Invariants of Singularities of Pairs

Lawrence Ein

1. Introduction

Let X be a smooth complex variety and Ybe a closed subscheme of X . We are inter-ested in invariants attached to the singular-ities of the pair (X, Y ). We discuss variousmethods to construct such invariants, comingfrom the log-resolutions of the pairs, and thegeometry of the space of arcs. We present sev-eral applications of these invariants to algebra,higher dimensional birational geometry and tosingularities. The general setup is to assumeonly that X is normal and Q-Gorenstein, asin Kollar’s paper [32]. However, several ofthe approaches we will discuss become par-ticularly transparent if we assume, as we do,the smoothness of the ambient variety.

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2. Multiplier ideals

Multiplier ideals were first introduced byJ. Kohn for solving certain partial differentialequations. Siu and Nadel introduced them tocomplex geometry. We discuss below theseideals in the context of algebraic geometry.

Let X be a smooth complex affine varietyand Y be a closed subscheme of X . Supposethat the ideal of Y is generated by f1, . . . , fm,and let λ be a positive real number. We definethe multiplier ideal of (X, Y ) of coefficient λas follows:

J (X,λ·Y ) =

g ∈ OX |

|g|2

(∑mi=1 |fi|2)λ

is locallyL1

.

Example 2.1. Let X = Cn and let Y bethe closed subscheme of X defined by f =xa11 · · · x

ann . Then

J (X,λ · Y ) = (xbλa1c1 . . . x

bλancn ),

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where bαc denotes the integer part of α.

We can use a log resolution of singulari-ties and the above example to give in gen-eral a more geometric description of the mul-tiplier ideals of (X, Y ). By Hironaka’s The-orem there is a log resolution of singularitiesof the pair (X, Y ), i.e. a proper birationalmorphism

µ : X ′ −→ X

with the following properties. The variety X ′

is smooth, µ−1(Y ) is a divisor, and the unionof µ−1(Y ) and the exceptional locus of µ hassimple normal crossings. The relative canon-ical divisor KX ′/X is locally defined by the

determinant of the Jacobian J(µ) of µ, We

write µ−1(Y ) =∑Ni=1 aiEi and KX ′/X =∑N

i=1 kiEi, where the Ei are distinct smooth

irreducible divisors in X ′ such that∑Ni=1Ei

has only simple normal crossing singularities.

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The local integrability of a function g on Xcan be expressed as a local integrability condi-tion on X ′ via the change of variable formula.This reduces us to a monomial situation, sim-ilar to that in Example 2.1. On deduces thatg ∈ J (X,λ · Y ) if and only if

ordEig ≥ bλaic − kifor every i. Equivalently, if we put bλµ−1(Y )c =∑ibλaicEi, then

(1)J (X,λ·Y ) = µ∗OX ′(KX ′/X−bλµ

−1(Y )c).

Note that because of the original definition,it follows that this expression for J (X,λ · Y )is independent of the choice of a resolution ofsingularities. We note that if λ1 ≥ λ2, then

J (X,λ1 · Y ) ⊆ J (X,λ2 · Y ).

If λ is small enough, then λai < ki + 1 fori = 1, . . . , N . This implies that

ordEi1 ≥ bλaic − kEi,

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hence J (X,λ · Y ) = OX . This leads us tothe definition of the log canonical threshold ofthe pair (X, Y ): this is the smallest λ suchthat J (X,λ · Y ) 6= OX , i.e.

c = lc(X, Y ) = mini

ki + 1

ai

.

We may regard 1c as a refined version of multi-

plicity. In general a singularity with a smallerlog canonical threshold tends to be more com-plex.

The first appearance of the log canonicalthreshold was in the work of Arnold, Gusein-Zade and Varchenko (see [2] and [48]), in con-nection with the behavior of certain integralsover vanishing cycles.

In the last decade this invariant has enjoyedrenewed interest due to its applications to bi-rational geometry. The following is probablythe most interesting open problem about logcanonical thresholds.

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Conjecture 2.2. (Shokurov’s ACC Conjec-ture) For every n, Consider the set

Tn = lc(X, Y ) | X dimX = n, Y ⊂ Xwhere X is a log-canonical variety. Then Tnsatisfies the Ascending Chain Condition: Theset Tn contains no infinite strictly increasingsequences.

