Divisorial Valuations via Arcs

44(2008), 425–448

Divisorial Valuations via Arcs

By

Tommaso deFernex^∗, LawrenceEin^∗∗and ShihokoIshii^∗∗∗

Abstract

This paper shows a finiteness property of a divisorial valuation in terms of arcs.

First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset of the arc space. Then we define the codimension for this subset and give a formula of the codimension in terms of “relative Mather canonical class”. By using this subset, we prove that a divisorial valuation is determined by assigning the values of finite functions. We also have a criterion for a divisorial valuation to be a monomial valuation by assigning the values of finite functions.

Introduction

Let X be a complex algebraic variety of dimension n ≥ 1. An impor- tant class of valuations of the function field C(X) of X consists of divisorial valuations. These are valuations of the form

v=qvalE:C(X)^∗−→Z,

where E is a prime divisor on a normal variety Y equipped with a proper, birational morphismf:Y →X, q=q(v) is a positive integer number called the multiplicity of v, and for every h ∈ C(X)^∗ that is regular at the generic point of f(E), valE(h) := ordE(h◦f) is the order of vanishing of h◦f at

Communicated by S. Mukai. Received January 30, 2007. Revised September 14, 2007.

2000 Mathematics Subject Classification(s): 14J17, 14M99.

∗Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 48112-0090, USA.

e-mail: defernex@math.utah.edu

∗∗Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan St., M/C.

249, Chicago, IL 60607-7045, USA.

e-mail: ein@math.uic.edu

∗∗∗Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, 152- 8551, Tokyo, Japan.

e-mail: shihoko@math.titech.ac.jp

c 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

the generic point of E. It is a theorem of Zariski that the set of valuation rings associated with this class of valuations coincides with the set of discrete valuation rings (R,mR) of C(X) with trdeg(R/mR : C) = n−1. Thanks to Hironaka’s resolution of singularities, divisorial valuations had acquired a fundamental role in singularity theory.

Since the publication of the influential paper of Nash [N] and the intro- duction of motivic integration (see, e.g., [K], [DL], [B]), it become apparent the close link between certain invariants of singularities related to divisorial valua- tions and the geometry of arc spaces. This link was first explored by Mustat¸ˇa in [M1], [M2], and then further studied in [ELM], [I2], [I3]. In particular, when the ambient varietyX is smooth, it is shown in [ELM] how one can reinterpret invariants such as multiplier ideals in terms of properties of certain subsets in the space of arcsX_∞ofX.

The main purpose of this paper is to extend the results of [ELM] to ar- bitrary varieties and to employ such results towards the characterization of divisorial valuations by evaluation against finite numbers of functions.

Naturally associated to the valuationv=qval_E, there is a subset W(v) =W(E, q)⊂X_∞,

constructed as follows. We can assume without loss of generality that bothY andE are smooth. Then W(v) is defined as the closure of the image, via the natural mapY_∞→X_∞, of the set of arcs onY with order of contact alongE equal toq. It turns out that

v= val_W(v),

where for every irreducible constructible subsetC⊂X_∞ that is not contained in the arc space of any proper closed subvariety of X, we define a valuation valC: C(X)^∗ → Z by taking the order of vanishing (or polarity) along the generic point of C (see Definition 2.3). Although the subsets of X_∞ of the formW(v) are quite special, there is a much larger class of subsetsC ⊆X_∞, such that the associated valuation val_C is a divisorial valuation. These sets are calleddivisorial sets. It was shown in [I3] that sets of the form W(v) are maximal (with respect to inclusion) among all divisorial sets defining the same valuation; for this reason they are calledmaximal divisorial sets.

Other important classes of subsets ofX_∞are those consisting, respectively, of (quasi)-cylindersand contact loci. The valuations associated to irreducible components of such sets are calledcylinder valuationsand contact valuations, respectively. As it turns out, maximal divisorial sets belong to these classes of sets. Generalizing to singular varieties some results of [ELM], we obtain the

following properties:

(a) every divisorial valuation is a cylinder valuation (Theorem 3.9);

(b) every cylinder valuation is a contact valuation (Proposition 3.10);

(c) every contact valuation is a divisorial valuation (Proposition 2.12).

Geometrically, the correspondence between divisorial valuations and cylinder valuations is constructed by associating to any divisorial valuationvthe subset W(v)⊂X_∞. The fact thatW(v) is a quasi-cylinder shows thatv, being equal to val_W(v), is a cylinder valuation. The other direction of the correspondence is more elaborate: starting from a cylinder valuation valC, one first realizes val_C as a contact valuation, that is, a valuation determined by an irreducible component of a contact locus, the definition of which involves certain conditions on the order of contact along some subscheme of X. The divisor E is then extracted by a suitable weighted blowup on a log resolution of this subscheme, andqis determined by the numerics involved in the construction.

