Ana J Reguera
email: areguera@agt.uva.es.

COORDINATES AT STABLE POINTS OF THE SPACE OF ARCS

Keywords: Space of arcs, divisorial valuation, graded algebra. MSC : 13A02, 13A18, 14B05, 14B25, 14E15, 14J17, 32S05

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The space of arcs X ∞ of a reduced separated scheme of finite type X over a perfect field k has finiteness properties when we localize at its stable points. The stable points of X ∞ were introduced in [START_REF] Reguera | A curve selection lemma in spaces of arcs and the image of the Nash map[END_REF], its definition is based on the stability property of the family of truncated arcs {j n (X ∞ )} n in Denef and Loeser's fundational article [DL]. Stable points are those fat points P of X ∞ for which the truncations of its zero set Z(P ) determine trivial fribrations of fiber A d k over an open set, being X equidimensional of dimension d. Equivalently, stable points of X ∞ are defined on an open subset of X ∞ by the radical of a finitely generated ideal, i.e. they are the generic points of the irreducible cylinders (see [EM], sec. 5). It was proved in [START_REF] Reguera | A curve selection lemma in spaces of arcs and the image of the Nash map[END_REF] that the ideal of definition of a stable point of X ∞ on an open subset of X ∞ , with the reduced structure, is finitely generated (see 2.4). This implies that the complete ring O X∞,P , P being stable, is a Notherian ring, which is the basis of a useful Curve Selection Lemma centered at the stable points [START_REF] Reguera | A curve selection lemma in spaces of arcs and the image of the Nash map[END_REF], corol. 4.8).

To compute the dimension of O X∞,P , P being stable, is an important problem. One upper bound of dim O X∞,P is the codimension as a cylinder of Z(P ) (see prop. 2.3). But the inequality is in general strict ( [IR], ex. 2.8). Another approach we have followed in some concrete examples [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF]5.6,5.16) is to describe the ring O X∞,P by generators and relations. The main purpose of this work is to provide coordinates in O X∞,P .

Divisorial valuations are closely related to stable points : the valuation ν P associated to a stable point P of X ∞ is divisorial [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF], (vii) in prop. 3.7). On the other hand, every divisorial valuation ν E defines a stable point P E on X ∞ and moreover, if we consider a multiple e ν E of ν E , we also have a stable point P eE which contains P E (see 2.7). A study of the graded algebra associated to the divisorial valuation ν E will be crucial in our study.

If X is smooth at the center P 0 of a stable point P of X ∞ , then we prove (prop. 2.6) that the local ring O X∞,P is regular and essentially of finite type over some field and dim O X∞,P equals the dimension as a cylinder of Z(P ). Hence, applying the Change of Variables in the Motivic Integration ( [DL], lemma 3.4), it follows that dim O X∞,PeE = e(k E + 1) where k E is the discrepancy of X with respect to E. Assuming that char k = 0, in this paper we recover this equality, and moreover, we give an explicit description of a minimal set of generators of P eE , i.e. we give a regular system of parameters of O X∞,PeE .

If X is not smooth at P 0 then, if X → A d k is a general projection, we have that a set of generators of the image of P eE in (A d ) ∞ provides a set of generators of P eE [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF], prop. 4.5). From this it follows that the embedding dimension of the ring O X∞,PeE is bounded from above by e( k E + 1), where k E is the Mather discrepancy of X with respect to E. A further work will be done to determine whether this defines a minimal system of coordinates of (X ∞ , P eE ).

One of the main ideas in our proof is to define some "approximate roots" in gr νE O A d ,P0 , being X → A d a general projection. In fact, if X is a curve then we have that embdim O X∞,P E = k E + 1, which in this case is equal to mult X + 1 ( [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF], corol. 5.7). We follow the same line as in the proof for the case of curves, being the more subtle part to define these approximate roots {q j,r } (j,r)∈J (definition 3.4). Although they do not generate gr νE O A d ,P0 in general (gr νE O A d ,P0 is not in general finitely generated for d ≥ 3), they generate a localization of gr νE O A d ,P0 modulo étale covering (theorem 3.8). This is done in section 3. In section 4 we describe minimal coordinates of (A d ) ∞ at the image P A d eE of P eE in (A d ) ∞ from the q j,r 's. From this we obtain a regular system of parameters of O X∞,PeE if X is smooth at P 0 (theorem 4.8), and a system of coordinates of (X ∞ , P eE ) for general X (corally 4.10).

Preliminaries

2.1.

Let k be a perfect field. For any scheme over k, let X ∞ denote the space of arcs of X. It is a (not of finite type) k-scheme whose K-rational points are the Karcs on X (i.e. the k-morphisms Spec K [[t]] → X), for any field extension k ⊆ K. More precisely, X ∞ := lim ← X n where, for n ∈ N, X n is the k-scheme of finite type whose K-rational points are the K-arcs of order n on X (i.e. the k-morphisms Spec K [[t]]/(t) n+1 → X). In fact, the projective limit is a k-scheme because the natural morphisms X n ′ → X n , for n ′ ≥ n, are affine morphisms. We denote by j n : X ∞ → X n , n ≥ 0, the natural projections.

Given P ∈ X ∞ , with residue field κ(P ), we denote by h P : Spec κ(P ) [[t]] → X the induced κ(P )-arc on X. The image in X of the closed point of Spec κ(P ) [[t]], or equivalently, the image P 0 of P by j 0 : X ∞ → X = X 0 is called the center of P . Then, h P induces a morphism of k-algebras h ♯ P : O X,j0(P ) → κ(P ) [[t]] ; we denote by ν P the function ord t h ♯ P : O X,j0(P ) → N ∪ {∞}.

The space of arcs of A N k = Spec k[x 1 , . . . , x N ] is (A N k ) ∞ = Spec k[X 0 , . . . , X n , . . .] where for n ≥ 0, X n = (X 1;n , . . . , X N ;n ) is an N -uple of variables. For any f ∈ k[x 1 , . . . , x N ], let ∑ ∞ n=0 F n t n be the Taylor expansion of f (

∑ n X n t n ), hence F n ∈ k[X 0 , . . . , X n ]. If X ⊆ A N
k is affine, and I X ⊂ k[x 1 , . . . , x N ] is the ideal defining X in A N k , then we have X ∞ = Spec k[X 0 , . . . , X n , . . .] / ({F n } n≥0,f ∈IX ).

2.2.

For r, m ∈ N, 0 ≤ r ≤ m, let A r,m k := k[[x 1 , . . . , x r ]][x r+1 , . . . , x m ] and let X ⊆ Spec A r,m k be an affine irreducible k-scheme. A point P of X ∞ is stable if there exist G ∈ O X∞ \ P , such that, for n >> 0, the map X n+1 -→ X n induces a trivial fibration j n+1 (Z(P )) ∩ (X n+1 ) G -→ j n (Z(P )) ∩ (X n ) G with fiber A d k , where d = dim X, Z(P ) is the set of zeros of P in X ∞ , j n (Z(P )) is the closure of j n (Z(P )) in X n and (X n ) G is the open subset X n \ Z(G) of X n . This definition is extended to any element X in X k , being X k the subcategory of the category of k-schemes defined by all separated k-schemes which are locally of finite type over some Noetherian complete local ring R 0 with residue field k ( [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] def. 3.3). Note that X k contains the separated k-schemes of finite type and it also contains the k-schemes Spec R, being R the completion of a local ring R which is a k-algebra of finite type. In [START_REF] Reguera | A curve selection lemma in spaces of arcs and the image of the Nash map[END_REF] and [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] a theory of stable points of X ∞ is developed. One important property of these points is the following : Proposition 2.3. [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF]prop. 3.7 (iv)) Let P be a stable point of X ∞ . For n ≥ 0, let P n be the prime ideal P ∩ O jn(X∞) , where j n (X ∞ ) is the closure of j n (X ∞ ) in X n , with the reduced structure. Then we have that dim O jn(X∞),Pn is constant for n >> 0, and since

dim O X∞,P ≤ sup n dim O jn(X∞),Pn it implies that dim O X∞,P < ∞.
And the main result in the theory of stable points is :

2.4. Finiteness property of the stable points ([Re1] th. 4.1,[START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] 3.10). Let P be a stable point of X ∞ , then the formal completion O (X∞) red ,P of the local ring of (X ∞ ) red at P is a Noetherian ring.

Moreover, if X is affine, then there exists G ∈ O X∞ \ P such that the ideal

P ( O (X∞) red ) G is a finitely generated ideal of ( O (X∞) red ) G .
In particular P O (X∞) red ,P is finitely generated.

Besides we have O X∞,P ∼ = O (X∞) red ,P [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] th. 3.13). Hence, from 2.4 it follows that the maximal ideal of O X∞,P is P O X∞,P , and even more, P n = P n O X∞,P for every n > 0 (see [Bo] chap. III, sec. 2, no. 12, corol. 2). Therefore, if P is a stable point of X ∞ then embdim O X∞,P = embdim O (X∞) red ,P .

Though this article we will consider étale morphisms. The following holds : Proposition 2.5. Let X, Z ∈ X k and let σ : X → Z be an étale k-morphism. Then we have

X ∞ ∼ = Z ∞ × Z X in particular, X ∞ is étale over Z ∞ . Therefore, the morphism σ ∞ : X ∞ → Z ∞ induces a map {stable points of X ∞ } → {stable points of Z ∞ }
and, if Q is a stable point of X ∞ and P its image by the previous map, then O Z∞,P → O X∞,Q is étale and

(1)

O X∞,Q ∼ = O Z∞,P ⊗ κ(P ) κ(Q).
Proof : We may suppose that Z = Spec A, X = Spec B where B = (A[x]/(f )) g , f, g ∈ A[x] and the class of f ′ (x) in B is a unit ( [Ra], chap. V, th. 1). Then the stability property in [DL] (see also [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] (8) in 3.4) implies that

X ∞ = Spec (A ∞ [X 0 ] / (F 0 )) G0 where A ∞ = O Z∞ . From this it follows that X ∞ ∼ = Z ∞ × Z X. Moreover, for n ≥ 0, we have X n = Spec (A n [X 0 ] / (F 0 )) G0 that is, X n ∼ = Z n × Z X.
From this, the stability property [DL], lemma 4.1, and the definition of stable point, it follows that, if

Q is a stable point of X ∞ then its image P by σ ∞ is a stable point of Z ∞ .
For the last assertion note that, if

X := Spec O X,Q0 , being Q 0 the center of Q in X, then Q induces a stable point Q in X ∞ because h Q : Spec κ(Q)[[t]] → X factorizes through X, and we have (2) O X∞,Q ∼ = O X∞, Q .
Analogously, O Z∞,P ∼ = O Z∞, P , where Z := Spec O Z,P0 and P is the stable point of Z ∞ induced by P . Therefore, in order to prove (1) we may suppose that Z = Spec A, X = Spec B where A and B are complete local rings and Ra] VIII corol. to lemme 2 and [Ha] III exer. 10.4). Now,

X → Z is local étale, hence B ∼ = A ⊗ κ(P0) κ(Q 0 ) ([
X ∞ ∼ = Z ∞ × Z X, therefore B ∞ ∼ = A ∞ ⊗ κ(P0) κ(Q 0 ) and (B ∞ ) Q ∼ = (A ∞ ) P ⊗ κ(P0) κ(Q 0 ). Thus (A ∞ ) P → (B ∞ ) Q is étale and hence (A ∞ ) P → (B ∞ )
Q is also étale, and from [Ra], VIII corol. to lemme 2, it follows that (B ∞ )

Q ∼ = (A ∞ ) P ⊗ κ(P ) κ(Q), therefore (1) holds.
The inequality in prop. 2.3 may be strict (see [IR] example 2.8). However, if X is nonsingular at P 0 , then we will next show that equality holds. Proposition 2.6. Let P be a stable point of X ∞ . If X is nonsingular at the center P 0 of P , then the ring O X∞,P is regular and essentially of finite type over a field, and we have dim O X∞,P = sup n dim O jn(X∞),(P )n .

Proof : The first statement is prop. 4.2 in [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF]. The second one also follows from the proof of [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF], prop. 4.2. In fact, by prop. 2.5 and since there exists an étale morphism from a neighborhood of P 0 to a subset of A r,d-r k , where d = dim X, we may suppose that X ⊆ A r,d-r k . In this case we have

O X∞ = O X [X 1 , . . . , X n , . . .] and O Xn = O X [X 1 , . . . , X n ], n ≥ 0 where X n = (X 1;n , . . . , X d;n ), n ≥ 1. By 2.4, there exist a finite number of polyno- mials G 1 , . . . , G s , G ∈ O X∞ such that P = ((G 1 , . . . , G s ) : G ∞ ) If n 0 ∈ N is such that O jn 0 (X∞) contains G 1 , . . . , G s , G, then k(X n0+1 , . . . , X n , . . .) ⊂ O X∞,P . This implies that O X∞,P ∼ = k(X n0+1 , . . . , X n , . . .) ⊗ k O jn 0 (X∞),Pn 0
hence we conclude the result.

2.7.

Let X be a reduced separated k-scheme of finite type and let ν be a divisorial valuation on X, i.e. ν is a divisorial valuation on an irreducible component of X. Then there exists a proper and birational morphism π : Y → X, with Y normal, such that the center of ν on Y is a divisor E of Y . We also denote by

ν E the valuation ν. Let π ∞ : Y ∞ → X ∞ be the morphism on the spaces of arcs indu- ced by π. Let Y Ereg ∞ be the inverse image of E ∩ Reg(Y ) by the natural projection j Y 0 : Y ∞ → Y , which is an irreducible subset of Y ∞ , and let N E be the closure of π ∞ (Y Ereg ∞ ). Then N E is an irreducible subset of X ∞ , let P E be the generic point of N E . More generally, for every e ≥ 1, let Y eEreg ∞ := {Q ∈ Y ∞ / ν Q (I E ) = e}, where I E is the ideal defining E in an open affine subset of Reg(Y ) (the set Y eEreg ∞ will be also denoted by Y eE ∞ if Y is nonsingular). Then Y eEreg ∞ is an irreducible subset of Y ∞ , let N eE be the closure of π ∞ (Y eEreg ∞
) and P eE (also denoted by P X eE ) be the generic point of N eE . Note that P eE only depends on e and on the divisorial valuation ν = ν E , more precisely, if π ′ : Y ′ → X is another proper and birational morphism, with Y ′ normal, such that the center E ′ of ν on Y ′ is a divisor, then the point P eE ′ defined by e and E ′ coincides with P eE . We have that P eE is a stable point of X ∞ [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF], prop. 4.1, see also [START_REF] Reguera | A curve selection lemma in spaces of arcs and the image of the Nash map[END_REF], prop. 3.8).

