0, x m1{n1 , x m1{n1 `xm2{pn1n2q , . . . , x m1{n1 `xm2{pn1n2q `¨¨¨`x m k´1 {pn1¨¨¨n k´1 q . Then the associated lotus is as represented in Figure 38. Using formula [START_REF] Lê | Sur les noeuds algébriques[END_REF] and the notations introduced in Example 5.30, we have: m j n j ´mj´1 " rp j , q j , . . .s, for all j P t1, . . . , ku. In fact, one gets the same lotus whenever C is an arbitrary branch with the previous characteristic exponents relative to L and the branches L j are semiroots of C (see [START_REF] Popescu-Pampu | Approximate roots[END_REF]Corollary 5.6]). This shows that our notion of completion of a reduced curve singularity C relative to a toroidal pseudo-resolution process is a generalization of the operation which adds to a branch a complete system of semiroots relative to L (see [99, Definition 6.4]).

6.6. Historical comments.

Historical information about the notion of characteristic exponent may be found in our paper [44, Introduction, Rem. 2.9].

In addition to the older Enriques diagrams and dual graphs of exceptional divisors of embedded resolutions, Kuo and Lu associated a third kind of tree to a curve singularity C " Zpf px, yqq in their 1977 paper [START_REF] Kuo | On analytic function germs of two complex variables[END_REF]. An example of such a tree, extracted from their paper, is shown in Figure 57. Their trees were rooted and their sets of leaves were in bijection with the set of Newton-Puiseux series ηpxq associated with the corresponding plane curve singularity C. They used their trees in order to relate the structure of C to that of its polar curve defined by the equation Bf By " 0. In his 1983 paper [START_REF] Eggers | Polarinvarianten und die Topologie von Kurvensingularitaeten[END_REF], Eggers showed that a kind of Galois quotient of the Kuo-Lu tree of f was more convenient for this purpose. Figure 57 shows the first example given in [START_REF] Eggers | Polarinvarianten und die Topologie von Kurvensingularitaeten[END_REF]. A variant of the Eggers [START_REF] García Barroso | Ultrametric spaces of branches on arborescent singularities[END_REF]Theorem 112], in the framework of Eggers-Wall trees defined relative to arbitrary coordinate systems. It is important to consider the Eggers-Wall tree of C relative to coordinate systems which are not necessarily generic relative to C. Indeed, this freedom is essential when one wants to compare the Eggers-Wall tree of C with that of its strict transform by a blow up or a more complicated toric modification, because after such a modification the natural coordinate x defines the exceptional divisor, and is not necessarily generic with respect to the strict transform. In his paper [START_REF] Popescu-Pampu | Sur le contact d'une hypersurface quasi-ordinaire avec ses hypersurfaces polaires[END_REF], extracted from his thesis [START_REF] Popescu-Pampu | Arbres de contact des singularités quasi-ordinaires et graphes d'adjacence pour les 3-variétés réelles[END_REF], the third author did not consider any genericity hypothesis, in order to extend the definition of this kind of tree to higher dimensional quasi-ordinary hypersurface singularities. This generalized notion of Eggers-Wall tree was further developed in connexion with the study of the associated polar hypersurfaces in the 2005 paper [START_REF] García Barroso | Decomposition in bunches of the critical locus of a quasi-ordinary map[END_REF] of the first and second authors. In turn, the notion of Kuo-Lu tree was extended to quasi-ordinary hypersurface singularities by the first author and Gwoździewicz in their 2015 paper [START_REF] García Barroso | Quasi-ordinary singularities: tree model, discriminant, and irreducibility[END_REF] and used again by them in [START_REF] García Barroso | Higher order polars of quasi-ordinary singularities[END_REF], in order to study the structure of higher order polars of such singularities.

The notations for elementary Newton polygons described in Definition 6.14 were introduced by Teissier in his 1977 paper [120, Section 3.6], where he restricted them to a, b P N ˚Yt8u. Allowing the two numbers in Definition 6.14 to be rational is convenient in order to express Newton polygons in terms of Eggers-Wall trees (see Corollary 6.17).

Let us consider now the valuative aspects of Eggers-Wall trees. Favre and Jonsson proved in their 2004 book [START_REF] Favre | The valuative tree[END_REF] that the set of semivaluations of the local C-algebra ÔS,o which are normalized by the constraint that a defining function x of the smooth germ L has value 1, has a natural structure of rooted real tree, which they called the valuative tree. In his 2015 survey [START_REF] Jonsson | Dynamics on Berkovich spaces in low dimensions[END_REF], Jonsson revisited part of the theory of [START_REF] Favre | The valuative tree[END_REF] with a more geometric approach which is valid for algebraically closed fields of arbitrary characteristic. Favre and Jonsson gave several descriptions of its tree structure. In our paper [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]Theorem 8.34] we gave a new description of it, as the universal Eggers-Wall tree of Definition 6.12. Namely, we proved that the valuative tree could also be obtained as a projective limit of Eggers-Wall trees. The main point of our proof is that Θ L pCq embeds naturally in the valuative tree, for any C. We showed also in [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]Theorem 8.18] that the triple pi L , 1 `eL , c L q is the pullback by this embedding of a triple of three natural functions on the valuative tree: the multiplicity, the log-discrepancy and the self-interaction.