A key point in the proof of these assertions, as well as a fundamental prop- erty for many applications, is a codimension formula for the maximal divisorial setW(v) associated to a divisorial valuation v. When X is a smooth variety, it was shown in [ELM] that

codim

W(v), X_∞

=kv(X) +q(v),

where k_v(X) := v(K_{Y /X}) is the discrepancy of X along v and q(v) is the multiplicity ofv. The definition of discrepancy needs to be modified whenX is singular. Given an arbitrary varietyX, we take a resolution of singularities f:Y →X that factors through the Nash blowup

ν:X −→X,

and define therelative Mather canonical divisorKY /X off (see Definition 1.1).

This divisor, which is defined in total generality, is always an effective integral divisor, and it coincides withK_{Y /X} when X is smooth (the two divisors are in general different forQ-Gorenstein varieties). The relative Mather canonical divisor plays a fundamental role in the geometry of arc spaces and the change- of-variables formula in motivic integration. Defining kv(X) := v(KY /X), the codimension formula of [ELM] generalizes to arbitrary varieties as follows.

Theorem 0.1. With the above notation, we have codim

W(v), X_∞

=k_v(X) +q(v).

Using the interpretation of divisorial valuations via arc spaces, we show that each divisorial valuation v is characterized by its values v(fi) = vi on a finite number of functionsfi. More precisely, we have the following result.

Theorem 0.2. Suppose that X = SpecA is an affine variety, and let v be a divisorial valuation over X. Then there exists elements f₁, . . . , fr∈ A andv₁, . . . , v_r∈Nsuch that for every f ∈A\ {0}

v(f) = min{v(f)|v is a divisorial valuation such that v(f_i) =v_i}. Theorem 0.2 is obtained by determining functionsfiand numbersvi such that

(1) W(v)⊂

r i=1

Cont^vi(fi),

with equality holding off a set contained in (SingX)_∞. This is the content of Theorem 4.2. A similar result can be obtained using MacLane’s results from [ML] (see Remark 4.6); it would be interesting to further investigate the con- nection between MacLane’s key polynomials and the functions f_i determined in the proof of the above results, and to study properties that the first ones satisfy with respect to the geometry of arc spaces.

In the case of monomial valuations on toric varieties, we apply Theorem 0.1 to give a precise characterization in terms of a system of parameters. For simplicity, we present here the result in the special case when the toric variety is equal toCⁿ.

Theorem 0.3. Letvbe a divisorial valuation ofC(x₁, . . . , x_n), centered at the origin0 of X =Cⁿ. Assume that there are positive integers a1, . . . , an

such that

(2) v(xi)≥ai and

ai≥kv(X) +q(v).

Then v is a monomial valuation determined by the weights ai assigned along the parametersx_i,q(v) = gcd(a₁, . . . , a_n), and equalities hold in both formulas in (2).

The more general version of this result, holding for arbitrary singular toric varieties, requires some additional notation, and is given in Theorem 5.1.

The authors express their hearty thanks to Bernard Teissier for his helpful advice, and would like to thank Mircea Mustat¸ˇa for many helpful discussions, in particular about the proof of Theorem 5.1.

§1. The Relative Mather Canonical Divisor

In this preliminary section, we review the construction of the Nash blowup and define a generalization of the relative canonical divisor to certain resolutions of arbitrarily singular varieties. This will give a geometric interpretation of a certain ideal sheaf that governs the dimension of fibers of maps between arc space, and consequently the change-of-variable formula in motivic integration.

We start with an arbitrary complex varietyX of dimensionn. Note that the projection

π:PX(∧ⁿΩX)−→X

is an isomorphism over the smooth locusX_reg ⊆X. In particular, we have a sectionσ:X_reg→PX(∧ⁿΩ_X).

Definition 1.1. The closure of the image of the sectionσis the Nash blowupofX, and is denoted byX:

PX(∧ⁿΩ_X)

π

⊇ X :=σ(Xreg)

X_reg  //

σ

99

X.

The Nash blowupX comes equipped with the morphism ν:=π|_Xb:X −→X

and the line bundle

KX :=O_PX(∧ⁿΩ_X)(1)|_Xb.

We call this bundle theMather canonical line bundleofX.