2.8.

With the notation in 2.7, the image of the canonical homomorphism dπ :

π * (∧ d Ω X ) → ∧ d Ω Y is an invertible sheaf at the generic point of E. That is, there exists a nonnegative integer k E such that the fibre at E of the sheaf dπ(π * (∧ d Ω X )) is isomorphic to the fibre at E of O Y (-k E E).
We call k E the Mather discrepancy of X with respect to the prime divisor E. Note that k E ̸ = 0 if and only if π is an isomorphism at the generic point of E, and that k E only depends on the divisorial valuation ν = ν E . We have :

(3) sup n dim O jn(X∞),(PeE )n = e ( k E + 1) ( [DL], lemma 3.4, [FEI], theorem 3.9). Hence by prop. 2.3 we have dim O X∞,P eE ≤ e ( k E + 1).

Moreover, let P be a stable point of X ∞ and let P 0 be its center. If P 0 is the generic point of X then ν P is trivial. Otherwise, ν P is a divisorial valuation ([Re2], (vii) in prop. 3.7 and prop. 3.8), i.e. there exists π : Y → X birational and proper such that the center of ν P on Y is a divisor E and there exists e ∈ N such that ν P = eν E . There exists a stable point P Y ∈ Y ∞ whose image by π ∞ is P ([Re2], prop. 4.1). Therefore P Y ⊇ P Y eE and P ⊇ P eE . Now, assume that X is nonsingular at P 0 , and recall that in this nonsingular case we have k E = k E , where k E is the discrepancy of X with respect to E, which is defined to be the coefficient of E in the divisor K Y /X with exceptional support which is linearly equivalent to K Y -π * (K X ) ( [EM], appendix). Applying prop. 2.6 and lemma 4.3 in [DL] we conclude Corollary 2.9. Let P be a stable point of X ∞ . Suppose that X is nonsingular at the center P 0 of P , and that P 0 is not the generic point of X and ν P = eν E . Then O X∞,P is a regular ring of dimension

dim O X∞,P = ek E + dim O Y∞,P Y .

In particular

dim O X∞,PeE = e(k E + 1).

The following question is open :

Question 2.10. Let P be a stable point of X ∞ and suppose that the local ring O X∞,P is regular. Is X nonsingular at the center P 0 of P ? 3. On the graded algebra of the local ring of a smooth scheme associated to a divisorial valuation From now on, let k be a field of characteristic 0. Through this article, we will denote by k < y 1 , . . . , y r > the henselization of the local ring k[y 1 , . . . , y r ] (y1,...,yr) , being y 1 , . . . , y r indeterminacies (see [Ra] for more details on henselization).

Let η : Y → A d k be a k-morphism dominant and generically finite, where Y is a nonsingular k-scheme, let E be a divisor on Y and let P 0 be the center on A d k of the valuation defined by E. In this section we will define elements {q j,r } (j,r)∈J in the fraction field of O A d ,P0 (prop. 3.3)

♯ : O V,η(y0) → O U ,y0 , then (4) x 1 = u m1 1 x 2 = ∑ 1≤i≤m2 λ 2,i u i 1 + u m2 1 u 2 x 3 = ∑ 1≤i≤m3 λ 3,i (u 2 ) u i 1 + u m3 1 u 3 . . . . . . . . . x δ = ∑ 1≤i≤m δ λ δ,i (u 2 , . . . , u δ-1 ) u i 1 + u m δ 1 u δ x δ+1 = u δ+1 . . . . . . . . . x d = u d where δ = codim A d η(ξ E ), m 1 ≤ ord u1 x j = min{i / λ j,i ̸ = 0} for 2 ≤ j ≤ d, 0 < m 1 ≤ m 2 ≤ . . . ≤ m d , λ j,i (u 2 , . . . , u j-1 ) ∈ k[[u 2 , . . . , u j-1 ]], for 2 ≤ j ≤ δ, 0 ≤ i ≤ m j , and, given j ′ < j, if i < m j ′ then λ j,i ∈ k[[u 2 , . . . , u j ′ -1 ]]. Moreover, since x j -u mj 1 u j belongs to k[[u 1 , . . . , u j-1 ]] and is integral over k[u 1 , . . . , u d ] (u1,...,u d )
, it is also integral over k[u 1 , . . . , u j-1 ] (u1,...,uj-1) . Therefore, after a possible replacement of y 0 by another point in an open subset of U ∩ E, we may suppose that, for 2 ≤ j ≤ δ and 0 ≤ i ≤ m j , λ j,i (u 2 , . . . , u j-1 ) belongs to the henselization k < u 2 , . . . , u j-1 > of the local ring k[u 2 , . . . , u j-1 ] (u2,...uj-1) , and, if i < m j ′ , j ′ < j, then λ j,i belongs to k < u 2 , . . . , u j ′ -1 >.

Besides, from the expression (4) it follows that there exists an open neighborhood of y 0 in E whose closed points y ′ 0 satisfy the same property, i.e. there exists a regular system of parameters of y ′ 0 and of η • φ(y ′ 0 ) for which (4) holds. In fact, replace

u i by u ′ i = u i + c i mod u 1 , for 2 ≤ i ≤ d, where (c i ) i lies in an open subset of k d-1
. Hence, we may suppose with no loss of generality that (5)

λ j,i (u 2 , . . . , u j-1 ) ∈ k < u 2 , . . . , u j-1 > for 2 ≤ j ≤ δ, 0 ≤ i ≤ m j if i < m j ′ , j ′ < j, then λ j,i ∈ k < u 2 , . . . , u j ′ -1 > if λ j,i (u 2 , . . . , u j-1 ) ̸ = 0 then it is a unit in k < u 2 , . . . , u j-1 > λ j,mj (u 2 , . . . , u j-1 ) is a unit, for 2 ≤ j ≤ d. Note that U is nonsingular. Note also that ∧ d Ω V is an invertible sheaf, hence the image of dη : η * (∧ d Ω V ) → ∧ d Ω U is an invertible sheaf. The order a E in E of the corresponding divisor is equal to the order in E of the image of d(η • φ) : (η • φ) * (∧ d Ω V ) → ∧ d Ω U .
So, from now on, after a possible replacement of Y by U and of η : Y → A d by η • φ : U → V , we will suppose that (4) is a local expression of η. Besides, from (4) it follows that : ( 6)

a E = m 1 + . . . + m δ -1.
Lemma 3.1. Let A be a finitely generated k-algebra and let θ : Y → Spec A[x, y] be a k-morphism, where x, y are indeterminacies. Let j, 2 ≤ j ≤ d + 1 and suppose that there exists a multiplicative system S j-1 of A[x] and there exist elements

l j ′ ∈ S -1 j-1 A[x] for 2 ≤ j ′ ≤ j -1 such that, if we set v j ′ := θ ♯ (l j ′ ) for 2 ≤ j ′ ≤ j-1, then {u 1 , v 2 , . . . , v j-1 , u j , . . . , u d } is a regular system of parameters of O Y,y0
. Suppose that the images of x, y by θ ♯ are given by x → u m1 1 and

(7) y → ∑ m1≤i≤m λ i (v 2 , . . . , v j-1 ) u i 1 + u m 1 ϱ mod (u 1 ) m+1 where m ≥ m 1 , ϱ ∈ O Y,y0 and λ i (v 2 , . . . , v j-1 ) ∈ k < v 2 , . . . , v j-1 > Set (8) e := g.c.d.({m 1 } ∪ {i / λ i ̸ = 0}), β 0 := m 1 , e 0 := β 0 β r+1 := min {i / λ i ̸ = 0 and g.c.d.{β 0 , . . . , β r , i} < e r } and e r+1 := g.c.d.{β 0 , . . . , β r+1 } for 1 ≤ r ≤ g -1, being g such that e g = e β g+1 := m.
Let n 0 = 1 and n r := er-1 er for 1 ≤ r ≤ g and let β 0 = β 0 and β r , 1 ≤ r ≤ g + 1 be defined by

(9) β r -n r-1 β r-1 = β r -β r-1 ,
hence we have

(10) β r > n r-1 β r-1 for 1 ≤ r ≤ g,
and β g+1 ≥ n g β g ; n r β r belongs to the semigroup generated by β 0 , . . . , β r-1 , 1 ≤ r ≤ g + 1. andthere exist {h 1 = y, h 2

Then, there exist an open subset U of Y containing ξ E and a sequence of integers {i

s } N s=1 such that (i) i 1 < i 2 < . . . < i N = β g+1 and {i s } N s=1 ⊂ {β 0 } ∪ ∪ g+1 r=1 (n r-1 β r-1 , β r ], (ii) {β r } g+1 r=1 is contained in {i 1 , . . . , i N }, that is, there exist s 1 < s 2 < . . . < s g+1 := N such that i sr = β r for 1 ≤ r ≤ g + 1, (iii) for each closed point y ′ 0 in U ∩ E there exist a regular system of parameters {u 1 , v ′ 2 , . . . , v ′ j-1 , u ′ j , . . . , u ′ d } of O Y,y ′ 0 , where v ′ i = v i + c i , u ′ i = u i + c i , (c i ) i ∈ k d-1 ,
, . . . , h N } satisfying : given s, let r, 1 ≤ r ≤ g + 1, be such that n r-1 β r-1 < i s ≤ β r (or r = 1 if s = 1 and i 1 = β 0 ), then (a) h s ∈ T -1 r-1 . . . T -1 0 S -1 j-1 A[x, y],
where T r ′ is the multiplicative part generated by q r ′ := h s r ′ (resp.

q 0 := x 1 ) for 1 ≤ r ′ ≤ r -1 (resp. r ′ = 0), (b) the image of h s in K(O Y,y ′ 0 ) belongs to O Y,y ′ 0 ,

and if we identify

h s with its image in O Y,y ′ 0 then (11) h s = ∑ is≤i≤m (r) λ s,i (v ′ 2 , . . . , v ′ j-1 ) u i 1 + γ s,m (r) (v ′ 2 , . . . , v ′ j-1 ) u m (r) 1 ϱ mod (u 1 ) m (r) +1 where λ s,i , γ s,m (r) ∈ k < v ′ 2 , . . . v ′ j-1 >, λ s,is ̸ = 0 for 1 ≤ s < N , γ s,m (r) is a unit and m (r) := m + (n 1 -1)β 1 + . . . + (n r-1 -1)β r-1 . Moreover, for r ≤ r ′ ≤ g, let β (r) r ′ := β r ′ + (n 1 -1)β 1 + . . . + (n r-1 -1)β r-1 then we have (12) min { i / λ s,i ̸ = 0 and g.c.d.{e r-1 , β (r) r , . . . , β (r) r ′ -1 , i} < e r ′ -1 } = β (r) r ′ and λ s,β (r) r ′ is a unit. (c) For s ≥ 2, if s = s r-1 + 1 (resp. s r-1 + 1 < s), then h s := q b s 0 0 • • • q b s ρ ρ P s ( µ s h q b s 0 0 • • • q b s ρ ρ , l 2 , . . . , l j-1
)

where h = (q r-1 ) nr-1 (resp. h = h s-1 ), ρ = r -2 (resp. ρ = r -1), the integers {b s r ′ } ρ r ′ =0 are the unique nonnegative integers satisfying b s r ′ < n r ′ , 1 ≤ r ′ ≤ ρ, and n r-1 β r-1 = ∑ 0≤r ′ ≤r-2 b s r ′ β r ′ (resp. i j,s-1 = ∑ 0≤r ′ ≤r-1 b s j,r ′ β j,r ′ ), µ s = (λ s1,β 1 ) b s 1 • • • (λ sρ,β ρ ) b s ρ is a unit, and P s ∈ k[z, v ′ 2 , . . . , v ′ j-1 ] is such that (13) P s (λ, v ′ 2 , . . . , v ′ j-1 ) = 0, ∂P s ∂z (λ, v ′ 2 , . . . , v ′ j-1 ) is a unit in k < v ′ 2 , . . . , v ′ j-1 >, where λ = (λ s-1,is-1 ) nr-1 (resp. λ = λ j,s-1,ij,s-1 ).
Proof : First note that (10) follows from ( 8) and ( 9) (see [Za] 2.2.1 in the Appendix). Note also that there exists an open neighborhood of

y 0 in E such that if y ′ 0 is a closed point on it and {u 1 , v ′ 2 , . . . , v ′ j-1 , u ′ j , . . . , u ′ d } is a regular system of parameters of O Y,y ′ 0 , where v ′ i = v + c i , u ′ i = u i + c i , (c i ) i ∈ k d-1
, then the integers defined by ( 8) and ( 9) for the expression of the image of y in terms of

{u 1 , v ′ 2 , . . . , v ′ j-1 , u ′ j , . . . , u ′ d }
are the same as the ones defined for the expression in (7). Thus, to prove the lemma, it suffices to show that, after a possible replacement of y 0 in an open subset U ∩ E of E, there exist {i s } N s=1 and {h s } N s=1 satisfying (i), (ii) and (a), (b) for the image of

h s in K(O Y,y0 ) (hence v ′ 2 = v 2 , . . . , v ′ j-1 = v j-1 in (11)) and (c).
We will define {i s } N s=1 and {h s } N s=1 by induction on s. First, after a possible replacement of y 0 in an open subset of E, we may suppose that, for every i such andthat (a) and(b) hold for s = 1. Now, let s ≥ 2 and suppose that i 1 < . . . < i s-1 and h 1 , . . . , h s-1 are defined and satisfy the required conditions. If i s-1 = β g+1 then set N := s -1. If not, then i s-1 < β g+1 . Thus, there exist r,

that λ i ̸ = 0 in (7), λ i is a unit in the ring R j-1 := k < v 2 , . . . , v j-1 > . Then, for s = 1, let i 1 := min{i / λ i (v 2 , . . . , v j-1 ) ̸ = 0}and h 1 := y. It is clear that β 0 ≤ i 1 ≤ β 1
1 ≤ r ≤ g + 1 such that i s-1 ∈ {β r-1 } ∪ (n r-1 β r-1 , β r ). Let s 1 < s 2 < . . . < s r-1 ≤ s -1 be such that i s r ′ = β r ′ for 1 ≤ r ′ ≤ r -1 and let q 0 := x, q r ′ := h s r ′ for 1 ≤ r ′ ≤ r -1. If i s-1 = β r-1 , recall that λ s-1,β r-1 (v 2 , . . . , v j-1 ) ∈ R j-1 \{0}, thus (λ s-1,β r-1
) nr belongs to R j-1 \ {0} and hence there exists an irreducible monic polynomial