An advantage of the identification of Θ L with the valuative tree is that it allows to get an interpretation of the points of Θ L which do not belong to any Θ L pCq as special infinitely singular semivaluations, in the language of [START_REF] Favre | The valuative tree[END_REF] and [START_REF] Jonsson | Dynamics on Berkovich spaces in low dimensions[END_REF].

Another advantage is obtained when the base algebraically closed field has positive characteristic. Let us define the functions i L , c L and e L on θ π pCq by the equalities appearing in Proposition 6. [START_REF] Cutkosky | Resolution of singularities[END_REF]. This provides a definition of a notion of Eggers-Wall tree in positive characteristic, where Newton-Puiseux series are not enough for the study of plane curve singularities (see Remark 2.17). The approach of Section 6.3 may be generalized to prove that in restriction to θ π pCq, the multiplicity function relative to L is equal to i L , the contact complexity function relative to L is equal to c L and the log-discrepancy function relative to L is equal to 1 `eL . This abstract Eggers-Wall tree may be associated with the ultrametric distance on the branches of C, as described in our paper [START_REF] García Barroso | Ultrametric spaces of branches on arborescent singularities[END_REF]. It may be seen also as a generalization of the notion of characteristic exponents in positive characteristic introduced in Campillo's book [START_REF] Campillo | Algebroid Curves in Positive Characteristic[END_REF], where the author computes these exponents using Hamburger-Noether expansions (see [START_REF] Campillo | Algebroid Curves in Positive Characteristic[END_REF]Section 3.3]), infinitely near points (see [START_REF] Campillo | Algebroid Curves in Positive Characteristic[END_REF]Remark 3.3.8]) or Newton polygons (see [START_REF] Campillo | Algebroid Curves in Positive Characteristic[END_REF]Section 3.4]).

Assume now that the germ C is holomorphic. Then the Enriques diagram and the weighted dual graph of the minimal embedded resolution, as well as the Eggers-Wall tree relative to generic coordinates encode the same information, which is equivalent to the embedded topological type of C. Proofs of this fundamental fact may be found in Wall's book [START_REF] Wall | Singular points of plane curves[END_REF]Propositions 4.3.8 and 4.3.9].

A basic problem is then to find methods to transform one kind of tree into the two other kinds. Noether described in [START_REF] Noether | Les combinaisons caractéristiques dans la transformation d'un point singulier[END_REF] how to pass from the characteristic exponents of an irreducible curve singularity C to the structure of the blow up process leading to an embedded resolution. Enriques and Chisini generalized this approach in [35, Libr. IV, Cap. I] to the case when C is not necessarily irreducible. Namely, they showed how to pass from the characteristic exponents of its branches and the orders of coincidence of pairs of branches in generic coordinates to the associated Enriques diagram.

Zariski and Lejeune-Jalabert proved by different methods in their 1971 paper [START_REF] Zariski | General Theory of Saturation and of Saturated Local Rings II: Saturated Local Rings of Dimension 1[END_REF] and 1972 thesis [START_REF] Lejeune-Jalabert | Sur l'équivalence des singularités des courbes algébriques planes. Coefficients de Newton[END_REF] respectively, that the characteristic exponents of the branches of C and the intersection numbers of its pairs of branches determine the embedded topological type of C and the combinatorics of its minimal embedded resolution. This may be seen as a proof of the fact that the weighted dual graph of the minimal embedded resolution is equivalent to the generic Eggers-Wall tree. Methods to pass from the knowledge of the characteristic exponents and intersection numbers to the dual graph were explained by Eisenbud and Neumann [34, Let us mention now several other trees which were associated to plane curve singularities. As explained in Subsection 4.5, the changes of variables considered by Puiseux (called sometimes Newton maps) were compositions of affine and of toric ones, which in general were not birational. Nevertheless, an algorithm of abstract resolution and of computation of Newton-Puiseux series may be developed also using them. A variant of the fan trees, adapted to this context and called Newton trees, was used by Cassou-Noguès in her papers mentioned in Subsection 4.5, written alone or in collaboration. The Newton trees encode also the toroidal pseudo-resolution processes described in the paper [START_REF] Cassou-Noguès | Multivariable Hodge theoretical invariants of germs of plane curves. II. In Valuation theory in interaction[END_REF] of Cassou-Noguès and Libgober. We refer the reader especially to the papers [START_REF] Cassou-Noguès | Invariants of plane curve singularities and Newton diagrams[END_REF] and [START_REF] Cassou-Noguès | Newton trees for ideals in two variables and applications[END_REF] for more details about this approach. The changes of coordinates [START_REF] Kempf | Toroidal embeddings. I[END_REF], which are very similar to Newton maps, were also used in the paper [START_REF] Kennedy | Monodromy of plane curves and quasi-ordinary surfaces[END_REF] of Kennedy and McEwan to study the monodromy of holomorphic plane curve singularities.