Remark 1.2. IfX is smooth, thenX =X andK_X is just the canonical line bundle ofX.

The original definition of Nash blowup is slightly different. Assuming the existence of an embeddingX →M into a manifold M, one can consider the Grassmann bundleG(ΩM, n) overM of rank n locally free quotients sheaves of ΩM. The map

x →

(Ω_X)_x(Ω_M)_x

∈G(Ω_M, n)_x,

defined for every smooth point xof X, gives a section σ:X_reg →G(Ω_M, n).

Then one takes the closure of the image of this section inG(Ω_M, n). As we prove next, the two constructions agree.

Proposition 1.3. Keeping the above notation, letX denote the closure of σ(X reg) in G(ΩM, n), and let ν: X → X the induced morphism. Then X=X andν =ν.

Proof. We have exact sequences

0 //S|_Xe ⁱ //ΩM|_Xe ^p //Q|_Xe //0

0 //ker(q) ^j //ΩM|_Xe ^q //ν^∗ΩX //0,

where Ω_M|_Xe is ^∗Ω_M|_Xe for the projection : G(Ω_M, n) → M and it also coincides with ν^∗(Ω_M|X). The top row is the restriction of the universal se- quence of the Grassmann bundleG(ΩM, n), and the second is the pull back of the sequence of differentials determined by the inclusion ofX inM. Over the smooth locus ofX we have

S|_eσ(X_reg)= ker(q)|_eσ(X_reg).

Then, since ker(q) is torsion free and the top sequence has a local splitting, the inclusion j factors through i and an inclusion ker(q) → S|Xe. Therefore pfactors through q and a surjection ν^∗ΩX Q|Xe. After taking wedges, we obtain a commutative diagram of surjections

(3) ∧ⁿΩ_M|Xe

&&

MM MM MM MM

M //∧ⁿQ|Xe

∧ⁿν^∗ΩX

OO .

Now we consider the inclusion overX

G(Ω_M|_Xe, n)→P_Xe(∧ⁿΩ_M|_Xe)

given by Pl¨ucker embedding. The factorization (3) implies that the image ofX under this embedding is contained inPXe(∧ⁿν^∗ΩX), when the latter is viewed as a subvariety ofP_Xe(∧ⁿΩM|_Xe) via the natural embedding. Therefore, by the compatibility of Pl¨ucker embeddings, the image ofX under the embedding

G(ΩM|X, n)→PX(∧ⁿΩM|X)

is contained inPX(∧ⁿΩ_X). Then, restricting over the regular locus of X, we have

X∩π⁻¹(Xreg) =π⁻¹(Xreg) =X∩π⁻¹(Xreg).

SinceX and X are irreducible varieties, both surjecting ontoX, we conclude that X =X. The equalityν=ν follows by the fact that the construction is carried overX.

Remark 1.4. The original construction of the Nash blowup using Grass- mann bundles can be given without using (or assuming) any embedding, by considering the Grassmann bundleG(ΩX, n) onX of ranknlocally free quo- tient sheaves of Ω_X. Notice that G(Ω_X, n) and PX(∧ⁿΩ_X) agree over the smooth locus ofX.

Remark 1.5. Mather used the above construction to propose a gener- alization to singular varieties of the notion of Chern classes of manifolds, by considering the classν_∗

c(Q^∗|Xe)∩[X]

inA_∗(X). This is known as theMather- Chern class of X. The push-forward ν_∗

c1(K_X⁻¹)∩[X]

is equal to the first Mather-Chern class ofX.

Now consider any resolution of singularities f:Y →X factoring through the Nash blowup ofX, so that we have a commutative diagram

Y fb

//

f

%%

X ν //X .

Definition 1.6. LetK_{Y /X} be the divisor supported on the exceptional divisor on Y and linearly equivalent to KY −f^∗KX. We call it the relative Mather canonical divisoroff.

Proposition 1.7. The relative Mather canonical divisor KY /X is an effective divisor and satisfies the relation:

df(f^∗∧ⁿΩX) =OY(−KY /X)· ∧ⁿΩY, wheredf:f^∗∧ⁿΩ_X → ∧ⁿΩ_Y is the canonical homomorphism.

Proof. By generic smoothness of X, the kernel of the morphism ν^∗∧ⁿ ΩX → KX is torsion, hence, pulling back to Y, we obtain a commu- tative diagram

f^∗∧ⁿΩX df //

∧ⁿΩY

f^∗KX δ

99r

rr rr rr r

.

Then, we have an effective divisorDwith the support on the exceptional divisor such that

im(δ) =OY(−D)· ∧ⁿΩY.