This conjecture attracted considerable in-terest due to its implications to the Termina-tion of Flips Conjecture (see [Bir] for a resultin this direction). For this talk, we’ll only dis-cuss about

T smn = lc(X, Y ) | X dimX = n, Y ⊂ Xwhere X is smooth variety The first uncon-ditional results on sequences of log canoni-cal thresholds on smooth varieties of arbitrarydimension have been obtained deFernex andMustata in [dFM] using nonstandard meth-ods from model theory, and they were subse-quently reproved and strengthened by Kollar

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in [Kol2] using the recent spectacular results ofBirkir, Cascini, Hacon and McKernan [BCHM]on existence of minimal models. In particu-lar, Kollar proves that the set of accumulationpoints of T smn is exactly the set T smn−1.

Theorem 2.3. (de Fernex, Ein and Mustata)T smn satisfies the ascending chain condition.

The proof is in fact rather elementary anddoes not need to use the results from [BCHM].I understand that Mustata will discuss theproblem in a later talk. I would leave the de-tails to his talk.

We can consider also higher jumping num-bers. In general, we say that λ is a jumpingnumber of (X, Y ), if

J (X,λ · Y ) ( J (X, (λ− ε) · Y )

for all ε > 0. If λai is not an integer, thenbλaic = b(λ− ε)aic for sufficiently small pos-itive ε. We see that a necessary condition for λ

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to be a jumping number is that λai is an inte-ger for some i. In particular, if λ is a jumpingnumber, then it is rational and has a boundeddenominator.

The following theorem gives a periodicityproperty of the jumping numbers.

Theorem 2.4. (i) If Y = D is a hypersur-face in X, then

J (X,λ ·D) · OX(−D) = J (X, (λ+ 1) ·D).

(ii) (Ein and Lazarsfeld [16]) For every Y de-fined by the ideal IY , if λ ≥ dimX − 1,then

J (X,λ ·D) · IY = J (X, (λ + 1) · Y ).

Corollary 2.5. Suppose that λ > dimX −1. Then λ is a jumping number for (X, Y )if and only if so is (λ + 1).

It follows from the sub-additivity theoremof multiplier ideals (Demailly, Ein and Lazars-feld [13] that the following result holds.

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Theorem 2.6. (Ein, Lazarsfeld, Smith andVarolin) Suppose c is the log-canonical thresh-old of (X, Y ). Let λ ≥ 0 be a non-negativereal number. Then there is a jumping λ′ inthe interval (λ, λ + c].

We conclude that the set of jumping num-bers of the pair (X, Y ) is a discrete subsetof Q and it is eventually periodic with periodone.

Example 2.7. If Y is a smooth subvariety ofX of codimension e, then the set of jumpingnumbers of the pair (X, Y ) is e, e + 1, · · · .In particular lc(X, Y ) = e.

Example 2.8. (Howald) LetX = Cn and letY be the closed subscheme defined by a mono-mial ideal a. If a = (a1, a2, . . . , an) ∈ Nn,we denote the monomial x

a11 · · · x

ann by xa.

Consider the Newton polyhedron Pa associ-ated with a: this is the convex hull of those

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a ∈ Nn such that xa ∈ a. Using toric geome-try Howald showed in [27] that

J (X,λ · Y ) = (xa | a + e ∈ λ · Int(Pa)),

where e = (1, . . . , 1). In particular, the logcanonical threshold c of (X, Y ) is character-ized by the fact c · e lies on the boundary ofPa.

Example 2.9. (Howald) Let X = Cn andlet Y be the closed subscheme defined by themonomial ideal a = (x

a11 , . . . , x

ann ). In this

case, the boundary of the Newton polyhedronPa is

u = (u1, . . . , un) ∈ Rn+ |n∑i=1

uiai

= 1.

Applying Howald’s theorem , one sees thatlc(X, Y ) =

∑i

1ai

.

One reason that multiplier ideals have beenvery powerful in studying questions in higherdimensional algebraic geometry is that they

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appear naturally in a Kodaira type vanishingtheorem. The following statement is the alge-braic version of a result due to Nadel. In ourcontext, it can be deduced from the Kawamata-Viehweg Vanishing Theorem (see [34]). LetX be a smooth complex variety and Y be aclosed subscheme of X . Let µ : X ′ −→ X bea log resolution of (X, Y ). Let E = µ−1Y

Theorem 2.10. (Nadel’s vanishing theorem)Suppose that L is a divisor on X and λ isa positive real number. Assume that µ∗L−λ · E is nef and big, then for every i > 0

Hi(X,OX(KX + L)⊗ J (X,λ · Y )) = 0.

3. Applications of multiplier ideals

One of the most important applications ofmultiplier ideals is the following theorem ofSiu (see [44] and [45]) on the deformation in-variance of plurigenera.

