COORDINATES AT STABLE POINTS OF THE SPACE OF ARCS

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COORDINATES AT STABLE POINTS OF THE SPACE OF ARCS

Ana J. Reguera

To cite this version:

Ana J. Reguera. COORDINATES AT STABLE POINTS OF THE SPACE OF ARCS. Journal of

Algebra, Elsevier, 2018. �hal-01305997�

ANA J. REGUERA

Universidad de Valladolid. Dpto. de ´Algebra, An´alisis Matem´atico, Geometr´ıa y Topolog´ıa.

47011 Valladolid, Spain. E-mail : areguera@agt.uva.es. Tel. : 34 983423048. Fax : 34 983423788

Abstract. Let X be a variety over a field k and let X

_∞

be its space of arcs.

Let P

E

be the stable point of X

_∞

defined by a divisorial valuation ν

E

on X.

Assuming char k = 0, if X is smooth at the center of P

E

, we make a study of the graded algebra associated to ν

E

and define a finite set whose elements generate a localization of the graded algebra modulo ´ etale covering. This provides an explicit description of a minimal system of generators of the local ring O

X_∞,PE

. If X is singular, we obtain generators of P

_E

/ P

_E²

and conclude that embdim O

(X_∞)_red,P_E

= embdim O \

X_∞,P_E

≤ b k

E

+ 1 where b k

E

is the Mather discrepancy of X with respect to ν

_E

.

¹

1. 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 [Re1], its definition is based on the sta- bility 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

^dk

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 [Re1] 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 ([Re1], 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 ([Re2], 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 coordi- nates in O \

X_∞,P

.

Divisorial valuations are closely related to stable points : the valuation ν

P

asso- ciated to a stable point P of X

_∞

is divisorial ([Re2], (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

1. Keywords :Space of arcs, divisorial valuation, graded algebra.

MSC :13A02, 13A18, 14B05, 14B25, 14E15, 14J17, 32S05.

1

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_∞,P_eE

= 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_∞,P_eE

.

If X is not smooth at P

₀

then, if X → A

^dk

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

([Re2], prop. 4.5). From this it follows that the embedding dimension of the ring O \

X_∞,P_eE

is bounded from above by e( b k

E

+ 1), where b 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

_ν

E

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

= b k

E

+ 1, which in this case is equal to mult X + 1 ([Re2], 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

(defini- tion 3.4). Although they do not generate gr

_ν

E

O

_A^d,P0

in general (gr

_ν

E

O

_A^d,P0

is not in general finitely generated for d ≥ 3), they generate a localization of gr

_ν

E

O

_A^d,P0

modulo ´ etale covering (theorem 3.8). This is done in section 3. In section 4 we describe minimal coordinates of ( A

^d

)

_∞

at the image P

_eE^Ad

of P

eE

in ( A

^d

)

_∞

from the q

j,r

’s. From this we obtain a regular system of parameters of O

X_∞,P_eE

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).

2. 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 K- arcs 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

^Nk

= Spec k[x

1

, . . . , x

N

] is ( A

^Nk

)

_∞

= 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 defi- ning X in A

^Nk

, 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

^dk

, 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 ([Re2]

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 b R b the completion of a local ring R which is a k-algebra of finite type. In [Re1] and [Re2] a theory of stable points of X

_∞

is developed. One important property of these points is the following :

Proposition 2.3. ([Re2], 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, [Re2] 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

([Re2] 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 c

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 ´ etale morphisms. The following holds : Proposition 2.5. Let X, Z ∈ X

k

and let σ : X → Z be an ´ etale k-morphism. Then we have

X

_∞

∼ = Z

_∞

×

Z

X

in particular, X

_∞

is ´ etale 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 ´ etale 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 [Re2] (8) in 3.4) implies that

X

_∞

= Spec (A

_∞

[X

₀

] / (F

₀

))

_G

0

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

))

_G

0

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 b := Spec O \

X,Q₀

, being Q

0

the center of Q in X , then Q induces a stable point Q b in X b

_∞

because h

_Q

: Spec κ(Q)[[t]] → X factorizes through X b , and we have

(2) O \

X_∞,Q

∼ = O \

Xb_∞,Qb

.