It follows that D is linearly equivalent to K_Y −f^∗K_X, therefore we obtain D=KY /X.

For the second statement, we should note thatν^∗∧ⁿΩX→KXis surjective asKX is relatively very ample with respect toπ:PX(∧ⁿΩX)→X. This gives the surjectivity off^∗∧ⁿΩX→f^∗KX and the second statement.

Note that K_{Y /X} is always an effective integral Cartier divisor, and in particular it is in general different from the relative canonical divisor defined in theQ-Gorenstein case. In fact, the following property holds.

Proposition 1.8. Let X be a normal and locally complete intersection variety, and letf:Y →X be a resolution of singularities factoring through the Nash blowup ofX. ThenK_{Y /X} =K_{Y /X} if and only ifX is smooth.

Proof. It follows by [EM1] that the differenceK_{Y /X}−K_{Y /X} is given by the vanishing order of the Jacobian ideal sheaf ofX.

Definition 1.9. For every prime divisorE onY, we define k_E(X) := ord_E(K_{Y /X}),

and call it the Mather discrepancy of X along E. More generally, if v is a divisorial valuation overX, then we can assume without loss of generality that v=qvalE for a prime divisorE onY and a positive integerq, and define the Mather discrepancyof X alongv to be

k_v(X) :=q·k_E(X).

IfX is smooth, then we denotekE(X) :=kE(X) andkv(X) :=kv(X).

§2. Contact Loci in Arc Spaces and Valuations

In this section we set up basic statements for contact loci and divisorial valuations.

Definition 2.1. Let X be a scheme of finite type over Cand K ⊃C a field extension. A morphismα: SpecK[[t]]→X is called an arc ofX. We denote the closed point of SpecK[[t]] by 0 and the generic point by η. For m∈N, a morphismβ: SpecK[t]/(t^m+1)→X is called anm-jetofX. Denote the space of arcs ofX byX_∞ and the space ofm-jets of X byX_m. See [M2]

for more details.

The concept “thin” in the following is first introduced in [ELM] and a “fat arc” is introduced and studied in [I2].

Definition 2.2. Let X be a variety over C. We say that an arc α: SpecK[[t]] → X is thin if αfactors through a proper closed subset of X. An arc which is not thin is called afat arc. An irreducible constructible subset C in X_∞ is called a thin set if the generic point of C is thin. An irreducible constructible subset inX_∞ which is not thin is called afat set.

Definition 2.3. Let α: SpecK[[t]] → X be a fat arc of a variety X and α^∗:OX,α(0) → K[[t]] be the local homomorphism induced from α. Sup- pose that the induced morphism SpecK → X is not dominant. By Propo- sition 2.5, (i) in [I2], α^∗ is extended to the injective homomorphism of fields α^∗:C(X)→K((t)), where C(X) is the rational function field of X. Define a map val_α:C(X)\ {0} →Zby

val_α(f) = ord_t(α^∗(f)).

The function val_α is a discrete valuation of C(X). We call it the valuation corresponding to α. If αis the generic point of an (irreducible and fat) con- structible setC, the valuation valα is also denoted by valC, and is called the valuation corresponding to C. From now on, we denote ordtα^∗(f) by ordα(f).

A fat arcαofX is called adivisorial arcif valα is a divisorial valuation over X. A fat set C is called a divisorial set if the valuation val_C is a divisorial valuation overX.

Remark 2.4. For every irreducible, fat setC ⊂ X_∞ and every regular functionf onX, we have valC(f) = min{ordγ(f)|γ∈C}.

Definition 2.5. Let ψm : X_∞ → Xm be the canonical projection to the space of m-jets Xm. A subset C ⊂ X_∞ is called a cylinder if there is a constructible set Σ⊂Xmfor some m∈Nsuch that

C=ψ_m⁻¹(Σ).

Definition 2.6 ([ELM]). For an ideal sheafaon a varietyX, we define Cont^m(a) ={α∈X_∞|ordα(a) =m}

and

Cont^≥m(a) ={α∈X_∞|ordα(a)≥m}.

These subsets are called contact loci of the ideal a. The subset Cont^≥m(a) is closed and Cont^m(a) is locally closed; both are cylinders. If Z = Z(a), then Cont^m(a) and Cont^≥m(a) are sometimes denoted by Cont^m(Z) and Cont^≥m(Z), respectively.