Newton maps and Newton trees have been used to study the singularities of quasi-ordinary hypersurfaces by Artal, Cassou-Noguès, Luengo and Melle Hernández (see for instance [START_REF] Artal Bartolo | Quasi-ordinary singularities and their zeta functions[END_REF] and [START_REF] Artal Bartolo | On quasi-ordinary singularities and Newton trees[END_REF]). In their 2014 paper [START_REF] González Pérez | Motivic Milnor fiber of a quasi-ordinary hypersurface[END_REF], the second author and González Villa compared the Newton maps with the toric morphisms appearing in a toroidal pseudo-resolution of an irreducible germ of quasi-ordinary hypersurface.

Newton trees are algebraic variants of the splice diagrams associated by Eisenbud and Neumann in their 1986 book [START_REF] Eisenbud | Three-dimensional link theory and invariants of plane curve singularities[END_REF] to any oriented graph link in an integral homology sphere, extending a graphical convention introduced by Siebenmann in his 1980 paper [START_REF] Siebenmann | On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology 3-spheres[END_REF]. In our recent paper [46, Section 5], we explained how to pass from the Eggers-Wall tree of a holomorphic plane curve singularity C relative to a smooth branch L to the splice diagram of the oriented link of L `C in S 3 .

In his 1993 papers [START_REF] Kapranov | Veronese curves and Grothendieck-Knudsen moduli space M 0[END_REF] and [START_REF] Kapranov | The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation[END_REF], Kapranov associated a version of Kuo and Lu's trees to finite sets of formal power series with complex and real coefficients respectively. He called them Bruhat-Tits trees.

A version of Kuo and Lu's trees was used recently by Ghys in his book [50] about the topology of real plane curve singularities. He associated two such trees, one for x ą 0 and another one for x ă 0 to any germ whose branches are smooth and transversal to the reference branch x " 0, and studied their relation, describing all the possible couples of such trees. In a theorem proved with Christopher-Lloyd Simon (see [50, Page 266]), Ghys extended this analysis to all plane curve singularities with only real branches. For this more general problem, it was not any more a variant of Kuo and Lu's tree which was crucial, but a real version of the dual graph of the associated minimal resolution. A different real version of the dual resolution graph was introduced before by Castellini in [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF]Chap. 3].

Ghys' version of Kuo and Lu's trees was also used by Sorea in her study [START_REF] Sorea | The shapes of level curves of real polynomials near strict local minima[END_REF] of curve singularities defined over R but without any real branch, that is, singularities of real analytic functions f px, yq in the neighborhood of a local maximum or minimum. Those trees were related in this work with another kind of tree, defined using Morse theory, the so-called Poincaré-Reeb tree of the function f relative to x.

Versions of our fan tree were considered by Weber in his 2008 survey [START_REF] Weber | On the topology of singularities[END_REF] about the embedded topological type of holomorphic plane curve singularities, based on the earlier 1985 preprint [START_REF] Michel | Topologie des germes de courbes planes à plusieurs branches[END_REF] of Michel and Weber, which contained also many examples. The reading of Weber's survey [START_REF] Weber | On the topology of singularities[END_REF] should facilitate the interpretations of the objects manipulated in this paper in terms of the embedded topological type of C.

Overview and perspectives

We begin this final section by an overview of the content of the paper. Then we formulate a few remarks about perspectives of development of the use of lotuses in the study of singularities. The final Subsection 7.3 contains a list of notations used in this paper.

Figure 1 .

 1 Figure 1. A lotus. It is part of Figure 36, which corresponds to Example 5.28.

  Chapter 7], Teissier [121, Section 1], [123, Sections 3-4], de Jong & Pfister [66, Section 5.1], Cutkosky [28, Section 2.1] or Greuel, Lossen & Shustin [59, Thm. I.3.3]): Theorem 2.20. (Newton-Puiseux theorem) Any non-zero monic polynomial f P Crrxssrys such that f p0, yq " y d has d roots in the ring Crrx 1{N ss. As a consequence, the quotient field of the ring Crrx 1{N ss is the algebraic closure of the quotient field of the ring Crrxss.

Theorem 2 .

 2 38. (Jacobian criterion) Let C be a reduced curve in an open set of C 2

Figure 3 .

 3 Figure 3. The weighted dual graph of the minimal embedded resolution of the semicubical parabola

x 3 y 3 x 7 y 2 K1K2Figure 4 .

 324 Figure 4. The Newton polygon of the series f px, yq " py 2 ´2x 3 qpy 3 ´x7 q

Figure 8 .Proposition 3 . 7 .Definition 3 . 8 .