Analogously, O \

Z_∞,P

∼ = O \

_Zb∞,Pb

, where Z b := Spec O \

Z,P0

and P b is the stable point of Z b

_∞

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 X → Z is local ´ etale, hence B ∼ = A ⊗

κ(P₀)

κ(Q

₀

) ([Ra] VIII corol. to lemme 2 and [Ha] III exer. 10.4).

Now, 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 ´ etale and hence (A \

_∞

)

P

→ (B \

_∞

)

Q

is also ´ etale, 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 [Re2]. The second one also follows from the proof of [Re2], prop. 4.2. In fact, by prop. 2.5 and since there exists an ´ etale morphism from a neighborhood of P

0

to a subset of A

^r,d_k ^−r

, where d = dim X , we may suppose that X ⊆ A

^r,d_k ^−r

. In this case we have

O

X_∞

= O

X

[X

₁

, . . . , X

_n

, . . .] and O

X_n

= 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

_n

0+1

, . . . , X

_n

, . . .) ⊂ O

X_∞,P

. This implies that

O

X_∞,P

∼ = k(X

_n0+1

, . . . , X

_n

, . . .) ⊗

k

O

_jn

0(X_∞),P_n0

hence we conclude the result.

2.7. Let X be a reduced separated k-scheme of finite type and let ν be a divi- sorial 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

^Y₀

: 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

_eE^X

) 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

_∞

([Re2], prop. 4.1, see also [Re1], 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 b 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

( − b k

E

E). We call b k

E

the Mather discrepancy of X with respect to the prime divisor E. Note that b k

E

̸ = 0 if and only if π is an isomorphism at the generic point of E, and that b k

E

only depends on the divisorial valuation ν = ν

E

. We have :

(3) sup

_n

dim O

_{jn(X∞),(PeE)n}

= e ( b k

_E

+ 1) ([DL], lemma 3.4, [FEI], theorem 3.9). Hence by prop. 2.3 we have

dim O

X_∞,P_eE

≤ e ( b 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

_eE^Y

and P ⊇ P

_eE

. Now, assume that X is nonsingular at P

0

, and recall that in this nonsingular case we have b 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

^dk

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

^dk

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) whose initial forms generate a localization of the graded algebra gr

_νE

O

_A^d,P0

modulo ´ etale covering. In section 4 we will prove that they have the property of determining a basis of P

_eE^Ad

/(P

_eE^Ad

)

²

, being P

_eE^Ad

the image by η

_∞

of the generic point of Y

_∞^eE

(see 2.7). From this and applying prop. 2.5, we 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). 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,ξ_E

is g u

^m1

, where m

1

> 0 and g is a unit in O

Y,ξ_E

. By restricting U and adding a m

1

-th root of g, we can define an ´ etale morphism φ : U e → U such that the image of x

1

in O

_Ue

is u

^m₁¹

where u

1

is a local equation of the strict transform E e of E in U. Moreover, since char e k = 0, and Ω

_Ad

⊗ K(Y ) ∼ = Ω

Y

⊗ K(Y ), we may restrict U e and U and define { u

1

, . . . , u

d

} ⊂ O

_Ue

, { x

1

, . . . , x

d

} ⊂ O

V

, where V is an open subset of X , determining respective regular systems of parameters in a closed point y

0

∈ E e and in η ◦ φ(y

0

), and such that, if we identify x

1

, . . . , x

d

with their images by η

^♯

: O

V,η(y₀)

→ O

U ,ye 0

, then

(4)

x

1

= u

^m₁¹

x

₂

= ∑

1≤i≤m₂

λ

_2,i

u

ⁱ₁

+ u

^m₁²

u

₂

x

₃

= ∑

1≤i≤m₃

λ

_3,i

(u

₂

) u

ⁱ₁

+ u

^m₁³

u

₃

. . . . . . .