Definition 2.7([ELM]). For a simple normal crossing divisor E = s

i=1E_i on a non-singular varietyX, we introduce the multi-contact locusfor a multi-indexν= (ν₁, . . . ν_s)∈Z^s_≥0:

Cont^ν(E) ={α∈X_∞|ord_α(I_Ei) =ν_i for every i},

whereIE_i is the defining ideal ofEi. If all intersections among theEi are irre- ducible, then each of these multi-contact loci Cont^ν(E) is irreducible whenever it is not empty.

Definition 2.8. Let f: Y →X be a resolution of the singularities of X, and suppose thatE is an irreducible smooth divisor onY. For anyq∈Z+, we define

W(E, q) =f_∞(Cont^q(E))

and call it a maximal divisorial set. For v = qvalE we denote sometimes W(E, q) byW(v). When we should clarify the spaceXwithW(E, q) =W(v)⊂ X_∞ we denoteW(E, q) byW_X(E, q) orW_X(v).

Remark2.9. It follows by [I3, Proposition 3.4] that the above definition of maximal divisorial set agrees with the one given in [I3, Definition 2.8].

Remark2.10. The setW(E, q) only depends on the valuationv=qvalE, and not on the particular modelY we have chosen; this justify the notation W(v). Moreover, let g: X → X be a proper birational morphism of normal varieties, and let U ⊂ X be an open subset intersecting the center of v on X. We consider v also as a divisorial valuation over X and over U. Since we can assume thatv =qvalE for some smooth divisor E on a resolution of X, it follows immediately from our definition of maximal divisorial set that WX(v) =g_∞(WX(v)) =g_∞(WU(v)) (cf. [I3, Proposition 2.9]).

Remark2.11. The set W(E, q) is a divisorial set corresponding to the valuation qval_E and has the following “maximality” property: any divisorial setC with valC =qvalE is contained inW(E, q) (see [I3]).

The following is a generalization of a result of [ELM].

Proposition 2.12. LetX = SpecAbe an affine variety, and leta⊂A be a non-zero ideal. Then, for any m∈N, every fat irreducible component of Cont^≥m(a)is a maximal divisorial set.

Proof. Let ϕ: Y → X be a log-resolution of a, and write a ·OY = OY(−s

i=1r_iE_i), where E =s

i=1E_i is a simple normal crossing divisor on Y. By [ELM, Theorem 2.1], we have

Cont^≥m(a)⊃

Pr_iν_i≥m

ϕ_∞(Cont^ν(E)),

where the complement in Cont^≥m(a) of the above union is thin.

We claim that there are only finite number of maximal divisorial sets ϕ_∞(Cont^ν(E))’s inX_∞, for all possible values ofν. This follows by the follow- ing two facts:

(i) We have ν ≤ ν if and only if Cont^ν(E)⊃ Cont^ν(E), where the partial order≤in Z^s_≥0 is defined by

(ν₁, . . . , ν_s)≤(ν₁, . . . , ν_s) if ν_i≤ν_i for alli.

(ii) The number of minimal elements of {ν ∈Z^s_≥0|

riνi ≥m} according to this order≤is finite.

Then the maximalϕ_∞(Cont^ν(E))’s are the fat components of Cont^≥m(a).

By [ELM, Corollary 2.6], Cont^ν(E)’s are divisorial sets. Therefore, a fat ir- reducible component of Cont^≥m(a) is a divisorial set. To show the maximal- ity, let C be a fat component of Cont^≥m(a) and α ∈ C the generic point.

Let valα =qvalF. Then, it is clear thatC ⊂ W(F, q) by the maximality of W(F, q). For the opposite inclusion, take the generic pointβ∈W(F, q). Then it follows that val_β = val_α, which means that ord_β(f) = ord_α(f) for every f ∈K(X). This givesβ∈Cont^≥m(a), and thereforeW(F, q) is contained in a fat irreducible component of Cont^≥m(a). In conclusion,C=W(F, q).

§3. Codimension of a Maximal Divisorial Set

In this section we give an extension of the formula on the codimension of maximal divisorial sets established in [ELM] to singular varieties. LetX be an arbitrary complex variety, and letn= dimX. LetJX ⊂OX be the Jacobian ideal sheaf ofX. In a local affine chart this ideal is defined as follows. Restrict

X to an affine chart, and embed it in someA^d, so that it is defined by a set of equations

f1(u1, . . . , ud) =· · ·=fr(u1, . . . , ud) = 0.

ThenJX is locally defined, in this chart, by the d−nminors of the Jacobian matrix (∂fj/∂ui). Let S ⊂ X be subscheme defined by JX. Note that S is supported precisely over the singular locus ofX.