P s ∈ k[z, v 2 , . . . , v j-1 ] such that P s ((λ s-1,β r-1 ) nr , v 2 , . . . , v j-1 ) = 0 and ∂P s ∂z ((λ s-1,β r-1 ) nr , v 2 , . . . , v j-1 ) ̸ = 0
Moreover, after a possible replacement of y 0 in an open subset of E, we may suppose that ( 14)

P s ((λ s-1,β r-1 ) nr , v 2 , . . . , v j-1 ) = 0 and ∂P s ∂z ((λ s-1,β r-1 ) nr , . . . , v j-1 ) is a unit in R j-1 . Analogously, if i s-1 ∈ (n r-1 β r-1 , β r )
, then after a possible replacement of y 0 in an open subset of E, we may suppose that there exists an irreducible monic polynomial

P s ∈ k[z, v 2 , . . . , v j-1 ] such that (15) P s (λ s-1,is-1 , v 2 , . . . , v j-1 ) = 0, ∂P s ∂z (λ s-1,is-1 , . . . , v j-1 ) is a unit in R j-1 . If i s-1 = β r-1 , let b s r ′ = b r-1,r ′ , 0 ≤ r ′ ≤ r-2, be the unique nonnegative integers satisfying b r-1,r ′ < n r ′ for 1 ≤ r ′ ≤ r -2, and n r-1 β r-1 = ∑ 0≤r ′ ≤r-2 b r-1,r ′ β r ′ , and let µ s := (λ s1,β 1 ) b s 1 • • • (λ sr-2,β r-2 ) b s r-2 , which is a unit in R j-1 , such that the image of q b s 0 0 • • • q b s r-2 r-2 by θ ♯ is equal to µ s u nr-1β r-1 1 mod (u 1 ) nr-1β r-1 +1 . Set (16) h s := q br-10 0 • • • q br-1r-2 r-2 P ′ s ( µ s (q r-1 ) nr-1 q br-10 0 • • • q br-1r-2 r-2
, l 2 , . . . , l j-1

) 14), ( 16) and Taylor's development for

and i s := (n r-1 -1)β r-1 + min { i / i > β r-1 , λ s-1,i ̸ = 0 } , unless we have λ s-1,i = 0 for all i > β r-1 , which implies r -1 = g, then set i s := β g+1 . From (
P s it follows that, if s < N (resp. s = N ) then the ν E -value of the image θ ♯ (h s ) of h s in O Y,y0 is i s > n r-1 β r-1 (resp. i s ≥ i N = β g+1 ),
and the exponents of u 1 in θ ♯ (h s ) with nonzero coefficient (see the left hand side of ( 11)) are determined by the ones in θ ♯ (h s-1 ) by adding (n r-1 -1)β r-1 , therefore n r-1 β r-1 < i s ≤ β r and ( 11) and (13) hold for s. Moreover, for r ≤ r ′ ≤ g,

the coefficient λ s,β (r) r ′ in u β (r) r ′ 1 of θ ♯ (h s ) is equal, modulo product by a unit, to (λ s-1,β r-1 ) nr-1-1 λ s-1,β (r-1) r ′
, therefore it is a unit, and (b) is satisfied. Besides,

h s ∈ T -1 r-2 . . . T -1 0 S -1 j-1 A[x, y],
hence (a) also holds.

If n r-1 β r-1 < i s-1 < β r then e r-1 divides i s-1 (by (b) applied to s -1) and there exist unique nonnegative integers

{b s r ′ } r-1 r ′ =0 satisfying b s r ′ < n r ′ for 1 ≤ r ′ ≤ r -1 and i s-1 = ∑ 0≤r ′ ≤r-1 b s r ′ β r ′ (because n r-1 β r-1 ≤ i s-1 ). Then, let µ s := (λ s1,β 1 ) b s 1 • • • (λ sr-1,β r-1 ) b s r-1 , which is a unit in R j-1 , such that the image of q b s 0 0 • • • q b s r-1 r-1 by θ ♯ is equal to µ s u is-1 1 mod (u 1 ) is-1+1 , and set (17) h s := q b s 0 0 • • • q b s r-1 r-1 P s ( µ s h s-1 q b s 0 0 • • • q b s r-1 r-1
, l 2 , . . . , l j-1

)
and i s := min {i / i > i s-1 , λ s,i ̸ = 0}, unless we have λ s-1,i = 0 for all i > β r-1 , which implies r -1 = g and then we set i s := β g+1 . It is clear that (a) holds and, from ( 15) and ( 17), it follows that, if s < N (resp.

s = N ), then the ν E -value of the image θ ♯ (h s ) of h s in O Y,y0 is i s > i s-1 > n r-1 β r-1 (resp. ≥ i N = β g+1 > n g β g ),
and the exponents of u 1 in θ ♯ (h s ) with nonzero coefficient are the same as the ones for by θ ♯ (h s-1 ), hence n r-1 β r < i s ≤ β r and ( 11) and ( 13) hold for s. Moreover,

for r ≤ r ′ ≤ g, the coefficient λ s,β (r) r ′ in u β (r) r ′ 1 of θ ♯ (h s ) is the same, modulo product by a unit, as the coefficient λ s-1,β (r) r ′ of θ ♯ (h s-1 ), therefore it is a unit, and (b) is satisfied. Besides note that β (r) r = β r for 1 ≤ r ≤ g + 1, hence from the previous construction it follows that {β r } g+1 r=1 ⊂ {i s } N s=1 , hence the result is proved. Corollary 3.2. Let j, 2 ≤ j ≤ δ. Set A := k[x 2 , . . . , x j-1 ], x = x 1 , y = x j , and let θ : Y → Spec A[x 1 , x j ] be the composition of η : Y → A d with the projection A d → Spec A[x 1 , x j ].
Suppose that the hypothesis in lemma 3.1 holds and let the image by η ♯ of x j be given by (18)

x j = ∑ m1≤i≤mj λ ′ j,i (v 2 , . . . , v j-1 ) u i 1 + u mj 1 u j mod (u 1 ) mj +1 . where λ ′ j,i (v 2 , . . . , v j-1 ) ∈ R j-1 = k < v 2 , . . . , v j-1 >. Let {β j,r } gj +1
r=0 , {e j,r } gj r=0 , {n j,r } gj r=0 and {β j,r } gj +1 r=0 be the integers defined by ( 8) and ( 9). Then there exist an open subset U of Y and, for each point y ′ 0 in U ∩ E, a regular system of parameters

{u 1 , v ′ 2 , . . . , v ′ j-1 , u ′ j , . . . , u ′ d } of O Y,y ′ 0 , where v ′ i = v + c i , u ′ i = u i + c i , (c i ) i ∈ k d-1
, and there exist elements {q j,0 = x 1 , q j,1 , . . . , q j,gj +1 } where

q j,r ∈ T -1 r-1 • • • T -1 0 S -1 j-1 [x 1 , x 2 , . . . , x j-1 , x j ]
being T r ′ the multiplicative part generated by q j,r ′ , such that the images of {q j,r

} gj +1 r=0 in O Y,y ′ 0 are given by (19) q j,r = µ j,r (v ′ 2 , . . . , v ′ j-1 ) u β j,r 1 mod (u 1 ) β j,r +1 for 0 ≤ r ≤ g j q j,gj +1 = µ j,gj +1 (v ′ 2 , . . . , v ′ j-1 ) u β j,g j +1 1 u j mod (u 1 ) β j,g 1 +1 +1
where

µ j,r (v ′ 2 , . . . , v ′ j-1 ) is a unit in k < v ′ 2 , . . . , v ′ j-1 > for 0 ≤ r ≤ g j + 1.
Proof : This is consequence of lemma 3.1. In fact, after a possible replacement of y 0 in an open subset of E, we may suppose that there exist {i s } N s=1 and {h s } N s=1 satisfying (i), (ii) and (a), (b) in lemma 3.1. Let q j,0 := x 1 , q j,1 := h s1 , . . . , q j,gj := h sg j . If λ sg j +1, β g+1 = 0 in the expression (11) for η ♯ (h sg j +1 ) then let q j,gj +1 := h sg j +1 . Otherwise, after a possible replacement of y 0 in an open subset of E, we may suppose that there exists an irreducible monic polynomial P ∈ k[z, v 2 , . . . , v j-1 ] such that P (λ sg j +1 ,β j,g j +1 , v 2 , . . . , v j-1 ) = 0 and ∂P ∂z (λ sg j +1,βg j +1 ) is a unit in R j-1 .

Then we proceed as in ( 17), that is we set

q j,gj +1 := q bj,0 j,0 • • • q bj,g j j,gj P   µ h sg j +1 q bj,0 j,0 • • • q bj,g j j,gj , l 2 , . . . , l j-1  
where b j,0 , . . . , b j,gj are nonnegative integers satisfying b j,r < n j,r , 1 ≤ r ≤ g j , and β j,gj +1 = ∑ 0≤r≤gj b j,r β j,r , and µ = (λ s1,β j,1 ) bj,1 • • • (λ sg j ,β j,1 ) bj,g j . Then {q j,r } gj +1 r=0

satisfy the required condition.

x 1 = u m1 x j = ∑ m1≤i≤mj λ j,i (v 2 , . . . , v j-1 ) u i + u mj v j mod (u) mj +1 , for 2 ≤ j ≤ δ x r = v r for δ + 1 ≤ r ≤ d.
where

0 < m 1 ≤ m 2 ≤ . . . ≤ m d and, for 2 ≤ j ≤ δ, if we set R j-1 := k < v 2 , . . . , v j-1 >, then λ j,i (v 2 , . . . , v j-1 ) ∈ R j-1 , λ j,i ̸ = 0 implies that it is a unit in R j-1 , and λ j,mj (v 2 , . . . , v j-1 ) is a unit in R j-1 and (20) if i < m j ′ , j ′ < j, then λ j,i ∈ R j ′ -1 . (ii) For 2 ≤ j ≤ δ, let B j := R j-1 [x 1 , x j ] (x1,xj ) , let ν j be the restriction of ν E to B j , let β j,0 = m 1 , β j,1 , .
. . , β j,gj be a minimal system of generators of the semigroup ν j (B j \{0}) and β j,gj +1 = ν j (I j ), being I j the complete ideal defined by the restriction of ν j to a general fibre of Spec B j → Spec R j-1 . Set

J * := {(1, 0)}∪{(j, r) / 2 ≤ j ≤ δ, 1 ≤ r ≤ g j }, J := J * ∪{(j, g j +1) / 2 ≤ j ≤ δ}
let us consider the lexicographic order in J and, for (j, r) ∈ J , let

J * j,r := {(j ′ , r ′ ) ∈ J * / (j ′ , r ′ ) < (j, r)}, J j,r := {(j ′ , r ′ ) ∈ J / (j ′ , r ′ ) < (j, r)} .
Then, there exist elements {q j,r } (j,r)∈J in k(x 1 , . . . , x j ), more precisely,

(21) q j,r ∈ ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ k[x 1 , . . . , x j ]
where, for (j ′ r ′ ) ∈ J * j,r , T j ′ ,r ′ is the multiplicative system generated by q j ′ ,r ′ , such that : (a.2) q 1,0 := x 1 and, for 2

≤ j ≤ δ, 0 ≤ r ≤ g j + 1, the image of q j,r in the fraction field K(O Y,y0 ) of O Y,y0 belongs to O Y,y0
and, if we identify q j,r with its image, then

(22) q j,r = µ j,r (v 2 , . . . , v j-1 ) u β j,r mod (u) β j,r +1 for 1 ≤ r ≤ g j q j,gj +1 = µ j,gj +1 (v 2 , . . . , v j-1 ) u β j,g j +1 v j mod (u) β j,g j +1 +1
where µ j,r (v 2 , . . . , v j-1 ) is a unit in R j-1 for 1 ≤ r ≤ g j + 1. (b.2) For 2 ≤ j ≤ δ, set q j,0 := q 1,0 = x 1 , e j,r := g.c.d.{β j,0 , . . . , β j,r }, n j,r := ej,r-1 ej,r for 1 ≤ r ≤ g j , and let b j,0 , . . . , b j,gj be the unique nonnegative integers satisfying (23) b j,r < n j,r for 1 ≤ r ≤ g j and β j,gj +1 = ∑ 0≤i≤gj b j,r β j,r , then, identifying q j,r with its image in O Y,y0 , we have q j,gj +1 q bj,0 j,0 . . . q bj,g j j,gj

= v j ∈ O Y,y0 .
(iii) Even more, for 2 ≤ j ≤ δ, there exist nonnegative integers N j and s j,1 < s j,2 < . . . < s j,gj +1 = N j , and elements {h j,s } Nj s=1 , such that q j,r = h j,sj,r for 1 ≤ r ≤ g j + 1, and besides the following holds : given s, let r, 1 ≤ r ≤ g j + 1 be such that s j,r-1 < s ≤ s j,r (resp. r = 1 if s ≤ s j,1 ), then we have :

(a.3) h j,s ∈ ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ k[x 1 , . . . , x j ] (b.3) the image of h j,s in K(O Y,y0
) belongs to O Y,y0 and, if we identify h j,s with its image in O Y,y0 then

h j,s = ∑ ij,s≤i≤m (r) j λ j,s,i (v 2 , . . . , v j-1 ) u i +γ j,s,m (r) j (v 2 , . . . , v j-1 )u m (r) j u j mod (u) m (r) j +1
where n j,r-1

β j,r-1 < i j,s ≤ β j,r , i j,s-1 < i j,s , i j,s = β j,r iff s = s r , λ j,s,i , γ j,s,m (r) j ∈ R j-1 , λ j,s,ij,s , γ j,s,m (r) j
is a unit, and m (r) j

:= m j + (n j,1 -1)β j,1 + . . . + (n j,r-1 -1)β j,r-1 . (c.3) If s = s j,r-1 + 1 (resp. s r-1 + 1 < s), then h j,s is equal to q b s j,0 j,0 • • • q b s j,ρ j,ρ P j,s   µ j,s h q b s j,0 j,0 • • • q b s j,ρ j,ρ , q 2,g2+1 q b2,0 2,0 . . . q b2,g 2 2,g2 , . . . , q j-1,gj-1+1 q bj-1,0 j-1,0 . . . q bj-1,g j-1 j-1,gj-1   where h = q nj,r-1 j,r-1 (resp. h = h j,s-1 ), ρ = r -2 (resp. ρ = r -1), the integers {b s j,r ′ } ρ r ′ =0 satisfy b s j,r ′ < n j,r ′ , 1 ≤ r ′ ≤ ρ, and n j,r-1 β j,r-1 = ∑ 0≤r ′ ≤r-2 b s j,r ′ β j,r ′ (resp. i j,s-1 = ∑ r ′ ≤r-1 b s j,r ′ β j,r ′ ), µ j,s = µ b s j,0 j,0 • • • µ b s j,ρ j,ρ
is a unit, and P j,s ∈ k[z, v 2 , . . . , v j-1 ] is irreducible and satisfies (24) P j,s (λ, v 2 , . . . , v j-1 ) = 0, ∂P j,s ∂z (λ, v 2 , . . . , v j-1 ) is a unit in R j-1 ;

being λ = (λ j,s-1,ij,s-1 ) nj,r-1 (resp. λ = λ j,s-1,ij,s-1 ).