 83738 Figure 8. The fan F p3{5, 2{1, 5{2q and the points p p3{5q, p p2{1q, p p5{2q

2 Figure 9 . 8 3. 2 .

 2982 Figure 9. The regularization F reg p3{5, 2{1, 5{2q of the fan of Figure 8

Figure 12 . 2 Figure 13 .

 12213 Figure 12. The toric morphism defined by the two regular cones of Example 3.26

  contains information about sources of toric geometry in papers by Demazure, Hochster, Bergman, Sumihiro and Miyake & Oda. Details about the development of toric geometry may be found in Cox, Little and Schenck's 2011 book [26, Appendix A].

Remark 4. 3 .Figure 14 .

 314 Figure 14. Illustration of Remark 4.3

Figure 15 .

 15 Figure 15. The Newton polygon, the tropicalization and the Newton fan of Example 4.10

  3. It produces a toroidal embedded resolution of C instead of a pseudo-resolution (see Definition 4.15). Proposition 4.20 means that if C is Newton non-degenerate relative to the cross pL, L 1 q chosen at Step 2 of Algorithm 4.22, then this algorithm stops after performing only one Newton modification. More generally, a fundamental property of Algorithm 4.22 is: Theorem 4.23. Algorithm 4.22 stops after a finite number of iterations.

Figure 18 .

 18 Figure 18. Third level of Newton modifications in Example 4.28

Figure 19 .0 2 ˙is

 192 Figure 19. The trunks associated to the toroidal pseudo-resolution of Example 4.28

Figure 20 .

 20 Figure 20. Construction of the fan tree of the toroidal pseudo-resolution of Example 4.28

Figure 21 .Figure 22 .Figure 23 .

 212223 Figure 21. The fan tree θ π pCq is isomorphic to the dual graph of the toroidal boundary BΣ

Figure 24 .

 24 Figure 24. Newton's ruler

e 1 Ð 2 0δpe 1 , 2 0Figure 25 .

 121225 Figure 25. Vocabulary and notations about petals

e 1 e 2 0δpe 1 , e 2 qFigure 26 .

 21226 Figure 26. Partial view of the universal lotus Λpe 1 , e 2 q relative to pe 1 , e 2 q

2 Figure 27 .

 227 Figure 27. The Newton lotuses Λ p3{5q, Λ p3{5, 2{1, 5{2q and their marked points

2 Figure 28 .

 228 Figure 28. The regularized fan F reg p3{5, 2{1, 5{2q and the Newton lotus Λ p3{5, 2{1, 5{2q

e 1 e 2 0δpe 1 , e 2 qFigure 29 . 1 e 2 e 1 e 2 Figure 30 .

 21229121230 Figure 29. Partial view of the Enriques subtree of the universal lotus Λpe 1 , e 2 q

Figure 33 .

 33 Figure 33. An illustration of Proposition 5.21 for p 2 {p 1 " 3{2

Ḟigure 35 . 28 Example 5 . 28 .

 3528528 The 2-dimensional Newton lotuses of Example 5.Consider the toroidal pseudo-resolution process of Example 4.28. The construction of the corresponding fan tree was explained in Example 4.36 and illustrated in Figure 20. The left column of Figure 35 represents the Newton fans produced each time one runs STEP 2 of Algorithm 4.22. The middle column shows the associated trunks and the right column the corresponding lotuses.

Figure 37 .

 37 Figure 37. Comparison of the fan tree and the lotus of Example 5.28

( 8 )

 8 The Enriques tree of Λ π pCq is the Enriques diagram of the constellation of infinitely near points at which are based the crosses introduced during the blow up process leading to the boundary BΣ reg . Proof. Points (1) and (2) result from Proposition 4.18. Points (3) and (4) result from Propositions 4.35, 5.10 and 5.25. Point (5) results from Corollary 2.28 and Proposition 2.37. A prototype of this result had been stated in [102, Thm. 6.2]. Points (6) and (7) result from Proposition 5.16. Point (8) results from Proposition 5.14.

38 .

 38 Figure 38. The lotus of toroidal pseudo-resolution for one branch from Example 5.30

Ḟigure 39 .

 39 The Enriques trees and the extended Enriques trees in Example 5.[START_REF] Dumas | Sur la résolution des singularités de surfaces[END_REF]

Figure 40 .Figure 41 .

 4041 Figure 40. The Enriques tree of the toroidal pseudo-resolution of Example 5.31

3 Figure 42 .CFigure 43 .

 34243 Figure 42. The Newton lotuses Λ 1 , Λ 2 and Λ 3 of Example 5.33

Figure 44 .Figure 45 .

 4445 Figure 44. The two semipetals and the axis of the petal δpe 1 , e 2 q

Figure 48 .Figure 49 .

 4849 Figure 48. Du Val's "proximity graphs"

Figure 50 .

 50 Figure 50. The general shape of Schulze-Röbbecke's divides

Figure 51 .

 51 Figure 51. The Eggers-Wall trees of Zpf px, yqq and Zpxyf px, yqq from Example 6.4

Figure 52 .