We decompose X_∞\S_∞=

∞ e=0

X_∞^e , where X_∞^e :={γ∈X_∞|ord_γ(JX) =e}, and let X_m,∞ := ψ_m(X_∞) and X_m,^e _∞ := ψ_m(X_∞^e ), whereψ_m:X_∞ →X_m is the truncation map. Also, let

X_∞^≤e:={γ∈X_∞|ord_γ(JX)≤e} and X_m,^≤e_∞:=ψ_m(X_∞^≤e).

We will need the following geometric lemma on the fibers of the truncation maps. A weaker version of this property was proven by Denef and Loeser in [DL, Lemma 4.1]; the sharper stated here is taken from [EM2, Proposition 4.1].

Lemma 3.1([DL], [EM2]). For m ≥ e, the morphism X_m+1,^e _∞ → X_m,^e _∞ is a piecewise trivial fibration with fibers isomorphic to Aⁿ.

Definition 3.2. LetCandCbe two constructible sets inX_∞that are not contained inS_∞. We denoteC∼C ifC∩(X_∞\S_∞) =C∩(X_∞\S_∞). A constructible setCinX_∞that is not contained inS_∞is called aquasi-cylinder if there is a cylinderC such thatC∼C.

Remark3.3. The relation∼is clearly an equivalence relation. Note also that the closure of a cylinder which is not contained inS_∞ is a quasi-cylinder.

LetC be an irreducible quasi-cylinder. By Lemma 3.1, we can also define the codimension ofC inX_∞. Indeed, for a cylinder C such thatC∼C, one can check that the codimension of

C^≤_m^e:=ψ_m(C)∩X_m,^≤e_∞

insideX_m,^≤e_∞ stabilizes forme(this is done in detail in Section 5 of [EM2]), and thus we can define

codim(C, X_∞) := codim(C^≤_m^e, X_m,^≤e_∞) for me.

Here, we note that the conditionC^≤_m^eis not empty is equivalent to the condition e≥ordα(JX), whereα∈C is the generic point, and thereforeC^≤_m^eis indeed nonempty fore0, since C⊆S_∞. Observe that C^≤_m^eis open, hence dense, in ψm(C). Note also that codim(C, X_∞) = codim(C, X_∞), since ψm(C)⊆ ψ_m(C)⊆ψ_m(C). Then we define

codim(C, X_∞) := codim(C, X_∞).

Notice that this definition does not depend on the choice of C. Indeed if C is another cylinder with C ∼ C, then we have C ∼ C. This implies that the symmetric difference between C and C is contained in S_∞. We deduce that codim(C, X_∞) = codim(C, X_∞) and hence codim(C, X_∞) = codim(C, X_∞).

Remark 3.4. IfC is an irreducible cylinder, then this definition of codi- mension coincides with the “usual” definition of codimension, i.e., the maximal lengthr of a sequence C = C₀ C₁ · · · C_r =X_∞ of irreducible closed subsets of X_∞. Indeed, if s the codimension of C as defined above, then the inequality

r≥s

is obvious by definition of s. For the opposite inequality we note that s = codim(ψm(C), Xm,∞) form0. Then the opposite inequality is obtained as follows: from the sequence,

C=C0C1· · ·Cr

of irreducible closed subsets ofX_∞, we have the sequence ψ_m(C) =ψ_m(C₀)ψ_m(C₁)· · ·ψ_m(C_r) form0, sinceCi= lim←−ψm(Ci). This yieldsr≤s.

For a non-singular varietyX, every component of a cylinder is fat ([ELM]), but this is no longer true in the singular case, as the following example show.

Example 3.5. Let X be the Whitney Umbrella, i.e. a hypersurface in C³ defined by xy²−z² = 0. For m ≥ 1, let αm: C[x, y, z]/(xy²−z²) → C[t]/(t^m+1) be the m-jet defined byαm(x) = t, αm(y) = 0, αm(z) = 0. Then, the cylinderψ_m⁻¹(αm) is contained in Sing(X)_∞, where Sing(X) = (y=z= 0).

Proposition 3.6. Let X be a reduced scheme. The number of irre- ducible components of a cylinder onX_∞ is finite.