Proof : The result is a consequence of lemma 3.1 and its corollary 3.2. First note that, given j, 2 ≤ j ≤ δ, if there exist {q j,r } gj +1 r=1 in k(x 1 , . . . , x j ) satisfying ( 22) and we define (25) l j := q j,gj +1 q bj,0 j,0 . . . q bj,g j j,gj

∈ k(x 1 , . . . , x j )
where q j,0 = x 1 and {b j,r } gj r=0 satisfy (23), and v j to be the image of l j , then v j belongs to O Y,y0 and besides (26) v j = γ j u j mod (u) where γ j is a unit in R j-1 .

In fact, with the notation in ( 22) we may take

γ j = µj,g 1 +1 µ b j,1 j,1 ...µ b j,g j j,g j .
Note also that, fixed j, 2 ≤ j ≤ δ, if (26) holds for every j ′ ≤ j -1, then the image of x j in O Y,y0 is given by

x j = ∑ m1≤i≤mj λ j,i (v 2 , . . . , v j-1 ) u i 1 + u mj 1 u j mod (u 1 ) mj +1
where λ j,i ∈ R j-1 , m j is the integer in (4), λ j,i ̸ = 0 implies that it is a unit in R j-1 , λ j,mj is a unit in R j-1 and ( 20) holds (recall the conditions in ( 5)). Moreover, the integers {β j,r } gj r=0 (resp. β j,gj +1 ) defined in ( 8) and ( 9) for the image of x j are a minimal system of generators of the semigroup ν j (B j \ 0) (resp. equal to ν j (I j )).

From this, and setting u := u 1 and v r to be the image of x r for δ + 1 ≤ r ≤ d (i.e. v r = u r , δ + 1 ≤ r ≤ d, with the notation in (4)), (i) would follow.

Hence, in order to prove (i) and (ii), it suffices to show that, for 2 ≤ j ≤ δ, there exist {q j,r } gj +1 r=1 satisfying ( 21) and ( 22), where R j-1 is defined taking v j ′ to be the image of l j ′ for 2 ≤ j ′ ≤ j -1 (see ( 25)). We argue by induction on j. For j = 2 the hypothesis in corol. 3.2 is clearly satisfied (we may take S 1 = {1}). Thus, by corol. 3.2, there exist {q 2,r } g2+1 r=1 satisfying ( 21) and ( 22). Now, let j, 2 ≤ j ≤ δ and suppose that, for 2 ≤ j ′ ≤ j -1, there exist {q j ′ ,r } g j ′ +1 r=0 satisfying ( 21) and ( 22). Since v j ′ is defined to be the image of l j ′ , 2 ≤ j ′ ≤ j -1, the hypothesis of corol. 3.2 is satisfied. In fact, there exists a multiplicative part

S j-1 of k[x 1 , . . . , x j-1 ] such that ∏ (j ′ ,r ′ )∈J * j,1 T -1 j ′ ,r ′ k[x 1 , . . . , x j-1 ] ∼ = S -1 j-1 k[x 1 , . . . , x j-1 ], hence l j ′ ∈ S -1 j-1 k[x 1 , . . . , x j-1 ] for 2 ≤ j ′ ≤ j -1. Thus, corol. 3.2 assures the existence of {q j,r } gj +1
r=1 satisfying ( 21) and ( 22). From this, we conclude (i) and (ii). Besides, from the proof of corol. 3.2 (see the proof of lemma 3.1), (iii) follows.

Definition 3.4. The local expression in prop.3.3 (i) (or in (4) at the beginning of this section) will be called a general transverse expression of η : Y → A d k with respect to E. The elements {q j,r } (j,r)∈J obtained in prop. 3.3 (ii) will be called a system of transverse generators for η :

Y → A d k with respect to E. Remark 3.5. For j = 2, B 2 = k[x 1 , x 2 ] (x1,x2
) is a two-dimensional regular local ring. Then q 2,0 , q 2,1 , . . . , q 2,g2 , q 2,g2+1 ∈ B 2 is a minimal generating sequence for ν 2 ( [Sp], theorem 8.6). In fact, since R 1 = k, if we apply lemma 3.1 to y = x 2 then all the λ s,i 's in (11) belong to k, hence we can take P s (z) = z -(λ s-1,is-1 ) nr (resp.

P s (z) = z -λ s-1,is-1 ) in (13). Hence q 2,r ∈ k[x 1 , x 2 ] for 0 ≤ r ≤ g 2 + 1, moreover we have q 2,0 = x 1 , q 2,1 = x 2 - ∑ i<β 2,1 λ 2,i q i β 2,0 2,0 and, for 1 ≤ r ≤ g 2 , q 2,r+1 = q n2,r 2,r -c 2,r q b2,r,0 2,0 . . . q b2,r,r-1 2,r-1 - ∑ γ=(γ0,...,γr) c 2,γ q γ0
2,0 . . . q γr 2,r b 2,r,i 's are the unique nonnegative integers satisfying b 2,r,i < n 2,i for 1 ≤ i ≤ r -1, and n 2,r β 2,r = ∑ 0≤i<r b 2,r,i β 2,i , the γ's are nonnegative integers satisfying γ i < n 2,i for 1 ≤ i ≤ r and n 2,r β 2,r < ∑ i γ i β 2,i , and c 2,r , c 2,γ ∈ k, c 2,r ̸ = 0 and c 2,γ ̸ = 0 only for a finite number of γ's.

Remark 3.6. Let j, 2 ≤ j ≤ δ. Set A := k[v 2 , . . . , v j-1 ], x = x 1 , y = x j and let θ : Y → Spec A[x 1 , x j ] be defined by the morphism of k-algebras given by v j ′ → v j ′ , 2 ≤ j ′ ≤ j -1, x i → η ♯ (x i ), i = 1, j (see (18)). Setting l j ′ = v j ′ , 2 ≤ j ′ ≤ j -1,
and S j-1 = {1}, the hypothesis in lemma 3.1 is satisfied. Let us apply lemma 3.1, then the integers defined in ( 8) and ( 9

q ′ j,r ∈ T ′ -1 j,r-1 • • • T ′ -1 j,0 k[v 2 , . . . , v j-1 , x 1 , x j ]
being T ′ j,r ′ the multiplicative part generated by q ′ j,r ′ , and such that the images by θ ♯ of {q ′ j,r } gj +1 r=0 are {η ♯ (q j,r )} gj +1 r=0 , thus given in (19). In fact, note that q j,r is obtained from q ′ j,r by replacing v i by qi,g i +1

q b i,0 i,0 ...q b i,g i i,g i , for 1 ≤ i ≤ j -1.
On the other hand, for 2 ≤ j ≤ δ, there exists a domain B j-1 which is an étale extension of k[v 2 , . . . , v j-1 ] and contains λ j,i (v 2 , . . . , v j-1 ), m 1 ≤ i ≤ m j (see (i) in prop. 3.3). Let ν j be the valuation on B j-1 [x 1 , x j ] extending ν j and such that ν j (ℓ) = 0 for all ℓ ∈ B j-1 (see (ii) in prop. 3.3). Let q j,1 , . . . , q j,gj +1 ∈ B j-1 [x 1 , x j ] be a minimal generating sequence for ν j defined as in remark 3.4, i.e. q j,0 = x 1 , q j,1 = x j -∑ i<β j,1 λ ′ j,i ( q j,0 ) i β j,0 and, for 1 ≤ r ≤ g j , (27) q j,r+1 = q nj,r j,r -c j,r q bj,r,0 j,0

. . . q bj,r,r-1 j,r-1 -∑ γ=(γ0,...,γr) c j,γ q γ0 j,0 . . . q γr j,r , 1 ≤ r ≤ g j where b j,r,i = b sj,r+1 j,i

, 1 ≤ i ≤ r -1, i.e. b j,r,i < n j,i and n j,r β j,r = ∑ 0≤i<r b j,r,i β j,i , we have ν j-1 ( q γ0 j,0 . . . q γr j,r ) > n j,r β j,r for each sequence γ of nonnegative integers in the right hand side, and c j,r , c j,γ ∈ B j-1 , c j,r ̸ = 0 and c j,γ ̸ = 0 only for a finite number of γ's.

Note that, for 1 ≤ r ≤ g j + 1, in the ring ∏ r-1 r ′ =0 T ′ -1 j,r ′ B j-1 [x 1 , x j ] we have (28) q ′ j,r = q j,r • ℓ + h where ℓ, h ∈ ∏ r-1 s=0 T ′ -1 j,s B j-1 [x 1 , x j ],
ℓ is a unit and ν( h) > β j,r .

3.7. Now, let X be a smooth k-scheme and let ν be a divisorial valuation on an irreducible component X 0 of X. Let P 0 be the center of ν on X and let R := O X,P0 . We consider the graded algebra associated with ν, that is,

gr ν R := ⊗ n∈Φ + ℘ n /℘ + n where Φ + := ν(R \ {0}
) is the semigroup of the valuation and, for n ∈ Φ + ,

℘ n = {h ∈ R / ν(h) ≥ n}, ℘ + n = {h ∈ R / ν(h) > n}.
Let π : Y → X 0 be a proper and birational morphism such that the center of ν on Y is a divisor E, and let η : Y → A d k be the composition of π with an étale morphism X 0 → A d k , where d = dim X 0 . Let us consider the notation introduced in this section for the morphism η, in particular, let {q j,r } (j,r)∈J be a system of transverse generators for η : Y → A d k with respect to E, (prop. 3.3 (ii)). Recall that the center of ν on A d k is (x 1 , . . . , x δ ) and let S := k[x 1 , . . . , x d ] (x1,...,x δ ) .

There exists a proper and birational morphism Z → A d k with Z smooth such that the center of ν on Z is a divisor F . Since O Z,F is the valuation ring of the restriction of ν to K(S), we have that O Z,F ≺ O Y,E , i.e. O Y,E dominates O Z,F , hence, after restricting to some open subset of Y , we may suppose that Y dominates Z, let σ : Y → Z denote the corresponding morphism. Note that we have q j,gj +1 q bj,0 j,0 . . . q bj,g j j,gj

∈ O Z,F for 2 ≤ j ≤ δ.
because these elements belong to K(S) and have ν-value equal to 0 ; we also denote by v j the element qj,g j +1 q b j,0 j,0 ...q b j,g j j,g j of O Z,F (see prop. 3.3 (ii)). Besides, the ramification

index e of O Y,E over O Z,F is equal to g.c.d.({β j ′ ,r ′ } (j ′ ,r ′ )∈J * ). Thus there exist {a j,r } (j,r)∈J * , a j,r ∈ Z, such that (29) z := ∏ (j ′ ,r ′ )∈J * q aj,r j,r ∈ O Z,F and ν (z) = ∑ (j ′ ,r ′ )∈J * a j,r β j,r = e.
Then,

ν(σ * (dz ∧ dv 2 ∧ . . . ∧ dv d )) = e -1
and hence, if k F (A d ) denotes the discrepancy of A d with respect ro F , we have ( 30)

a E = ek F (A d ) + e -1
Since S ≺ R, the initial forms of the elements of k[x 1 , . . . , x d ] are well defined elements in gr ν R, and since q 1,0 = x 1 , applying (21) in prop. 3.3, by recurrence on (j, r) we can define the initial form q j,r of q j,r for every (j, r) ∈ J . We have

q j,r ∈ ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ (gr ν R)
where, for (j ′ , r ′ ) ∈ J * j,r , T j ′ ,r ′ is the multiplicative system generated by q j,r . Let k[{q j,r } (j,r)∈J ] be the k-subalgebra of the fraction field K(gr ν R) of gr ν R generated by the q j,r 's and, for δ + 1 ≤ j ≤ d, let x j be the initial form of x j . With this notation, the following holds :

Theorem 3.8. The system of transverse generators {q j,r } (j,r)∈J satisfy the following properties :

(i) We have an isomorphism of graded rings

G := ∏ (j,r)∈J * T -1 j,r k [ {q j,r } (j,r)∈J , x δ+1 , . . . , x d ] Φ ∼ = A[u e , u -e ]
where deg(u) = 1, and A is a k-algebra which is étale over the polynomial ring in d -

1 variables k[v 2 , . . . , v d ], being deg(v j ) = 0, 2 ≤ j ≤ d. (ii) We have an isomorphism ∏ (j,r)∈J * T -1 j,r gr ν R ∼ = B[u e , u -e ]
whose restriction to G is Φ, where A ⊗ k κ(P 0 ) ⊆ B and the extension is étale. Besides, the fraction field

K(B) of B is κ(E). (iii) For 2 ≤ j ≤ δ, the isomorphism Φ in (i) restricts to G j := ∏ (j,r)∈J * j,g j +1 T -1 j,r k [ {q j ′ ,r ′ } (j ′ ,r ′ )∈Jj,g j +1∪{(j,gj +1)} ] Φ ∼ = A j-1 [v j ][u ej , u -ej ]
where e j := g.c.d.

{β j ′ ,r ′ / (j ′ , r ′ ) ∈ J * j,gj +1 }, A 1 = k and A j-1 is étale over k[v 2 , . . . , v j-1 ] for 2 < j ≤ δ. (iv) For 2 ≤ j ≤ δ, there exists a domain B j-1 étale over A j-1 such that B j-1 [ {q 1,0 } ∪ {q j,r } gj +1 r=1 ] ∼ = B j-1 [ y 1,0 , y j,2 . . . , y j,gj +1 ] / J j
where the y j,r 's are indeterminacies and J j is a prime ideal which is generated by {y

nj,r j,r -c j,r y bj,r,0 1,0 • y bj,r,1 j,1 . . . y bj,rr-1 j,r-1 } gj r=1 , being c j,r ∈ B j-1 .
In particular, the previous ring is a domain which is a complete intersection over B j-1 .