 52 Figure 52. The Eggers-Wall tree of the curve singularities C 1 and C of Example 6.5

Figure 53 .

 53 Figure 53. The elementary Newton polygons t a b u, t a 8 u, t 8 b u

1 *

 1 ˚Y t8u, b P Q ˚and any d P N ˚, one has: d whenever b P N ˚. The elementary Newton polygons are generators of the semigroup of Newton polygons, with respect to Minkowski sum. In fact one has more: Proposition 6.15. Each Newton polygon N may be written in a unique way, up to permutations of the terms, as a Minkowski sum of elementary Newton polygons with pairwise distinct inclinations. Their compact edges are translations of the compact edges of N .

8 ΘFigure 55 .

 855 Figure[START_REF] González Pérez | Motivic Milnor fiber of a quasi-ordinary hypersurface[END_REF]. The fan tree θ π pCq and the corresponding Eggers-Wall tree Θ L p Ĉπ q in Example 6.[START_REF] Deligne | Intersections sur les surfaces régulières[END_REF]

  e L pP q " d 1 c 1 `d2 c 1 c 2 `d3 c 1 c 2 c 3 `¨¨¨`d k c 1 ¨¨¨c k .Example 6.32. Let us specialize Corollary 6.31 to the case where P is a leaf of θ π pCq " Θ L p Ĉπ q, labeled by a branch C. Therefore the characteristic exponents of a Newton-Puiseux series of C relative to L are:

Figure 56 .

 56 Figure 56. A Kuo-Lu tree

Figure 57 .

 57 Figure 57. An Eggers tree

  Appendix to Ch. 1], Brieskorn and Knörrer [15, Section 8.4], Michel and Weber [86], de Jong and Pfister [66, Section 5.4] and an algorithm was described by the first author in [42, Sect. 1.4.6].

7. 1 .Figure 58 .

 158 Figure 58. Overview of the constructions of the paper

  .4.9, Cor. 4.4.10]): Theorem 2.14. A Noetherian local ring of Krull dimension one is normal if and only if it is regular.

  Building iteratively the minimal embedded resolution of the semicubical parabola Let us compute now the weighted dual graph of π 3 . For every i P t0, 1, 2u, denote by E i ãÑ S i`1 the exceptional divisor of the blow up of the point O i P S i . If 0 ď i ă j ď 2, denote by E i,j the strict transform of E i on the surface S j`1 by the modification ψ j ˝¨¨¨˝ψ i : S j`1 Ñ S i . By Corollary 2.28, Equations (6) and[START_REF] A'campo | Geometry of plane curves via Tschirnhausen resolution tower[END_REF] imply that O 1 P E 0 and O 2 P E 1 X E 0,1 , because in the chart C 2

	one has E 2 0 " E 2 1 " E 2 2 " ´1.

  ' If o 1 and o 2 are two infinitely near points of o, then one says that o 2 is proximate to o 1 , written o 2 Ñ o 1 , if o 2 belongs to the strict transform of the irreducible rational curve created by blowing up o 1 . If moreover there is no point o 3 such that o 2 Ñ o 3 Ñ o 1 , one says that o 1 is the parent of o 2 . ' A finite constellation (above o) is a finite set C of infinitely near points of o, closed under the operation of taking the parent.

  .1.6]. Let us just sketch the main idea. Fix a local coordinate system px, yq on pS, oq, such that L " Zpxq. Start from a normalization of the branch A of the form u Ñ pu n , ζpuqq (see the explanations leading to formula (2)). Therefore, ζpx 1{n q is a Newton-Puiseux root of A. By Theorem 6.1, one has a defining function of the branch B of the form ś ηpxqPZxpBq py ´ηpxqq. Proposition 2.8 implies that:

	A ¨B " ν u	¨ź ηpxqPZxpBq pζpuq ´ηpu n qq '"	ÿ

ηpxqPZxpBq ν u pζpuq ´ηpu n qq .

Acknowledgements. This research was partially supported by the French grants ANR-17-CE40-0023-02 LISA and Labex CEMPI (ANR-11-LABX-0007-01), and also by the Spanish grants MTM2016-80659-P, MTM2016-76868-C2-1-P and SEV-2015-0554. We are grateful to Bernard Teissier for having invited us, when we were his PhD students, to explore the fascinating combinatorial aspects of plane curve singularities, and for everything he taught us about this world. We are grateful to Ana Belén de Felipe and Roberto Castellini for their remarks on lotuses. We thank Patricio Almirón, Pierrette Cassou-Noguès, Manuel González Villa, Carlos Guzmán Durán, Lê Dũng Tráng and Camille Plénat for their comments on preliminary versions of this paper. We are grateful to Dale Cutkosky for several explanations concerning morphisms in the toroidal category. We thank Jose Seade for the invitation to publish this work as a chapter of the "Handbook of Geometry and Topology of Singularities", and for his remarks which were very helpful in adapting it to the spirit of this handbook. We thank also the two anonymous referees for their useful recommendations and remarks.