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Theorem 3.1. Let f : X −→ T be a smoothprojective morphism of relative dimensionn between two smooth irreducible varieties.If we denote by Xt the fiber f−1(t) for eacht ∈ T , then for every fixed m > 0, the di-mension of the cohomology group H0(Xt, (Ω

nXt

)⊗m)is independent of t.

The techniques of extending sections of linebundle from a divisor involved in the proof ofthis theorem have been recently extended byHacon and McKernan to study existence offlips. Using these results and other importanttechniques introduced by Shokurov, we havethe following theorem.

Theorem 3.2. (Birkir, Cascini, Hacon andMcKernan) If X is a smooth complex pro-jective variety. Then the canonical ring ofX is finitely generated.

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Remark 3.3. Siu independently using moreanalytic techniques proved the same result forvarieties of general type.

The following extension theorem is essen-tially due to Hacon-McKernan.

Theorem 3.4. (Hacon-McKernan, Ein-Popa)Let (X,∆) be a log-pair, with X a nor-mal projective variety and ∆ an effectiveQ-divisor with [∆] = 0. Let S ⊂ X be anirreducible normal effective Cartier divisorsuch that S 6⊂ Supp(∆), and A a big andnef Q-divisor on X such that S 6⊆ B+(A).Let k be a positive integer and M a Cartierdivisor such that M ∼Q k(KX+S+A+∆).Assume the following:

• (X,S + ∆) is a plt pair.•M is pseudo-effective.

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• the restricted base locus B−(M) doesnot contain any irreducible closed sub-set W ⊂ X with minimal log-discrepancymld(µW ;X,∆) < 1 which intersects S.• the restricted base locus B−(MS) does

not contain any irreducible closed sub-set W ⊂ S with minimal log-dicrepancymld(µW ;S,∆S) < 1.1

Then the restriction map

H0(X,OX(mM)) −→ H0(S,OS(mMS))

is surjective for all m ≥ 1.

In a different direction, there are applica-tions of multiplier ideals to singularities of thetadivisors on abelian varieties. Let (X,Θ) be aprincipally polarized abelian variety, i.e. Θ isan ample divisor on an abelian variety X suchthat dimH0(X,OX(Θ)) = 1. The followingresult is due to Ein and Lazarsfeld [17].

1By inversion of adjunction, these last two conditions can be expressed in a unified was as sayingthat B−(M) does not contain any irreducible closed subset W ⊂ X with mld(µW ;X,∆ + S) < 1 whichintersects S but is different from S itself.

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Theorem 3.5. Let (X,Θ) be a principallypolarized abelian variety.

(i) (Kollar) (X,Θ) is log-canonical.(ii) (Ein and Lazarsfeld)If Θ is irreducible,

then Θ has at most rational singulari-ties.

Proof. (i) Observe that (X,Θ) is log-canonicalis equivalent to J (X, (1− ε)Θ) = OX for allε > 0. Suppose for contradiction that IZ =J (X, (1− ε)Θ) is a nontrivial ideal for someε > 0. Observe that Z is closed subschemecontains in the singular locus of Θ. Observethat KX is trivial. Now Nadel’s vanishingsays that Hi(IZ ⊗OX(Θ + x0)) = 0 for i >0 and any x0 ∈ X . Observe that χ(IZ ⊗OX(Θ)) = h0(IZOX(Θ)) > 0, since Z ⊂Θ. It follows from the vanishing theorem thatχ(IZ ⊗ OX(Θ + x0)) = h0(IZ ⊗ OX(Θ +

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x0)) > 0. We conclude that

Z ⊂⋂x0∈X

(Θ + x0) = ∅

This is a contradiction.

Corollary 3.6. Let (X,Θ) be a principallypolarized abelian variety of dimension g,with Θ irreducible. If

Σk(Θ) = x ∈ X | multx(Θ) ≥ k,then for every k ≥ 2 we have codim(Σk(Θ), X) ≥k + 1. In particular, Θ is a normal varietyand multx(Θ) ≤ g − 1 for every singularpoint x on Θ.

Remark 3.7. The fact that Θ is normal wasfirst conjectured by Arbarello, De Concini andBeauville. When X is the Jacobian of a curve,the fact that Θ has only rational singulari-ties was proved by Kempf. It was Kollar whofirst observed in [31] that one can use vanish-ing theorems to study the singularities of thetheta divisor.

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Multiplier ideals have been applied in sev-eral other directions: to Fujita’s problem onadjoint linear systems [3], to Effective Nullstel-lensatz, to Effective Artin-Rees Theorem [20].Building on work of Tsuji, recently Hacon andMcKernan and independently, Takayama haveused multiplier ideals to prove a very inter-esting result on boundedness of pluricanonicalmaps for varieties of general type (see [24] and[47]). We end this section with an applicationto commutative algebra due to Ein, Lazarsfeldand Smith [19].