Moreover, for any domain C, any ideal in C[y 1,0 , y j,2 . . . , y j,gj +1 ] generated by {y nj,r j,r -c j,r y bj,r,0 1,0

• y bj,r,1 j,1 . . . y bj,rr-1 j,r-1 } gj r=1 , c j,r ∈ C, is a prime ideal. Proof : First, we have that R = O X,P0 ⊇ k[x 1 , . . . , x d ] (x1,...,x δ ) =: S is étale, hence R ∼ = S ⊗ k κ(P 0 )
where we denote by R (resp. S) the completion with respect to the maximal ideal. Since the valuation ν on R (resp. on S) can be extended to a valuation ν on R (resp. on S) and we have gr ν R = gr ν R (resp. gr ν S = gr ν S) we conclude that gr ν R ∼ = gr ν S ⊗ k κ(P 0 ). Therefore, in (ii) we may suppose that

X = A d k , i.e. R = S.
Keep the notation in prop. 3.3. The morphism S → O Z,F induces an inclusion Φ :

gr ν S → gr ν O Z,F ∼ = κ(F ) [u e ]
where κ(F ) is the residue field of F on Y , which contains k(v 2 , . . . , v d ), and

u, v 2 , . . . , v d are indeterminacies, v j , 2 ≤ j ≤ d (resp. u) is the initial form of v j (resp. u), hence deg(v j ) = 0, deg(u) = 1. We have g2 ∏ r ′ =0 T -1 2,r ′ k [{q 1,0 } ∪ {q 2,r ′ } g2 r ′ =1 ] Φ ∼ = k[u e2,g 2 , u -e2,g 2 ] and hence G 2 Φ ∼ = k[v 2 ][u e2,g 2 , u -e2,g 2 ].
More precisely, the image of the ring in the left hand side in the fraction field

K(gr ν O Z,F ) of gr ν O Z,F is in fact in gr ν O Z,F
and is equal to the ring in the hand side. Arguing by recurrence on j, 2 ≤ j ≤ δ, it follows that ∏

(j ′ ,r ′ )∈J * j,g j +1 T -1 j ′ ,r ′ k [ {q j ′ ,r ′ } (j ′ ,r ′ )∈Jj,g j +1 ] Φ ∼ = A j-1 [u ej , u -ej ]
where e j := g.c.d.{e 2,g2 , . . . , e j,gj } = g.c.d.

{β j ′ ,r ′ / (j ′ , r ′ ) ∈ J * j,gj +1 } and A j-1 is the minimal subring of κ(F ) containing k[v 2 , . . . , v j-1 ] and µ j ′ ,r ′ (v 2 , . . . , v j ′ -1 ), µ j ′ ,r ′ (v 2 , . . . , v j ′ -1 ) -1 for (j ′ , r ′ ) ∈ J * j,gj +1 , hence A j-1 is étale over k[v 2 , . . . , v j-1 ]. Therefore G j Φ ∼ = A j-1 [v j ][u ej , u -ej ] and G = G δ ⊗ k k[x δ+1 , . . . , x d ] Φ ∼ = A[u e δ , u -e δ ]
where A = A δ-1 [v δ , . . . , v d ], hence (i) and (iii) hold.

In order to prove (ii), let B be the minimal subring of κ(F ) containing k[v 2 , . . . v d ] and {λ j,i (v 2 , . . . , v j-1 )} 2≤j≤d,m1≤i≤mj . From the construction of the h j,s 's in prop. 3.3 (iii) (see the proof of (iii) in lemma 3.1) it follows that, for every (j, i), 2 ≤ j ≤ d, m 1 ≤ i ≤ m j , there exists h ∈ ∏ (j,r)∈J * T -1 j,r S such that the initial form of h is λ j,i (v 2 , . . . , v d )u e . Now, let h ∈ S = k[x 1 , . . . , x δ ] (x1,...,x δ ) and let a := ν(h). Then e δ divides a and the image of h in O Y,y0 is equal to λ(v 2 , . . . , v δ )u a modulo u a+1 , where λ(v 2 , . . . , v δ ) ∈ B. Hence the initial form of h belongs to B[u e δ ]. Besides, it follows that the set of elements of K(S) of degree 0 is precisely K(B), that is, κ(F ) = K(B). From this (ii) follows.

For (iv), recall that, given n ∈ N, a field F containing a primitive n-th root of unity ξ and an element b

∈ F * = F \ {0}, if the class of b in F * /F * n has order m, then there exists d ∈ F such that X m -d is an irreducible polynomial in F [X] and moreover X n -b = ∏ n/m i=0 (X m -ξ i d) is the decomposition in F [x] of X n -b in irreducible
factors (see for instance prop. 9.6 in [Mo]). In particular, if A is a domain containing a primitive nth root of unity and b ∈ A is such that

(31) b 1 n ′ ̸ ∈ A for every n ′ > 1, n ′ |n, then X n -b is irreducible in A[x].
For j = 2, with the notation in remark 3.5, let J 2 is the ideal of k [y 1,0 , y 2,1 , . . . y 2,g2 ] generated by {y

n2,r 2,r -c 2,r y b2,r,0 1,0 y b2,r,1 2,1 . . . y b2,r,r-1 2,r-1 } g2 r=1
, where the y 2,r 's are indeterminacies. Let B 1 = A 1 = k and let us consider the morphism of k-algebras

k[y 1,0 , y 2,1 , . . . , y 2,g2+1 ] / J 2 → k[{q 1,0 } ∪ {q 2,r } g2+1 r=1 ]
sending y 2,r , 1 ≤ r ≤ g 2 + 1 (resp. y 1,0 ) to q 2,r (resp. q 1,0 ). Since k[{q 1,0 } ∪ {q 2,r } g2+1 r=1 ] is a 2-dimensional domain, to prove the isomorphism it suffices to show that for 1 ≤ r ≤ g 2 the element y

n2,r 2,r -c 2,r y b2,r,0 1,0 y b2,r,1 2,1 . . . y b2,r,r-1 2,r-1 is irreducible in ( k [y 1,0 , . . . , y 2,r-1 ] / ( {y n 2,r ′ 2,r ′ -c 2,r ′ y b 2,r ′ ,0 1,0 . . . y b 2,r ′ ,r ′ -1 2,r ′ -1 } r-1 r ′ =1
))

[y 2,r ]

i.e. y b2,r,0 1,0 . . . y b2,rr-1 2,r-1 does not have a n ′ -root for any n ′ > 1 dividing n 2,r . In fact, suppose that (32)

y b2,r,0 1,0 . . . y b2,r,r-1 2,r-1 =   ∑ a∈Z r ≥0 λ a y a0 1,0 . . . y ar-1 2,r-1   n ′ mod ( {y n 2,r ′ 2,r ′ -c 2,r ′ y b 2,r ′ ,0 1,0 . . . y b 2,r ′ ,r ′ -1 2,r ′ -1 } r-1 r ′ =1
)

where n ′ |n 2,r , λ a ∈ k, the sum in the right hand side term is finite, then we may suppose that ( 32) is homogeneous with respect to the degree, that is, for each a in (32), we have

n ′ ( ∑ r-1 i=0 a i β 2,i ) = n 2,r β 2,r .
Since there exists at least one a in (32) and we have n 2,r = e2,r-1 e2,r where e 2,l = g.c.d.(β 1,0 , . . . , β 2,i ), i = r -1, r, and n ′ |n 2,r , we conclude that n ′ e 2,r divides β 2,r and also e 2,r-1 , hence n ′ e 2,r divides e 2,r , that is n ′ = 1. Now, let j, 2 < j ≤ δ. Let us consider the notation in remark 3.6. We have

B j-1 [ {q 1,0 } ∪ {q j,r } gj +1 r=1 ] ∼ = B j-1 [ {q ′ 1,0 } ∪ {q ′ j,r } gj +1 r=1
] . Besides, from (28) it follows that, for 1 ≤ r ≤ g j +1, the initial form q ′ j,r of q ′ j,r belongs to

gr νj (B j-1 [x 1 , x j ]), although q ′ j,r ∈ ∏ r-1 r ′ =0 T ′ -1 j,r ′ B j-1 [x 1 , x j ].
It also follows that

B j-1 [ {q ′ 1,0 } ∪ {q ′ j,r } gj +1 r=1 ] ∼ = B j-1 [ y 1,0 , y j,1 , . . . , y j,gj +1 ] / J j
where J j is the ideal generated by {y nj,r j,r -c j,r y bj,r,0 1,0 . . . y bj,r,r-1 j,r-1 } gj r=1 . In fact, from the same argument as in before it follows that, for 1 ≤ r ≤ g j and for any n ′ dividing n j,r , c j,r y bj,r,0

1,0 • • • y bj,r,r-2 j,r-2
does not have a n ′ -root in the ring

B j-1 [y 1,0 , . . . , y j,r-2 ] / ( {y n j,r ′ j,r ′ -c j,r ′ y b j,r ′ ,0 1,0 . . . y b j,r ′ ,r ′ -1 j,r ′ -1 } r-1 r ′ =1
) More precisely, (b j,r,0 , . . . , b j,r,r-2 ) ̸ = (0, . . . , 0) and y bj,r,0

1,0 • • • y bj,r,r-2 j,r-2
does not have a n ′ -root in any ring of the form

C [y 1,0 , . . . , y j,r-2 ] / ( {y n j,r ′ j,r ′ -c j,r ′ y b j,r ′ ,0 1,0 . . . y b j,r ′ ,r ′ -1 j,r ′ -1 } r-1 r ′ =1
)

where C is a domain and the c j,r ′ 's are in C. Hence J j is a prime ideal and (iv) holds. This concludes the proof.

Remark 3.9. Similar ideas to the ones in (ii) in thorem 3.8 appear in [Pi], proof of th. 1.3.8.

Restricting to dimension 3, but considering any valuation ν of rational rank 1 and dimension 3, i.e. ν centered in a regular 3-dimensional ring R, in [Ka] an (infinite) generating sequence {q n } n∈N of ν in R is constructed. Our construction in prop. 3.3 is different to the one in [Ka] and we do not reach a generating sequence. Generating sequences in higher dimensional complete local rings are considered in [LMSS].

Defining coordinates at stable of the space of arcs

Let η : Y → A d k be a k-morphism dominant and generically finite, where Y is a nonsingular k-scheme, let E be a divisor on Y and e ≥ 1, and keep the notation in section 3.

Let P Y eE be the generic point of Y eE ∞ (see 2.7), and let P A d eE be the image by η ∞ of P Y eE , which is a stable point of (A d ) ∞ [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] prop. 4.5). We will first prove (prop. 4.5) that a system of transverse generators for η with respect to E induces a regular system of parameters of O (A d )∞,P A d eE . Then we will conclude theorem 4.8 and corollary 4.10.

Given a finitely generated k-algebra A, let us denote by A ∞ the ring of (Spec A) ∞ . Given l ∈ A, we denote by

∑ ∞ n=0 L n t n the image of l by the morphism of k-algebras A → A ∞ [[t]].
Lemma 4.1. [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] proof of prop. 4.1 (iii)) Let A ⊆ B be finitely generated kalgebras and let θ : Spec B → Spec A be the induced dominant morphism. Let P ′ be a stable point of Spec B ∞ and let P be its image by θ ∞ in Spec A ∞ . Let h ∈ B belonging to the fraction field K(A) of A, h = l/q where l, q ∈ A. Then, there exist

{H n } n≥0 in (A ∞ ) P such that (33) H n ≡ H n mod P ′ (recall that H n ∈ B ∞ for n ≥ 0). Even more, there exists c ∈ N such that Q 0 , . . . , Q c-1 ∈ P , Q c ̸ ∈ P
and there exist polynomials S n on 2(n + 1) indeterminacies with coefficients in k, for n ≥ 0, such that,

H n := S n (L c , . . . , L n+c , Q c , . . . , Q n+c ) (Q c ) n+1 ∈ (A ∞ ) P

satisfies (33).

Proof : First note that P is a stable point of Spec A ∞ [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] prop. 4.5), hence the existence of c such that Q 0 , . . . , Q c-1 ∈ P , Q c ̸ ∈ P ([Re2], th. 3.7 (i)). Then, the result follows from the following observation : given [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] proof of prop. 4.1).

h = l/q, l, q ∈ A, if Q 0 , . . . , Q c-1 ∈ P , Q c ̸ ∈ P , then we have Q c H n + . . . + Q n+c H 0 ≡ L n+c mod P ′ for n ≥ 0. ([

Lemma 4.2. Suppose that the assumptions in lemma 3.1 hold and suppose besides that

θ : Y → Spec A[x, y] is dominant. Let P = P A[x,y] eE be the image of P Y eE by θ ∞ , which is a stable point of Spec A[x, y] ∞ . Let y 0 , the regular system of parame- ters {u, v 2 , . . . , v d } of O Y,y0 and {h 1 = y, h 2 , . . . , h N } satisfy (a) to (c) in 3.1. For 2 ≤ j ′ ≤ j -1, let {L j ′ ;n } n≥0 in (A[x, y] ∞ ) P be such that L j ′ ;n ≡ L j ′ ;n mod P Y eE (see lemma 4.1). Then, there exists a multiplicative system S j-1 of A[x] ∞ such that L j ′ ;n ∈ S -1 j-1 A[x] ∞ for 2 ≤ j ′ ≤ j -1, n ≥ 0 and there exist elements {H s;n } 1≤s≤N,n≥0 in (A[x, y] ∞ ) P , n ≥ 0, satisfying : (i) H s;n ≡ H s;n mod P Y eE , therefore H s;n ∈ P (A[x, y] ∞ ) P for 0 ≤ n ≤ ei s -1 and H s;eis ̸ ∈ P (A[x, y] ∞ ) P . (ii) Let r, 1 ≤ r ≤ g + 1 be such that n r-1 β r-1 < i s ≤ β r (resp. r = 1 if s = 1 and i 1 = β 0 ). Set Q 0;n := X n for n ≥ 0, Q r ′ ;n := H s r ′ ;n , for 1 ≤ r ′ < r, n ≥ 0 and let T r ′ is the multiplicative part generated by Q r ′ ;eβ r ′ , 0 ≤ r ′ < r.
Then, for n ≥ e(β r -β r ), we have :