Next proposition explains that the symmetric binary operation ^on Q ˚allows to describe the intersection of two lotuses of the form Λpλq: Proposition 5. [START_REF] Cassou-Noguès | The Newton tree: geometric interpretation and applications to the motivic zeta function and the log canonical threshold[END_REF]. For any λ, µ P Q ˚, one has: Λpλq X Λpµq " Λpλ ^µq.

Therefore, the lotus Λpλ, µq is isomorphic as a simplicial complex with an oriented base to the triangulated polygon obtained by gluing ∆pλq and ∆pµq along ∆pλ ^µq.

Proof. Assume that λ " ra 1 , . . . , a k s. Proposition 5. [START_REF] Cassou-Noguès | Invariants of plane curve singularities and Newton diagrams[END_REF] shows in particular that the lotus Λpλq has n :" a 1 `¨¨¨`a k petals. Denote by pλ i q 1ďiďn the sequence of positive rationals such that the successive non-basic vertices of the petals of Λpra 1 , . . . , a k sq are the primitive vectors ppλ 1 q, . . . , ppλ n q. The sequence of continued fraction expansions of pλ i q 1ďiďn is: [START_REF] García Barroso | An approach to plane algebroid branches[END_REF] r1s, r2s, . . . , ra 1 s, ra 1 , 1s, ra 1 , 2s, . . . , ra 1 , a 2 s, ra 1 , a 2 , 1s, . . . , ra 1 , . . . , a k s.

One may prove this fact at the same time as Proposition 5.20, by making now an induction on the number n of petals of Λpra 1 , . . . , a k sq, instead of the number k of terms of the continued fraction.

The proposition results then by combining the previous fact with formula [START_REF] García Barroso | Higher order polars of quasi-ordinary singularities[END_REF].

Example 5.24. Let us consider the two rational numbers λ " r4, 2, 5s and µ " r3, 2, 1, 4s of Example 5.19. Then j " 0, k " 3, l " 4, therefore j `1 ă mintk, lu and λ ^µ " r3 `1s " 4. The lotus Λpλ, µq is therefore isomorphic to the triangulated polygon with an oriented base of the right side of Figure 34.

ppr4, 2, 5sq

Figure 34. The abstract lotus ∆pr4, 2, 5s, r3, 2, 1, 4sq

Iterating the gluing operation, one may construct an abstract lotus ∆pλ 1 , . . . , λ k q combinatorially equivalent to any given Newton lotus Λpλ 1 , . . . , λ k q, seen as a triangulated polygon with marked points and oriented base. One gets an abelian monoid of (abstract) lotuses, the monoid operation Z generalizing the gluing operation of Figure 34. Namely, if E 1 and E 2 are finite subsets of Q `Y t8u, then:

The neutral element of this monoid is the segment rA 1 , A 2 s " ∆pHq " ∆p0q " ∆p8q " ∆pt0, 8uq.

The lotus of a toroidal pseudo-resolution.

In this subsection we reach a second level of explanation of the subtitle of this article, the first level having been reached in Subsection 5.1 above. Namely, we define a new kind of lotus by gluing the lotuses associated to the Newton fans produced by Algorithm 4.22 (see Definition 5.26). We illustrate this definition by our recurrent example (see Example 5.28) and by the case of an arbitrary branch (see Example 5.30). Finally, we show how this lotus allows to visualize many objects associated to the regularized algorithm and with the decomposition into blow ups of points of the embedded resolution produced by it (see Theorem 5.29).

Consider again a reduced curve singularity C on the smooth germ of surface pS, oq. Fix a smooth branch L on pS, oq, and run Algorithm 4.22. Denote as before by π : pΣ, BΣq Ñ pS, L `L1 q a resulting Since pL, L 1 q is a cross on pS, oq and W P Θ L pL 1 q, one has that i L pW q " 1. Therefore, by Definition 6.9, the contact complexity of W is: [START_REF] Greco | Normal varieties[END_REF] c L pW q " d w c w .

Recall that A ^L B denotes the infimum of the points A and B of the universal Eggers-Wall tree Θ o,L relative to the partial order ĺ L induced by the root L. We need the following lemma: Lemma 6.18. Let A be a branch on pS, oq different from L, L 1 . The following properties are equivalent:

(1) The strict transform A w of A by π w intersects E w z pL w Y L 1 w q.

(2) The fan F L,L 1 pAq is the subdivision of σ 0 along the ray R `w.