x

_δ

= ∑

1≤i≤m_δ

λ

_δ,i

(u

₂

, . . . , u

_δ−1

) u

ⁱ₁

+ u

^m₁^δ

u

_δ

x

_δ+1

= u

_δ+1

. . . . . . . x

_d

= u

_d

where δ = codim

_Ad

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

^m₁^j

u

j

belongs to

k[[u

₁

, . . . , u

_j−1

]] and is integral over k[u

₁

, . . . , u

_d

]

_(u1,...,ud)

, 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 ∩ E, we may suppose that, for 2 e ≤ 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 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,m_j

(u

2

, . . . , u

j−1

) is a unit, for 2 ≤ j ≤ d.

Note that U e 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 e of the image of d(η ◦ φ) : (η ◦ φ)

^∗

( ∧

^d

Ω

V

) → ∧

^d

Ω

_Ue

. So, from now on, after a possible replacement of Y by U e and of η : Y → A

^d

by η ◦ φ : U e → 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

_j⁻₋¹₁

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 7→ u

^m₁¹

and

(7) y 7→ ∑

m₁≤i≤m

λ

_i

(v

₂

, . . . , v

_j−1

) u

ⁱ₁

+ u

^m₁

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_e

r

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.

Then, there exist an open subset U of Y containing ξ

_E

and a sequence of integers { i

s

}

^Ns=1

such that

(i) i

₁

< i

₂

< . . . < i

_N

= β

_g+1

and { i

_s

}

^Ns=1

⊂ { β

₀

} ∪ ∪

^g+1r=1

(n

_r−1

β

_r−1

, β

_r

], (ii) { β

_r

}

^g+1r=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,y₀^′

, where v

_i^′

= v

i

+ c

i

, u

^′_i

= u

i

+ c

i

, (c

i

)

i

∈ k

^d−1

, and there exist { h

₁

= y, h

₂

, . . . , 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

_r⁻₋¹₁

. . . T

₀⁻¹

S

_j⁻₋¹₁

A[x, y], where T

_r′

is the multiplicative part ge- nerated by q

_r′

:= h

_sr′

(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

ⁱ₁

+ γ

_s,m(r)

(v

^′₂

, . . . , v

_j^′₋₁

) u

^m₁^(r)

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

₀^bs0

· · · q

ρ^bsρ

, 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′

}

^ρr′=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

_s

1,β₁

)

^bs1

· · · (λ

_s

ρ,β_ρ

)

^bsρ

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,i_s−1

)

^nr−1

(resp. λ = λ

j,s−1,i_j,s−1

).

Proof : First note that (10) follows from (8) and (9) (see [Za] 2.2.1 in the Ap- pendix). 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

}

^Ns=1

and { h

_s

}

^Ns=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

}

^Ns=1

and { h

s

}

^Ns=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

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

and that (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 requi- red 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, 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,i_s−1

, v

2

, . . . , v

j−1

) = 0, ∂P

s

∂z (λ

s−1,i_s−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

_s

1,β₁

)

^bs1

· · · (λ

_s

r−2,β_r−2

)

^bsr−2

, which is a unit in R

_j−1

, such that the image of q

₀^bs0

· · · q

^b

sr−2

r−2

by θ

^♯

is equal to µ

_s

u

ⁿ₁^r−1βr−1

mod (u

₁

)

^{nr−1βr−1+1}

. Set (16) h

s

:= q

^b₀^r−10

· · · q

_r^br−1r−2₋₂

P

_s^′

(

µ

_s

(q

r−1

)

^nr−1

q

₀^br−10

· · · q

_r^br−1r−2₋₂

, l

2

, . . . , l

j−1

)

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

, there- fore 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

_r⁻₋¹₂

. . . T

₀⁻¹

S

_j⁻₋¹₁

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′

}

^rr^−′=0¹

satisfying b

^s_r′

< n

r^′

for 1 ≤