H s;n ∈ T -1 r-1 . . . T -1 0 S -1 j-1 A ∞ [X 0 , X 1 , . . . , X n , . . . , Y 0 , Y 1 , . . . , Y n-e(β r -βr) ]. (iii) If s = 1 then H 1;n = Y n for n ≥ 0. If s > 1 then H s;n ∈ ({Q r ′ ;n } r ′ ≤r-1 n<eβ r ′ ∪ {H s-1;n } n<eis-1 ) for 0 ≤ n < max {en r-1 β r-1 , ei s-1 } H s;n = u s,n Y n-(β r -βr) + ρ s,n for n > max {en r-1 β r-1 , ei s-1 } where u s,n , ρ s,n ∈ B r ∞ [Y eβr-1+1 , . . . , Y n-e(β r -βj,r)-1 ] and u s,n is a unit. (iv) Suppose that s > 1. If i s-1 = β r-1 (resp. i s-1 ∈ (n r-1 β r-1 , β r )) then H s;enr-1β r-1 (resp. H s;eis-1 ) is equal to Q b s 0 0;eβ 0 • • • Q b s ρ ρ;eβ ρ • P s    c s H Q b s 0 0;eβ 0 • • • Q b s ρ ρ;eβ ρ , L 2;0 , . . . , L j-1;0    where H = (Q r-1;eβ r-1 ) nr-1 (resp. H = H s-1;eis-1 ), c s ∈ k \ {0} and ρ, {b s r ′ } ρ r ′ =0 and P s are as in (c) in 3.1. (v) Fixed r, 1 ≤ r ≤ g + 1, the following ideals in T -1 r-1 . . . T -1 0 S -1 j-1 A[x, y] ∞ are equal : ( {Q r ′ ;n } 0≤r ′ ≤r 0≤n≤eβ r ′ -1 ) = ( {Q r ′ ;n } 0≤r ′ ≤1 0≤n≤eβ r ′ -1 ∪ {Q r ′ ;n } 2≤r ′ ≤r en r ′ -1 β r ′ -1 ≤n≤eβ r ′ -1
) and also the ideal generated by

{Q 0;n } em1-1 n=0 ∪ {H 1;n } ei1-1 n=0 ∪ ( ∪ s1 s=2 {H s;n } eis-1 e is-1 ) ∪ ∪ r r ′ =2 ( {H s r ′ -1 +1;n } eis r ′ -1 +1-1 n=en r ′ -1 β r ′ -1 ∪ ( ∪ s ′ r s=s r ′ -1 +2 {H s;n } eis-1 n=eis-1
)) .

Proof : The existence of S j-1 follows from lemma 4.1 ; in fact, it suffices to ask S j-1 to contain the elements Q c where q ∈ S j-1 and c is such that Q 0 , . . . , Q c-1 ∈ P and Q c ̸ ∈ P . Now, let us define the elements

{H s;n } n≥0 , 1 ≤ s ≤ N , by induction on s. For s = 1, h 1 = y ∈ A[x, y], so H 1;n ∈ A[x, y] ∞ for n ≥ 0. We set H 1;n := H 1;n = Y n ∈ A[x, y] ∞ for n ≥ 0. It is clear that (i) to (iii) are satisfied. Now, let s, 2 ≤ s ≤ N ,
and suppose that H s ′ ;n ∈ (A[x, y] ∞ ) P are defined, for 1 ≤ s ′ < s, n ≥ 0, and satisfy the conditions. Let r, 1 ≤ r ≤ g j + 1 be such that i

s-1 ∈ {β r-1 } ∪ (n r-1 β r-1 , β r ). Therefore {Q r ′ ;n } 0≤r ′ <r,n≥0 in (A[x,
y] ∞ ) P are defined, and satisfy :

Q r ′ ;n ∈ P (A[x, y] ∞ ) P for 0 ≤ n ≤ eβ r ′ -1 and Q r ′ ;eβ r ′ ̸ ∈ P (A[x, y] ∞ ) P .
Hence, for every l in the k-algebra k[{q r ′ } 0≤r ′ <r ∪ {h s-1 }] generated by q r ′ , 0 ≤ r ′ < r, and h s-1 , and for every n ≥ 0, there exists a polynomial function

L n on {Q r ′ ;n } r ′ <r,n≥eβ j ′ ,r ′ ∪ {H s-1;n } n≥eij,s-1 such that L n ≡ L n mod P Y eE . Moreover, given (34) h = l q ∈ O Y,y0 where l ∈ k[{q r ′ } 0≤r ′ <r ∪ {h s-1 }], q = ∏ 0≤r ′ <r q a r ′ r ′ being a r ′ ∈ N ∪ {0}, let c = ∑ 0≤r ′ <r a r ′ eβ r ′ , so that Q 0 , . . . , Q c-1 ∈ P , Q c ̸ ∈ P and set H n := S n (L c , L c+1 , . . . , L n+c , Q c , Q c+1 , . . . , Q n+c ) (Q c ) n+1 ∈ (A[x, y] ∞ ) P
where S n is the polynomial in lemma 4.1 ; then H n ≡ H n mod P Y eE . From this and (c) in lemma 3.1, which expresses h s as a polynomial in elements of the form (34), the definition of {H s;n } n≥0 ⊂ (A[x, y] ∞ ) P follows. They satisfy (i) and, from the expression in 3.1 (c) and the induction hypothesis, it follows that (ii) holds and that the first statement in (iii) and also (iv) are satisfied. In (iv), c s is the class of µ s ∈ R j-1 , hence c s ̸ = 0. The second statement in (iii) is obtained from the expression in 3.1 (c) and the induction hypothesis, applying also (13) in lemma 3.1. Finally, (v) can also be proved by induction, applying the same argument as before.

Let A 1,0 ∞ := k and, for 2

≤ j ≤ δ, let A j,1 ∞ := k[X j-1 0 , . . . X j-1 n , . . .], A j,r ∞ := A j,1 ∞ [X j;0 , . . . , X j,eβj,r-1 ] , 2 ≤ r ≤ g j + 1
, where X j-1 n := (X 1;n , . . . , X j-1;n ). Let {q j,r } (j,r)∈J be a system of transverse generators for η : Y → A d k with respect to E, as in 3.3 (ii). Even more, for 2 ≤ j ≤ δ, let us consider the elements {h j,s } Nj s=1 in 3.3 (iii) and set h 1,0 := q 1,0 = x 1 ∈ A. Let

I := {(1, 0)} ∪ {(j, s) / 2 ≤ j ≤ δ, 1 ≤ s ≤ N j }.
Then we have :

Lemma 4.3. There exist elements {H j,s;n } (j,s)∈I,n≥0 in O (A d )∞,P A d eE , n ≥ 0, satis- fying : (i) H j,s;n ≡ H j,s;n mod P Y eE , therefore H j,s;n ∈ P A d eE for 0 ≤ n ≤ j,s -1 and H j,s;eij,s ̸ ∈ P A d eE (ii) We have H 1,0;n = X 1;n for n ≥ 0. For 2 ≤ j ≤ δ, let r, 1 ≤ r ≤ g j + 1
be such that n j,r-1 β j,r-1 < i j,s ≤ β j,r (resp. r = 1 if s = 1 and i j,1 = β j,0 ). For (j ′ , r ′ ) ∈ J j,r , set Q j ′ ,r ′ ;n := H j ′ ,s r ′ ;n , n ≥ 0 and, for (j ′ , r ′ ) ∈ J * j,r , let T j ′ ,r ′ be the multiplicative system generated by Q j ′ ,r ′ ;eβ j ′ ,r ′ . Then, for n ≥ e(β j,r -β j,r ) we have :

H j,s;n ∈ ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ A j,r
∞ [X j;eβj,r-1+1 , . . . , X j;n-e(β j,r -βj,r) ]

(if r = 1, replace X j;eβj,r-1+1 by X j;0 in the previous equality). (iii) For 2 ≤ j ≤ δ, if s = 1 then H j,s;n = X j;n for n ≥ 0. If s > 1 then :

H j,s;n ∈ ( {Q j,r ′ ;n } r ′ ≤r-1 n<eβ r ′ ∪ {H j,s-1;n } n<eis-1
) for 0 ≤ n < max {en j,r-1 β j,r-1 , ei j,s-1 } and H j,s;n = u j,s,n X j;n-e(β j,r -βj,r) + ρ j,s,n for n > max {en j,r-1 β j,r-1 , ei j,s-1 } where u j,s,n , ρ j,s,n ∈ ∏

(j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ A j,r
∞ [X j,eβj,r-1+1 , . . . X j;n-e(β j,r -βj,r)-1 ] and u j,s,n is a unit. (iv) Suppose that j, s ≥ 2. If i j,s-1 = β j,r-1 (resp. i j,s-1 ∈ (n j,r-1 β j,r-1 , β j,r ))

then H j,s;enj,r-1β j,r-1 (resp. H j,s;eij,s-1 ) is equal to

Q b s j,0 1,0;eβ j,0 • Q b s j,1 j,1;eβ j,1 • • • Q b s j,ρ j,ρ;eβ j,ρ • •P j,s    c j,s H Q b s j,0 1,0;eβ j,0 • • • Q b s j,ρ j,ρ;eβ j,ρ , . . . , Q j-1,gj-1+1;eβ j-1,g j-1 +1 Q bj-1,0 1,0;eβ j-1,0 . . . Q bj-1,g j-1 j-1,gj-1;eβ j-1,g j-1   
where H = (Q j,r-1;eβ j,r-1 ) nj,r-1 (resp. H = H j,s-1;eij,s-1 ), c j,s ∈ k \ {0} and ρ, {b s j,r ′ } ρ r ′ =0 and P s are as in (c.3) in 3.3. (v) Set G 1,0 := {H 1,0;n / 0 ≤ n ≤ e m 1 -1} and, for 2 ≤ j ≤ δ, G j,1 := {H j,1;n / 0 ≤ n ≤ e i j,1 -1} ∪ ∪ s1 s=2 {H j,s;n / e i j,s-1 ≤ n ≤ e i j,s -1} G j,r := {H j,sr-1+1;n / e n j,r-1 β j,r-1 ≤ n ≤ e i j,sr-1+1 -1} ∪ ∪ sr s=sr-1+2 {H j,s;n / e i j,s-1 ≤ n ≤ e i j,s -1} for 2 ≤ r ≤ g j + 1.

then, for 2 ≤ j ≤ δ and 1 ≤ r ≤ g 1 + 1, we have (

{Q j,r ′ ;n } 0≤r ′ ≤1 0≤n≤eβ r ′ -1 ∪ {Q j,r ′ ;n } 2≤r ′ ≤r en j,r ′ -1β j,r ′ -1 ≤n≤eβ r ′ -1 ) ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ A j+1,1 ∞ = = (G 1,0 ∪ G j,1 ∪ . . . G j,r ) ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ A j+1,1 ∞ .
Proof : Let us prove, by induction on j, 1 ≤ j ≤ δ, the existence of {H j,s;n } (j,s)∈I n≥0 satisfying the required conditions. For j = 1, (j, s) = (1, 0), h 1,0

:= q 1,0 = x 1 ∈ O A d ,η(y0) , so, if we set H 1,0;n := H 1,0;n = X 1;n ∈ (A d )∞,P A d eE
for n ≥ 0 then it is clear that (i) to (iii) are satisfied. Now, let j, 2 ≤ j ≤ δ, and suppose that

H j ′ ,s ′ ;n ∈ O (A d )∞,P A d eE
are defined, for j ′ < j, (j ′ , s ′ ) ∈ I, n ≥ 0, and satisfy the conditions. Then the result follows applying lemma 4.2 to Y → Spec A[x 1 , x j ], where A = k[x 2 , . . . , x j-1 ], and the following remark : since

l j ′ = q j ′ ,g j ′ +1 q b j ′ ,0 1,0 q b j ′ ,1 1,1 . . . q b j ′ ,g j ′ j ′ ,g j ′ for 2 ≤ j ′ ≤ j -1
we may take S j-1 = {Q j ′ ,r ′ ;eβ j ′ ,r ′ } (j ′ ,r ′ )∈J * j-1,g j-1 +1 and

L j ′ ;0 = Q j ′ ,g j ′ +1;eβ j ′ ,g j ′ +1 Q b j ′ ,0 1,0;eβ j ′ ,0 • Q b j ′ ,1 j ′ ,1;eβ j ′ ,1 • • • Q b j ′ ,g j ′ j ′ ,g j ′ ;eβ j ′ ,g j ′ .
From this, (i) to (iv) follow for j. This concludes the proof.

Remark 4.4. Let j, 2 ≤ j ≤ δ. Let { q j,r } gj +1 r=0 in B j-1 [x 1 , x j ] be as in remark 3.6, and Q j,r;n ∈ B j-1 [x 1 , x j ] ∞ , n ≥ 0, as in the beginning of this section. Arguing by recurrence and applying (27), we obtain that, for 1 ≤ r ≤ g j + 1,

(35) Q j,r;n ∈ ( { Q j ′ ,r ′ ;n } 0≤r ′ ≤r-1 0≤n≤β j ′ ,r ′ -1 ) 2 B j-1 [x 1 , x j ] ∞ for 0 ≤ n < e ( (n r-1 -1)β r-1 + . . . + (n 1 -1)β 1 )
= e(β j,r -β j,r ). Set ϵ( q j,r ) := e(β j,r -β j,r ).

Analogously, for {q

′ j,r } gj +1 r=0 , q ′ j,r ∈ T ′-1 j,r-1 • • • T ′-1 j,0 k[v 2 , . . . , v j-1 , x 1 , x j ] (see re- mark 3.6), let {Q ′ j,r;n } n≥0 in ∏ 0≤s≤r-1 T ′ -1 j,s k[v 2 , . . . , v j-1 , x 1 ,
x j ] ∞ be obtained applying lemma 4.2. Given r, 0 ≤ r ≤ g j + 1, let {a s } 0≤s≤r-1 be nonnegative integers such that

z ′ j,r := q ′ j,r • ∏ 0≤s≤r-1 q ′ j,s as ∈ k[v 2 , . . . , v j-1 , x 1 , x j ]. and let Z ′ j,r;n ∈ k[v 2 , . . . , v j-1 , x 1 , x j ] ∞ [[t]]
, n ≥ 0, as before. Arguing by recurrence and applying (c) in lemma 3.1 it follows that (36)

Z ′ j,r;n ∈ ( {Q ′ j,s;n } 0≤s≤r-1 0≤n≤β j,s -1 ) 2 ∏ 0≤s≤r-1 T ′ -1 j,s k[v 2 , . . . , v j-1 , x 1 , x j ] ∞ for 0 ≤ n < ϵ(z ′ j,r ) := e ( ν(z ′ j,r ) -β j,r
) .