(3) A ^L L 1 " W . In addition, if these properties hold, then the order of vanishing of A along E w is equal to d w i L pAq and the intersection number

Proof. The equivalence of these three properties is immediate from Propositions 4.18 and 6.16. Recall that the order of vanishing ord Ew pAq is by definition the multiplicity of E w in the divisor pπ ẘLq, that is, the value taken by the divisorial valuation ord Ew defined by E w on a defining function f of A. Thanks to Proposition 4.18, this value is equal to trop A L,L 1 pwq, which may be written d w i L pAq by Lemma 6.16. By Proposition 4.18, E w ¨Aw is equal to the integral length of the compact edge of the Newton polygon N L,L 1 pAq. The equality E w ¨Aw " i L pAq{c w follows by using Lemma 6.16 again. Lemma 6.19. Let A and B be two branches on pS, oq. Consider the point W P Θ L pL 1 q fixed above, determined by relation [START_REF] Graham | Concrete Mathematics. A foundation for computer science[END_REF]. Assume that W " A ^L L 1 " B ^L L 1 inside the universal Eggers-Wall tree Θ L . Then the following conditions are equivalent:

(1) A ^L B " W .

(2) A ¨B " d w c w pL ¨AqpL ¨Bq.

(

Proof.

Proof of (1) ñ (2). This implication is a consequence of Formulae ( 54) and [START_REF] Greco | Normal varieties[END_REF].

Proof of (2) ñ (1). Let us denote by W 1 the point A ^L B. The assumption W ĺ L A, W ĺ L B implies that W ĺ L W 1 . By Formula (54), we get c L pW 1 q " d w {c w " c L pW q. Since the function c L is strictly increasing on rL, As, the inequalities

Proof of (1) ô (3). Let px, yq be a system of local coordinates defining the cross pL, L 1 q. Denote by f A a defining function of A with respect to this system and by K A the compact edge of the Newton polygon N L,L 1 pAq. By the proof of Lemma 6.16, if α A is the coefficient of x dw{cw in a fixed Newton-Puiseux series of A, then the restriction of f A to the compact edge K A is equal to:

We consider similar notations for the branch B. By Proposition 4.18, the point of intersection of the strict transform of A w with E w is parametrized by the coefficient α cw A . The desired equivalence follows since α cw A ‰ α cw B if and only if for every γ P C with γ cw " 1, one has that α A ‰ γ ¨αB , which is also equivalent to the property k L pA, Bq " d w {c w by the definition of the order of coincidence (see Definition 6.2). Proposition 6.20. Let A and B be two branches on S. Consider the point W P Θ L pL 1 q fixed above. Assume that W " A ^L L 1 " B ^L L 1 . Then:

(1) L ¨A " c w pE w ¨Aw q.

(2) A ¨B " A w ¨Bw `dw c w pL ¨AqpL ¨Bq.

(3) A w ¨Bw ą 0 if and only if W ă L A ^L B.

(16) The fan tree θ π pCq is also isomorphic with the Eggers-Wall tree Θ L p Ĉπ q (see Definition 6.3) of the completion of C relative to the toroidal modification π (see Theorem 6.27). The triple of functions (index i L , exponent e L , contact complexity c L ) defined on Θ L p Ĉπ q is determined by the slope function S π on the fan tree through explicit formulae (see Proposition 6.28). ( 17) If pL, L 1 q is a cross on S, then the Eggers-Wall tree Θ L pC `L1 q determines the Newton polygon N L,L 1 pCq (see Corollary 6.17).

Perspectives.

In this subsection we give a few perspectives on possible uses of lotuses. We believe that the lotuses of plane curve singularities may be useful in the following research topics:

(1) In the study of the topology of δ-constant deformations of such singularities. As mentioned in Subsection 6.6, Castellini's work [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF] gives a first step in this direction. An important advantage of lotuses in this context is that the lotuses of the singularities appearing in the deformations constructed in [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF] by A'Campo's method embed in the lotus of the original singularity. This embedding relation is much more difficult to express in terms of classical tree invariants of plane curve singularities. A crucial question is to understand whether this embedding property is specific to A'Campo type deformations, or if it extends to other kinds of δ-constant deformations. (2) In the analogous study for real plane curve singularities. One should probably describe real variants of the lotuses, embedded canonically up to isotopy in an oriented real plane. Again, Castellini's work [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF]Sect. 3.3.2] gives a first step in this direction. (3) In the extension of the distributive lattice structures described by Pe Pereira and the third author in [START_REF] Pereira | Fibonacci numbers and self-dual lattice structures for plane branches[END_REF] to arbitrary finite constellations, and in the application of those structures to the problem of adjacency of plane curve singularities. The natural operad structure on the set of finite lotuses associated to toroidal pseudo-resolution processes (defined by gluing the base of one lotus to an edge of the lateral boundary of another lotus) could be also useful in this direction. (4) In the study of complex surface singularities through the Hirzebruch-Jung method (see [START_REF] Popescu-Pampu | Introduction to Jung's method of resolution of singularities[END_REF]).

This method starts from a finite projection to a germ of smooth surface, and considers then an embedded resolution of the discriminant curve. The lotuses of such discriminant curves could be used as supports for encoding information about the initial finite projection, from which one could read invariants of the surface singularity.

List of notations.