Proof. First, we show the number of irreducible components of a cylinder that are not contained inS_∞ is finite. Let C =ψ⁻_m¹(Σ) be a cylinder over a constructible set Σ⊂Xm. Let ϕ:Y →X be a resolution of the singularities ofX, and assume thatϕis an isomorphism away fromS. Asϕis isomorphic away fromS, by the valuative criteria for the properness, the generic point of each component of C not contained in S_∞ can be lifted to the generic point of a component of the cylinder ϕ⁻_∞¹(C) = (ψ_m^Y)⁻¹ϕ⁻_m¹(Σ). The finiteness of the components of C not contained in S_∞ follows from the finiteness of the components of (ψ_m^Y)⁻¹ϕ⁻_m¹(Σ), asY is non-singular.

Now, to prove the proposition, we use induction of the dimension. If dimX = 0, then the assertion is trivial, since X_∞ X is a finite points set.

If dimX =n≥1 and assume that the assertion is true for a reduced scheme of dimension≤n−1. Let F1, . . . , Fr be the irreducible components of C not contained inS_∞. Let

C:=ψ_m

C\

i

F_i

be the closure in Σ∩Sm. Then, every irreducible component F of C other than Fi’s is contained in (ψ^S_m)⁻¹(C). As F is an irreducible component of C =ψ⁻_m¹(Σ) which contains (ψ^S_m)⁻¹(C), F is also an irreducible component of the cylinder (ψ_m^S)⁻¹(C) of a lower dimensional varietyS. By the induction hypothesis, we obtain the assertion of our proposition.

The second part of the following corollary also appears as [EM2, Lemma 5.1].

Corollary 3.7. Every fat irreducible component of a cylinder is a quasi- cylinder, and every thin component of a cylinder is contained inS_∞.

Proof. LetC₁, . . . C_r be the irreducible components of a cylinderC. As C_i= lim←−^mψ_m(C_i), form0 it follows that

ψ_m(C_i)⊂ψ_m(C_j) for i=j.

Then the non-empty open subsetCi\ _j=iψ⁻_m¹(ψm(Cj))

ofCi is a cylinder, therefore ifCi is fat, thenCiis an irreducible quasi-cylinder.

For the second assertion, assumeC₁ is thin. IfC₁ is not contained inS_∞, there ise >0 such thatX_∞^≤e∩C₁=∅which is open inC₁. Let

U :=X_∞^≤e\

j=1

ψ_m⁻¹(ψm(Cj))

and Um:=X_m,^≤e_∞\

j=1

ψm(Cj)

.

Then

U∩C1=ψ⁻_m¹(Um∩ψm(C1))

is a non-empty open subset of C₁ for every m 0. By Lemma 3.1, the codimension ofUm∩ψm(C1) in Xm,∞ is bounded. But it is in contradiction with the fact thatC1 is thin by [M1, Lemma 3.7].

The truncation morphisms induce morphisms fm:Ym−→Xm,∞⊆Xm.

Indeed, the inclusionfm(Ym)⊆Xm,∞ is implied by the fact thatYm,∞=Ym, which follows by the smoothness ofY. An important ingredient of the proof for our codimension formula, as well as the key ingredient for the change-of- variable formula in motivic integration, is the following geometric statement on the fibers of these morphisms, due to Denef and Loeser [DL, Lemma 3.4]. For a more precise statement of the following lemma, we refer to [EM2, Theorem 6.2 and Lemma 6.3].

Lemma 3.8 ([DL]). Let γ ∈ Y_∞ be any arc such that τ :=

ordγ(K_{Y /X})<∞. Then for anym≥2τ, lettingγm=ψ_m^Y(γ), we have f_m⁻¹

f_m(γ_m)∼=A^τ.

Moreover, for every γ_m∈f_m⁻¹(fm(γm)) we haveπ^Y_m,m−τ(γm) =π^Y_m,m−τ(γ_m ), whereπ_m,m^Y _−τ :Ym→Ym−τ is the canonical truncation morphism.

We obtain the following results.

Theorem 3.9. Let f :Y →X be a resolution of the singularities such thatEappears as a smooth divisor onY. Then,W(E, q)is a quasi-cylinder of X_∞ of codimension

codim(W(E, q), X_∞) =q·(ordE(K_{Y /X}) + 1), Proof. Let

Cont^q(E)⁰:={γ∈Y_∞|ordγ(E) =q,ordγ(Ex(f)\E) = 0}.

This is an open subset of Cont^q(E). Note thatτ := ordγ(K_{Y /X}) is the same for allγ∈Cont^q(E)⁰. Then, by Lemma 3.8, one can see that

f_m⁻¹ fm

ψ_m^Y(Cont^q(E)⁰)

=ψ_m^Y(Cont^q(E)⁰) for all m0.