Now, with the assumptions and notation in lemma 4.3, given (j, r) ∈ J , let {a j ′ ,r ′ (j, r)} (j ′ ,r ′ )∈J * j,r be any sequence of nonnegative integers such that

z j,r := q j,r • ∏ (j ′ ,r ′ )∈J * j,r q a j ′ ,r ′ (j,r) j ′ ,r ′ ∈ k[x 1 , . . . , x j ]
and let α j,r := ν(z j,r ) and let Z j,r;n ∈ k[x 1 , . . . , x j ] ∞ , n ≥ 0, as before. Then we have (

{Z j ′ ,r ′ ;n } (j ′ ,r ′ )∈J j,r 0≤n≤eα j ′ ,r ′ -1 ) ∏ (j ′ ,r ′ )∈J * j,r S -1 j ′ ,r ′ k[x 1 , . . . , x j ] ∞ = =   {Q j ′ ,r ′ ;n } (j ′ ,r ′ )∈J j,r 0≤n≤eβ j ′ ,r ′ -1   ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ k[x 1 , . . . , x j ] ∞
where S j ′ ,r ′ is the multiplicative part generated by Z j ′ ,r ′ ;eα j ′ ,r ′ . Moreover, arguing by recurrence and applying (c.2) in prop. 3.3 and also the condition (20), it follows that (37) Z j,r;n ∈ (

{Z j ′ ,r ′ ;n } (j ′ ,r ′ )∈J j,r 0≤n≤eα j ′ ,r ′ -1 ) 2 ∏ (j ′ ,r ′ )∈J * j,r S -1 j ′ ,r ′ k[x 1 , . . . , x j ] ∞
for 0 ≤ n < ϵ(z j,r ) := e (ν(z j,r ) -β j,r ). In fact, the proof is based on the one for (36), taking into account condition (20).

Let G := ∪ (j,r)∈J G j,r where the G j,r 's are defined in lemma 4.3 (v). Note that the cardinal of G 1,1 is em 1 and, for 2

≤ j ≤ δ, ♯ ( ∪ gj +1 r=1 G j,r
)

= e β j,1 + (e β j,2 -e n j,1 β j,1 ) + . . . + (e β j,gj +1 -e n j,gj

β j,gj ) = e ( β j,1 + ( β j,2 -β j,1 ) + . . . + (β j,gj +1 -β j,gj ) ) = e β j,gj +1 = e m j .
Hence, applying ( 6) and ( 30) we obtain ♯G = e (a E + 1) = e e (k F (A d ) + 1).

Proposition 4.5. We have

P A d eE O (A d )∞,P A d eE = (G) O (A d )∞,P A d eE moreover, there exists L ∈ O (A d )∞ \P A d eE such that P A d eE (O (A d )∞ ) L = (G) (O (A d )∞ ) L . Besides,
G j,r ⊂ ∏ (j ′ ,r ′ )∈J * j,r T -1 j ′ ,r ′ A j,r
∞ [X j;eβj,r-1+1 , . . . , X j;eβj,r-1 ]

(if r = 0 or 1, replace X j;eβj,r-1+1 by X j;0 and set β 1,0 := m 1 ). Then, for each j, 2 ≤ j ≤ δ, there exists M j ∈ N such that

Q j ′ ,r ′ ∈ ∏ (j ′′ ,r ′′ )∈J * j ′ ,r ′ T -1 j ′′ ,r ′′ k[X j 0 , . . . , X j Mj ] for every (j ′ , r ′ ) ∈ J * j,gj +1
and, if we set

B j ∞ := ∏ (j ′ ,r ′ )∈J * j,g j +1 T -1 j ′ ,r ′ k[X j 0 , . . . , X j Mj ] then G j := ∪ (j ′ ,r ′ )∈Jj,g j +1 ∪{(j,gj +1)} G j ′ ,r ′ ⊂ B j ∞
(in fact, M j can be taken to be equal to em j ). Set P j := B j ∞ ∩ P A d eE . We will prove, by induction on j, 2 ≤ j ≤ δ, that there exists L j ∈ B j ∞ \ P j such that the ring (B j ∞ ) Lj /(G j ) is a domain. For j = 2, we have h 1,0 = x 1 , thus G 1,0 = {X 1;0 , . . . , X 1;em1-1 } and, applying remark 3.5 and (iii) in lemma 4.3 to Q 2,r;n , en j,r-1 β 2,r-1 < n < eβ 2,r and (iv) in lemma 4.3 to Q 2,r;eβ 2,r , we obtain that

B 2 ∞ / (G 2 ) is isomorphic to ( S -1 2 k [y 2,0 , y 2,2 . . . , y 2,g2+1 ] / J 2 ) [{X 1;n } em1<n≤M2 ∪ {X 2;n } eβ2,g 2 +1 <n≤M2 ]
where the image of y 2,r , 1 ≤ r ≤ g 2 + 1 (resp. y 2,0 ) is Q 2,r;eβ 2,r (resp. X 1,em1 ), J 2 is the ideal in th. 3.8 (iv) and S 2 is the multiplicative part generated by {y 2,r } g2 r=0 , therefore B 2 ∞ /(G 2 ) is a domain by th. 3.8.

Let j, 3 ≤ j ≤ δ, and suppose that the result holds for j -1. Applying (iii) in lemma 4.3 to H j,s;n , for ei j,s-1 < n ≤ ei j,s -1 (resp. en j,r-1 β j,r-1 < n ≤ ei j,sr-1+1 -1) if s r-1 + 2 ≤ s ≤ s r (resp. s = s r-1 + 1) and applying (iv) in 4.3 to H j,s;eij,s-1 (resp. H j,s;enj,r-1β j,r-1 ), we obtain that there exists an étale extension

B j ∞ of B j ∞ containing the image of P Y eE , i.e. the contraction of P Y eE to B j ∞ is a prime ideal P j ̸ = B j ∞ , and such that B j ∞ /(G j ) B j ∞ is isomorphic to a localization of ( S -1 j D j-1 [y j,1 , . . . , y j,gj +1 ] / J j ) [ {X j;n } eβj,g j +1 <n≤Mj
] where D j-1 is a domain which is an étale extension of B j-1 ∞ /(G j-1 ), S j is the multiplicative part generated by {y j,r } gj r=1 and J j is an ideal generated by {y nj,r j,rc j,r y bj,r,0

1,0 • y bj,r,1 j,1 . . . y bj,rr-1 j,r-1 } gj r=1 , being c j,r ∈ D j-1 and y 1,0 = X 1,em1 ∈ D j-1 . Here y j,r is identified with Q j,r;eβ j,r . Applying th. 3.8 (iv) we conclude that B j ∞ /(G j ) is a domain. Since the morphism (B j ∞ ) Pj /(G j ) → ( B j ∞ ) Pj /(G j ) B j ∞ is local étale, hence an inclusion of local rings, we conclude that (B j ∞ ) Pj /(G j ) is a domain. Therefore, there exists L j ∈ B j ∞ \ P j such that (B j ∞ ) Lj /(G j ) is a domain (recall that B j ∞ is the localization of a finitely generated k-algebra).
In particular, it follows that there exists eE , we have eβ j,r ≤ ν P ′ (q j,r ) ≤ ν(q j,r ) = eβ j,r for (j, r) ∈ J .

L δ ∈ B δ ∞ \ P δ ⊂ O (A d )∞ \ P A d eE such that the ideal generated by G in ( ∏ (j,r)∈J * T -1 j,r O (A d )∞ ) L δ is a prime ideal. From this it follows that there exists L ∈ O (A d )∞ \P A d eE such that (G) (O (A d )∞ ) L is a prime ideal, in fact, we may take L = L δ • ∏ (j,r)∈J * Q aj,
Therefore ν P ′ (q j,r ) = eβ j,r for every (j, r) ∈ J and hence

ν P ′   q j,gj +1 q bj,0 j,0 . . . q bj,g j j,gj   = 0 for 2 ≤ j ≤ δ and ν P ′ (z) = ∑ (j ′ ,r ′ )∈J *
a j,r e β j,r = e e (recall (29) in 3.7). From this it follows that the morphism of k-algebras h

♯ P ′ : O X,P0 → κ(P ′ )[[t]] induced by the arc h P ′ extends to O Z,F . That is, h P ′ : Spec κ(P ′ )[[t]
] → X lifts to (Z, F ), more precisely, since ν P ′ (z) = ee, this lifting defines a point in Z eeF ∞ . Therefore P ′ ∈ {P A d eeF }, hence we conclude that P ′ = P A d eeF = P A d eE . Finally, since ♯G = e e (k F (A d ) + 1), the end of the proof follows from prop. 2.6 and equality (3), which is in fact lemma 3.4 in [DL].

Remark 4.6. Alternatively, in the proof of prop. 4.5 it can be proved by induction on j, 2 ≤ j ≤ δ, and applying (iii) and (iv) in lemma 4.3, not only that (G j ) is a prime ideal of B j ∞ , but also that the elements in G j are independent in (G j ) / (G j ) 2 . Then, lemma 3.4 in [DL] can be recovered (at least for X smooth) from propositions 4.5 and 2.6. Therefore, prop. 4.5 can be seen as a new version of lemma 3.4 in [DL], which is in fact the change of variables theorem in the motivic integration. Definition 4.7. Let η : Y → A d k be a k-morphism dominant and generically finite, where Y is a nonsingular k-scheme, let E be a divisor on Y and let e ≥ 1. Let {q j,r } (j,r)∈J be a system of transverse generators for η with respect to E (def. 3.4), and let {Q j,r;n } (j,r)∈J ,n≥0 defined as in lemma 4.3. We call Q := {Q j,r;n } (j,r)∈J , enj,r-1β j,r-1 ≤n≤eβ j,r -1 a regular system of parameters of O (A d )∞,P A d eE associated to {q j,r } (j,r)∈J .

In fact, note that by prop. 4.5 (see also lemma 4. Theorem 4.8. Assume that char k = 0. Let X be a nonsingular k-scheme, let ν be a divisorial valuation on an irreducible component X 0 of X, and let e ∈ N. Let π : Y → X 0 be a proper and birational morphism such that the center of ν on Y is a divisor E, and let η : Y → A d k be the composition of π with an étale morphism X 0 → A d k , where d = dim X 0 . Let Q = {Q j,r;n } (j,r)∈J , 0≤n≤eβ j,r -1 be a regular system of parameters of O (A d )∞,P A d eE associated to a system of transverse generators for η with respect to E. Then Q is also a regular system of parameters of O X∞,P X eE , that is . Then it follows from prop. 4.5. In fact, for the first assertion note that in this case k E (A d k ) is equal to the discrepancy k E of X with respect to E. For the second assertion, let {q j,r } (j,r)∈J be a system of transverse generators for η with respect to E. For each (j, r) ∈ J there exists a sequence of nonnegative integers {a j ′ ,r ′ (j, r)} (j ′ ,r ′ )∈J * j,r , such that z j,r := q j,r • ∏ (j ′ ,r ′ )∈J * j,r q a j ′ ,r ′ (j,r) j ′ ,r ′ ∈ O A d ,P0 .

P X eE O
(see prop. 3.3). Then,from prop. 4.5,(38) follows. This concludes the proof. Remark 4.9. Let P be any stable point of X ∞ and suppose that X is nonsingular at the center P 0 of P and that P 0 is not the generic point of X. There exists a birational and proper morphism π : Y → X such that the center of ν P on Y is a divisor E, and e ∈ N such that ν P = eν E ([Re2], (vii) in prop. 3.7). Let P Y ∈ Y ∞ whose image by π ∞ is P , then we have dim O X∞,P = ek E + dim O Y∞,P Y (corol. 2.9). Since P Y ⊇ P Y eE and P ⊇ P X eE , with the notation in theorem 4.8 and prop. 3.3, {U 0 , . . . , U e-1 } is part of a regular system of parameters of O Y∞,P Y and Q = {Q j,r;n } (j,r)∈J , 0≤n≤eβ j,r -1 is part of a regular system of parameters of O X∞,P . Moreover, suppose that {U 0 , . . . , U e-1 , G 1 , . . . , G s } is a regular system of parameters of O Y∞,P Y . To describe a regular system of parameters of O X∞,P we add to Q the following elements : By lemma 4.1 and since π is birational, for each y ∈ O Y and for each n, there exists Y n ∈ O X∞,P such that

Y n ≡ Y n mod P.
Then, let G i ∈ O X∞,P , 1 ≤ i ≤ s be obtained from G i by replacing U n and V j;n by U n andV j;n , for n ≥ 0, 2 ≤ j ≤ d. We have

G i ≡ G i mod P.
and Q ∪ {G 1 , . . . , G s } is a regular system of parameters of O X∞,P . Now let us consider a reduced separated k-scheme of finite type X and a divisorial valuation ν on X centered on Sing X. There exists a resolution of singularities π : Y → X (i.e. π is a proper, birational k-morphism, with Y smooth, such that the induced morphism Y \ π -1 (Sing X) → X \ Sing X is an isomorphism) such that the center of ν on Y is a divisor E. Corollary 4.10. Assume that char k = 0. Let X be a reduced separated k-scheme of finite type, let ν be a divisorial valuation on an irreducible component X 0 of X centered on Sing X and let e ∈ N. Let π : Y → X be a resolution of singualrities such that the center of ν on Y is a divisor E, and let η : Y → A d k be the composition of π with a general projection µ : X 0 → A d , where d = dim X 0 . Let Q = {Q j,r;n } (j,r)∈J , 0≤n≤eβ j,r -1 be a regular system of parameters of O (A d )∞,P A d eE associated to a system of transverse generators for η with respect to E. Then Q is a system of coordinates of (X ∞ , P X eE ), that is, P X eE O X∞,P X eE = ( {Q j,r;n } (j,r)∈J , enj,r-1β j,r-1 ≤n≤eβ j,r -1

) O X∞,P X eE . Therefore embdim O (X∞) red ,P eE = embdim O (X∞) red ,P eE ≤ ♯Q = e ( k E + 1).

where k E is the Mather discrepancy of X with respect to E.

Moreover, there exist elements z 1 , . . . z s ∈ O X,P0 and L ∈ O X∞ \ P X eE such that P X eE (O X∞ ) L = (Z 1;0 , . . . Z 1;eα1-1 , . . . , Z s;0 , . . . Z s;eαs-1 ) (O X∞ ) L where α i = ν(z i ) for 1 ≤ i ≤ s.