In order to help browsing through the text, we list the notations used for the main objects met in it: ! a b

) : Elementary Newton polygon (see Definition 6.14). ra 1 , . . . , a k s: Continued fraction with terms a 1 , . . . , a k (see Definition 5.17). c m pf q:

Coefficient of the monomial χ m in the series f (see Definition 4.1). c L :

Contact complexity function (see Definition 6.9). C L,L 1 :

Strict transform of C by the Newton modification ψ C L,L 1 (see Definition 4.14). Ĉπ :

Completion of C relative to the toroidal pseudo-resolution π (see Definition 4.15). ConvpY q:

Convex hull of a subset Y of a real affine space. χ m :

Monomial with exponent m P M (see the beginning of Subsection 3.2).

BX:

Toric boundary of the toric variety X (see Definition 3.18), or toroidal boundary of the toroidal variety X (see Definition 3.29). B `Λπ pCq:

Lateral boundary of the lotus Λ π pCq (see Definition 5.5). e L :

Exponent function (see Definition 6.3 and Notations 6.7). f K :

Restriction of f to the compact edge K of its Newton polygon (see Definition 4.2). Fpf q:

Newton fan of the non-zero series f P Crrx, yss (see Definition 4.9). F L,L 1 pCq:

Newton fan of C relative to the cross pL, L 1 q (see Definition 4.14). F reg :

Regularization of the fan F (see Definition 3.8). ΓpCq:

Enriques diagram of the finite constellation C (see Definition 4.31).

H f,ρ : Supporting half-plane of the Newton polygon N pf q determined by the ray ρ Ă σ 0 (see Proposition 4.7). i L :

Index function (see Definition 6.3 and Notations 6.7). k x pξ, ξ 1 q:

Order of coincidence of two Newton-Puiseux series (see Definition 6.2). k x pC, C 1 q:

Order of coincidence of two distinct branches, relative to a local coordinate system px, yq (see Definition 6.2). l Z :

Integral length (see Definition 3.1). pL, L 1 q:

Cross on a germ of smooth surface (see Definition 3.31). ΛpFq:

Lotus of the Newton fan F (see Definition 5.4). Λpλ 1 , . . . , λ r q: Lotus associated to the finite set tλ Monomial lattice associated to the cross pL, L 1 q, (see Definition 3.32).

N:

Set of non-negative integers.

N ˚:

Set of positive integers. N L,L 1 :

Weight lattice associated to the cross pL, L 1 q (see Definition 3.32). N pf q:

Newton polygon of the non-zero series f P Crrx, yss (see Definition 4.2). N L,L 1 pCq:

Newton polygon of C relative to the cross pL, L 1 q (see Definition 4.14). O ρ :

Toric orbit associated to the cone ρ of a fan (see the relation ( 22)).

ÔS,o :

Completed local ring of the complex surface S at the point o (see Definition 2.5). π ˚pC q :

Total transform of a plane curve singularity C by a modification π (see Definition 2.31). ψ F σ : Toric morphism from X F to X σ associated to any fan F which subdivides the cone σ (see relation [START_REF] Castellini | La topologie des déformations de A'Campo des singularités : une approche par le lotus[END_REF]). ψ C L,L 1 : Newton modification defined by C relative to the cross pL, L 1 q (see Definition 4.14).

R `:

Set of non-negative real numbers. Spf q:

Support of the power series f P Crrx, yss (see Definition 4.1). S π :

Slope function of the toroidal pseudo-resolution π of C (see Definition 4.33). σ 0 :

Regular cone generated by the canonical basis of the lattice Z 2 . σ L,L 1 0 :

Regular cone generated by the canonical basis of the lattice N L,L 1 (see Definition 3.32). t w :

One parameter subgroup of the algebraic torus T N , corresponding to the weight vector w P N (see the beginning of Subsection 3.2). T N :

Complex algebraic torus with weight lattice N (see formula [START_REF] Brocot | Calcul des rouages par approximation, nouvelle méthode[END_REF]). trop f :

Tropicalization of the non-zero power series f P Crrx, yss (see Definition 4.4). trop C L,L 1 : Tropical function of the curve singularity C relative to the cross pL, L 1 q (see Definition 4.14). θpFq:

Trunk of the fan F (see Definition 4.32). θ π pCq:

Fan tree of the toroidal pseudo-resolution π of C (see Definition 4.33). Θ L pCq:

Eggers-Wall tree of the plane curve singularity C relative to the smooth branch L (see Definition 6.3 and Notations 6.7). Θ L :

Universal Eggers-Wall tree (see Definition 6.12). X σ :

Affine toric variety defined by the fan consisting of the faces of the cone σ (see Definition 3.14). X F :

Toric variety defined by the fan F (see Definition 3.15).

Z:

Operation of the monoid of abstract lotuses (see formula (50)).

^:

Operation on the set Q ˚allowing to describe the intersection of Newton lotuses (see formula [START_REF] García Barroso | Higher order polars of quasi-ordinary singularities[END_REF]).

Zpf q:

Zero-locus of a holomorphic function f or of a formal germ f P ÔS,o . Z x pCq:

Set of Newton-Puiseux roots of a plane curve singularity C relative to a local coordinate system px, yq (see Definition 6.2).