The fact thatf_∞(Cont^q(E)⁰) is a quasi-cylinder inX_∞follows by the equalities fm◦ψ_m^Y =ψm◦f_∞ and the fact that Cont^q(E)⁰ is a cylinder inY_∞.

Let ordEK_{Y /X} = k and e = q · k. Then ψ_m^Y(Cont^q(E)) ⊂ ψ_m^Y(Cont^e(K_{Y /X})). By Lemma 3.8, the morphism ψ^Y_m(Cont^e(K_{Y /X})) → f_m

ψ^Y_m(Cont^e(K_{Y /X}))

induces a morphism ψ_m^Y(Cont^q(E))→f_m

ψ^Y_m(Cont^q(E))

with irreduciblee-dimensional fibers for m e. Note that ψ^Y_m(Cont^q(E)) is an irreducible closed subset of codimensionq inYm. Then

dim f_m

ψ_m^Y(Cont^q(E))

= dim

ψ^Y_m(Cont^q(E))

−e

=m(n+ 1)−q−e=m(n+ 1)−q(k+ 1).

The formula on codimension follows.

Proposition 3.10. The valuation corresponding to a fat irreducible component of a cylinder is the valuation corresponding to an irreducible com- ponent of a contact locus.

Proof. LetC⊂X_∞ be any fat irreducible component of a cylinder. For everyk∈N, define

ak!:={f ∈OX |valC(f)≥k!} and Bk!:={γ∈X_∞|ordγ(ak!)≥k!}. Note that we have a chain of inclusions

B_n!⊇B_(n+1)!⊇ · · · ⊇C.

For eachk, letc_kbe the smallest codimension inX_∞of irreducible components ofBk! containingC, and letnk be the number of such components. Since

ck ≤c_k+1≤codim(C, X_∞)<∞ and ifc_k =c_k+1, then n_k ≥n_k+1>0,

the sequences{ck}and{nk}stabilize. Therefore we find a closed subset W ⊂ X_∞containingCand equal to an irreducible component ofB_k!for everyk0.

We clearly have val_C ≥ val_W on regular functions because of the inclusion C ⊆W. Conversely, let h∈ OX be an arbitrary nontrivial element. We can arrange to have valC(h^m) =k! for some m, k∈N. This means thath^m∈ak!, hence we have valW(h^m)≥k! by the definition ofW. Then we conclude that

val_W(h) =val_W(h^m)

m ≥val_C(h^m)

m = val_C(h).

§4. Determination of a Divisorial Valuation by Finite Data Let X = SpecA be an affine variety, and let v be a divisorial valuation overX, i.e.,v=qvalEforq∈Nand a divisorEoverX. For a subsetV ⊂X_∞ we denote the set of fat arcs inV byV^o.

Lemma 4.1. With the above notation, letx1, . . . , xmbe elements inA, and denote by

ϕ:Y = SpecA x₂

x1, . . . ,x_m x1

→X = SpecA the canonical birational morphism. If

Σ = r j=1

(Cont^vj(fj))^o

is an irreducible subset in Y_∞ for some f₁, . . . , f_r ∈ A x₂

x1, . . . ,^x_x^m

1

and v₁, . . . , v_r∈N, then

(4) ϕ_∞(Σ) =



^r

j=1

Cont^vj(f_j)



∩ _m

i=1

Cont^pi(xi) o

for somev_j, pi ∈Nandf_j ∈A.

Proof. First we definev_j, pi andf_j (i= 1, . . . , m, j = 1, . . . , r). Let ˜αbe the generic point of Σ andα=ϕ_∞( ˜α). Letp_i = ord_α(x_i)≥0. As αhas the lifting ˜αonY, we have ord_α(x_i)−ord_α(x₁) = ord_α˜(x_i/x₁)≥0, which means p1≤pi for every i= 2, . . . , m.

Now for fj ∈A x2

x₁, . . . ,^x_x^m

1

, let the minimalasuch thatfjx^a₁∈Abeaj

and letf_j =fjx^a₁^j. Next letv_j=ajp1+vj. Then it is clear that

α∈



^r

j=1

Cont^vj(f_j)



∩ _m

i=1

Cont^pi(x_i) o

.

Therefore, the inclusion of the left side of (4) in the right side follows. For the converse inclusion, take any arcβ∈ ^r_j=1Cont^vj(f_j)

∩ ^m_i=1Cont^pi(xi)o

. Then, by the condition p1 ≤ pi (i > 1), β has the lifting ˜β on Y. Hence, ordβ˜(fj) = ordβ(f_j)−ordβ(x^a₁^j) =vj which implies that ˜β∈Σ.