Proof : We may suppose that π : Y → X dominates the Nash blowing up of X. We may suppose that X is affine, let X ⊆ A

  ) are {β j,r } gj +1 r=0 , {e j,r } gj r=0 , {n j,r } gj r=0 and {β j,r } gj +1 r=0 (see prop. 3.3 or corol. 3.2). We denote by {q ′ j,r } gj +1 r=0 the elements {q r = h sr } gj +1 r=0 in 3.1 (iii).(a), hence satisfying

  3 (v)), O (A d )∞,P A d eE is a regular local ring of dimension the cardinal of Q whose maximal ideal P A d eE O (A d )∞,P A d eE is generated by Q.

  N k = Spec k[y 1 , . . . , y N ]. Then, a general projection ρ : X ⊆ A N k → A d k , y → (x 1 , . . . , x d ) satisfies ord E π * (dx 1 ∧ . . . ∧ dx d ) = k E . Let P A d eE be the image of P Y eE by η ∞ .Then the result follows from prop. 4.5 applied to P A d eE and prop. 4.5 (iii) in[START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] applied to ρ : X → A d k .

where V is an open subset of X, determining respective regular systems of parameters in a closed point y 0 ∈ E and in η • φ(y 0 ), and such that, if we identify x 1 , . . . , x d with their images by η

  whose initial forms generate a localization of the graded algebra gr νE O A d ,P0 modulo étale covering. In section 4 we will prove that they have the property of determining a basis of P A d First, we may suppose that Y is an affine k-scheme. In fact, we may replace Y by an open affine subset which contains the generic point ξ E of E. Let u ∈ O U , U being an open subset of Y that contains ξ E , such that u defines a local equation of E. Since η is dominant and generically finite, there exist local coordinates x 1 , . . . , x d in an open subset of A d that contains η(ξ E ) such that the image of x 1 in O Y,ξE is g u m1 , where m 1 > 0 and g is a unit in O Y,ξE . By restricting U and adding a m 1 -th root of g, we can define an étale morphism φ : U → U such that the image of x 1 in O U is u m1 1 where u 1 is a local equation of the strict transform E of E in U . Moreover, since char k = 0, and Ω A d ⊗ K(Y ) ∼ = Ω Y ⊗ K(Y ), we may restrict U and U and define {u 1 , . . . , u d } ⊂ O U , {x 1 , . . . , x d } ⊂ O V ,

	eE /(P A d eE ) 2 , being P A d eE the image ∞ (see 2.7). From this and applying prop. 2.5, we by η ∞ of the generic point of Y eE
	will conclude analogous results for a smooth surface X and a divisorial valuation
	on X (theorems 3.8 and 4.8).
	Let us apply the description of the morphism η appearing in [Re2], proof of prop.
	4.5 (see (4) below).

  Proposition 3.3. There exist a point y 0 ∈ E, a regular system of parameters {u, v 2 , . . . , v d } of O Y,y0 and a regular system of parameters {x 1 , . . . , x d } of O A d ,η(y0) such that the following holds : (i) If we identify x 1 , . . . , x d with their images in O Y,y0 then

  the images of the elements of G in P A d First note that, by (i) in lemma 4.3, we haveG ⊂ P A d eE . Let us prove that (G) O (A d )∞,P A d eEis a prime ideal. By (ii) in lemma 4.3, for (j, r) ∈ J , we have

eE /(P A d eE ) 2 O (A d )∞,P A d eE are independent, hence define a basis as κ(P A d eE )-vector space. In particular, we obtain dim O (A d )∞,P A d eE = ♯G = e (a E + 1).

Proof :

  ∞ → (A d ) ∞ . Since e is the ramification index of O Y,E over O Z,F , P Z eeF is the image of P Y eE by σ ∞ : Y ∞ → Z ∞ and hence P A d eeF = P A d eE .Now, by the definition of G, and since P ′ ⊆ P A d

			r j,r;eβ j,r	for some positive integers
	eE {a j,r } (j,r)∈J * . Hence (G) O (A d )∞,P A d	is a prime ideal.
	Let us denote by P ′ the prime ideal of O A d ∞ such that (G) O (A d )∞,P A d eE	=
	P ′ O (A d )∞,P A d eE	. We will next prove that P ′ = P A d eE . In fact, with the notation in 2.7
	and 3.7, let P Z eeF be the generic point of Z eeF ∞ and let P A d eeF be the image of P Z eeF by
	the morphism Z

  X∞,P X eE = ( {Q j,r;n } (j,r)∈J , enj,r-1β j,r-1 ≤n≤eβ j,r -1 ) O X∞,P X eE . and O X∞,P X eE is a regular local ring of dimension dim O X∞,P X eE = ♯Q = e (k E + 1). where k E is the discrepancy of X with respect to E. Moreover, there exist elements z j,r ∈ O X,P0 , (j, r) ∈ J , and L ∈ O X∞ \ P X Recall that P Y eE is the generic point of Y eE ∞ (see 2.7) and that P X eE (resp. P A d eE ) is the image of P Y eE by π ∞ (resp. η ∞ ). By prop. 2.5 (see also corol. 2.9) it suffices to prove the result for the point P A d eE in (A d ) ∞

						eE
	such that (38)	P X eE (O X∞ ) L =	(	{Z j,r;n } (j,r)∈J ,0≤n<eαj,r	)	(O X∞ )

L

where α j,r = ν(z j,r ) for (j, r) ∈ J .

Proof :

Acknowledgements : I thank O. Piltant for his suggestions and comments.


Thomas Holterbach

Cristel Pelsser

Randy Bush

Laurent Vanbever

Quantifying interference between measurements on the RIPE Atlas platform

Keywords: Measurement, Delay, RIPE Atlas

Public measurement platforms composed of low-end hardware devices such as RIPE Atlas have gained significant traction in the research community. Such platforms are indeed particularly interesting as they provide Internet-wide measurement capabilities together with an ever growing set of measurement tools. To be scalable though, they allow for concurrent measurements between users. This paper answers a fundamental question for any platform user : Do measurements launched by others impact my results ? If so, what can I do about it ? We measured the impact of multiple users running experiments in parallel on the RIPE Atlas platform. We found that overlapping measurements do interfere with each other. We found that increasing hardware CPU greatly helped in limiting interference on the measured delays.

Introduction

Public measurement platforms composed of many low-end devices or probes, such as RIPE Atlas [RIPc], are increasingly used by researchers and network operators. To scale and be practical, measurement platforms schedule measurements in parallel, without providing feedback to the user. When put together with the limited hardware and software capabilities, this raises the question of measurement interferences. What is the impact of an increased load on the precision of measurements performed ? Do the measurements performed by one participant impact the results obtained by others ? If so, by how much ? Can this have an impact on previous research results ? [START_REF] Holterbach | Quantifying interference between measurements on the ripe atlas platform[END_REF] answers these questions empirically for the RIPE Atlas platform. Here we present of subset of these results.

As of April 2015, RIPE Atlas is composed of over 6,700 public probes scattered in 197 countries. Three versions of the probes exist, differing only by their hardware. Version 1 and version 2 are identical except for the amount of RAM they have. Both are Lantronix XPort Pro with a 167MHz CPU, 8MB or 16MB of RAM, respectively and a 16MB flash. The version 3 probe is a revamped TP-Link TL-MR3020 router with a 400MHz CPU, 32MB of RAM and a 4MB NAND. The v3 probes are therefore more powerful.

The number of v3 probes increased rapidly after they started to be distributed in 2013. While the number and proportion of v1 and v2 probes is decreasing, they remain non-negligible, accounting for 28.2% of the probes in April 2015.

RIPE Atlas provides different types of measurements. As of 2015, RIPE Atlas offers four § types of measures to its users : ping, traceroute, DNS and SSL [RIPb]. In RIPE Atlas, a measurement is defined by a type, a frequency and set of probes. It can therefore refer to an arbitrary number of individual measurements performed from multiple probes. Users can also provide a start date and an end date. If none † The credits go to IIJ for supporting Thomas and Cristel's work. ‡ We thank both the RIPE Atlas and the NLNOG RING support teams for accommodating our measurements and promptly replying to our questions.

§. HTTP measurements are possible but are restricted to researchers and other interested users on a case-by-case basis.

is provided, the measurement will start as soon as possible and has to be stopped manually. Measurements can be repeated or run only once (one-off ). One-off measurements are near real-time if no start time is defined : users should expect results within 10 seconds [RIPb].

Atlas probes are popular sources of measurements and are increasingly used in research. Since its inception, Atlas performed almost 30 million individual measurements ¶ . V3 probes hosted 2/3 of the measurements, while v1 and v2 probes hosted the rest. In March 2015, the user who used the most credits spent 83.3 million credits [RIPa]. This is enough to perform more than 2,700,000 traceroutes. During the same month, the most used Atlas probe (a v1) provided 608,824 results [RIPa], one every 4 seconds. Finally, the number of concurrent measurements is important. As an illustration, the platform was executing 592,000 concurrent individual measurements when we collected these statistics. An increasing number of research papers use RIPE Atlas [MTS15, FPA15, FSC15, CJR + 15].

Some Atlas probes are more used than others. Due to their geographical position, IPv6 capability or a NAT gateway, some probes are more attractive than others. As a result some probes are highly loaded.

Quantifying interference

We describe how we quantify interference between measurements performed on a RIPE Atlas probe. We take the perspective of one user λ and one probe ρ and measure the effects on the results reported by ρ to λ when ρ originates concurrent measurements. In particular, we look at changes in the delay reported by ρ when concurrent one-off traceroutes are originated on ρ. We launch an increasing number of one-off traceroutes on the probe. We use NL Ring nodes [NLN] as destinations of the pings sourced on ρ. We want to focus on the behavior of the probe and avoid network interferences. For each experiment, we measure the delay between the tested Atlas probe and a colocated Ring node in the same LAN (i.e. there is no IP hop between them). Because packets between the Atlas probe and its colocated Ring node always stay in the same LAN, we prevent our measurements from being polluted by Internet variations (Figure 1). We obtained these pairs of colocated Atlas probe and Ring node by a traceroute campaign between each Ring node and Atlas probes in the same AS. The results depicted in Table 2 all come from measurements done between an Atlas probe and its colocated NL Ring node.

Decreased precision

We now use our methodology ( §2) to measure the decrease in precision of delay-based measurements. We performed our measurements on multiple probes (at least two per version) to ensure conformity. As ¶. As a measurement may involve a large number of probes, the number of individual measurements is more representative of the load of the platform. FIGURE 2: Quantification of interferences for v1, v2 and v3 probes. On the left, the probe is loaded by sourcing 100 one-off traceroutes. We look at the impact of a load on the ping delay reported by the probe. The probe sends 1.4 ping/s. With more powerful hardware, v3 probes are less sensitive to load than v1 and v2. On the right, we see that delays measured from a v2 probe systematically increase when concurrent one-off traceroutes are launched on this probe.

Quantifying interference between measurements on the RIPE Atlas platform

their number is not negligible and their decrease in precision is serious, the next figure only focusses on v2 probes.

Delays measured from the probe increase when concurrent measurements are launched on it. We launched ping measurements from the Atlas probe and towards eight random Ring nodes plus the colocated Ring node. The ping rate towards each destination is 9 ping/min, averaging 1.4 ping/s over all destinations. We increase the load on the probe by launching successively 10, 25, 50, 100, 250, and 500 one-off traceroutes.

Figure 2 shows the impact of the concurrent one-off traceroutes on the delay measured from a v2 probe. The blue points are RTTs between the Atlas probe and its colocated Ring node, while the red points are RTTs between the Atlas probe and another Ring node. The gray areas are the periods when one-off traceroutes are running. The number above each gray area is the number of one-off traceroutes executed. To quantify the impact, we compare the median, 95th percentile, and standard deviation of the ping measurements before the one-off traceroutes (the white area preceding the gray area) and during the one-off traceroutes execution time (gray area). The difference is reported in Table 2.

Delays measured from the probe systematically increase when one-off traceroutes are performed. Starting 100 one-off traceroutes increases the median delay of the concurrent pings by more than 1 ms. For v1 and v2 Atlas probes, the standard deviation is seriously impacted : +16.3 ms (v1) and +7.4 ms (v2). Atlas probes v3 show less effect, the median is only increased by 0.06 ms while the standard deviation is not impacted ; this is due to v3 probes having more power. Key points. We observed significant interferences on delay measurements for v1 and v2 probes. These probes compose 28% percent of the platform. An important portion (34%) of the public experiments available result from experiments on v1 and v2 probes.

Impact for researchers. RIPE Atlas makes publicly available all the results collected with the platform since its inception in 2010 [RIPc]. Researchers using these data should consider the impact of interferences. Especially for data collected before 2013-prior to v3 probes. We suggest researchers to be very careful when using publicly available delay measurements.

Provide feedback to users with a measurement confidence index. A fundamental problem with Atlas is that the user has no visibility on the concurrent load of the platform. For that, we argue that RIPE can return a "confidence index" along with each result. The index would be function of the platform concurrent load. High (resp. low) load would lead to low (resp. high) confidence.

Other researchers have observed measurement interference and its impact on RTT. As an example, the effects virtualization can produce on measured delays have already been pointed out [START_REF] Whiteaker | Explaining packet delays under virtualization[END_REF][START_REF] Spring | Using planetlab for network research : Myths, realities, and best practices[END_REF]. As a number of large-scale platforms use VMs [Pla, ML, NLN], tools such as [START_REF] Park | Comon : A mostly-scalable monitoring system for planetlab[END_REF] for PlanetLab, provide users with information about the state of these platforms and their nodes. In contrast, RIPE Atlas does not use virtualization and relies on a scheduler to share resources among users.

In [START_REF] Bajpai | Lessons learned from using the ripe atlas platform for measurement research[END_REF], Bajpai et al. showed that RTTs from v1 and v2 probes to the first hop router are consistently higher than for v3 probes. They do not however study the relation between the measured delays and the load of the probes.

Conclusion

We presented the first measurement study of user-induced interferences on the RIPE Atlas platform. We found that measurements do interfere with each other. Delays reported from the probe increase and vary more when they compete with concurrent measurements.

Our findings also bring up new, non-trivial research questions : how can we design measurement platforms that provide more isolation between users, while still being efficient (i.e., not requiring a global lock). We plan to explore this direction in the future.

FIGURE 1 :

 1 FIGURE 1: We load the probe with traceroutes and pings to distant targets while we measure the delay toward local destinations.