Let X be a smooth n-dimensional varietyand Y ⊆ X defined by the reduced sheaf ofideals a. Themthth symbolic power of a is thesheaf a(m) of functions on X that vanish withmultiplicity at least m at the generic pointof every irreducible component of Y . If Y issmooth, then the symbolic powers of a agreewith the usual powers, but in general they arevery different.

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Theorem 3.8. If X is a smooth n-dimensionalvariety and if a is a reduced sheaf of ideals,then a(mn) ⊆ am for every m.

Multiplier ideals have been applied in sev-eral other directions: to Fujita’s problem onadjoint linear systems [3], to Effective Nullstel-lensatz, to Effective Artin-Rees Theorem [20]and to the symbolic powers of ideals. [19].

4. Bounds on log canonicalthresholds and birational

rigidity

In this section we compare the log canon-ical threshold with the classical Samuel mul-tiplicity. We give then an application of theinequality between these two invariants to aclassical question on birational rigidity. LetX be a smooth complex variety and x ∈ Xa point. Denote by R the local ring of X at

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x, and by m its maximal ideal. The follow-ing result was proved by de Fernex, Ein andMustata in [10].

Theorem 4.1. Let a be an ideal in R thatdefines a subscheme Y supported at x. Letc be the log canonical threshold of (X, Y ),l(R/a) be the length of R/a and e(a) bethe Samuel multiplicity of R along a. Ifn = dimR, then we have the following in-equalities.

(i) l(R/a) ≥ nn

n!·cn.

(ii) e(a) ≥ nn

cn . Furthermore, this is an equal-ity if and only if the integral closure ofa is equal to mk for some k.

Example 4.2. Suppose that a = (xa11 , . . . , x

ann ).

In this case e(a) =∏ni=1 ai and lc(a) =

∑ni=1

1ai

.

The inequality in Theorem 4.1(ii) becomesn∏i=1

ai ≥nn(∑ni=1

1ai

)n.

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This is equivalent to1

n

n∑i=1

1

ai

n

≥n∏i=1

1

ai,

which is just the classical inequality betweenthe arithmetic and the geometric mean.

Theorem 4.1 is used in [11] to study thebehavior of the log canonical threshold undera generic projection. Using the above theo-rems and some beautiful geometric ideas ofPukhlikov [42], one gives in [11] a simple uni-form proof for the following result.

Theorem 4.3. If X is a smooth hypersur-face of degree N in CPN , with 4 ≤ N ≤ 12,then X is birationally super-rigid. In par-ticular, every birational automorphism ofX is bi-regular.

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Remark 4.4. Consider the group AutC(C(X)),the automorphism group of the field of ratio-nal functions of X . This is naturally isomor-phic to BirC(X), the group of birational au-tomorphisms of X . If X is birationally super-rigid, then BirC(X) ' AutC(X), the auto-morphism group of X . When X is a hyper-surface of degree N in PN , X has no nonzerovector fields and therefore AutC(X) is a finitegroup. If C(X) is purely transcendental, thenAutC(X) will contain a subgroup isomorphicto the general linear group GLn. In partic-ular this shows that these hypersurfaces arenot rational. In a recent preprint, de Fernexhas overcame some of the technical difficultiesthat we encountered in [11]. He is now ableto extend the arguments to all N ≥ 4. WhenN=4, it is a classical theorem of Iskovskikhand Manin that X is birationally rigid [28].They used this to show that the function field

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of a suitable quartic threefold provides a coun-terexample to the classical Luroth’s problem.

5. Spaces of arcs and contact loci

Let X be a smooth n-dimensional complexvariety. Given m ≥ 0, we denote by

Xm = Hom(

Spec C[t]/(tm+1) , X)

the space of mth order jets on X . This car-ries a natural scheme structure. Similarly wedefine the space of formal arcs on X as

X∞ = Hom(

Spec C[[t]] , X).

These constructions are functorial, hence toevery morphism µ : X ′→ X we associate cor-responding morphisms µm and µ∞. Thanksto the work of Kontsevich, Denef, Loeser andothers on motivic integration, in recent yearsthese spaces have been very useful in con-structing invariants of singular algebraic va-rieties. For instance, the following is one ofthe applications.

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Definition 5.1. Suppose thatX1 andX2 aretwo smooth projective varieties. We say thatX1 is K−equivalent to X2, if there are properbirational morphisms from a smooth projec-tive variety Y , φi : Y −→ Xi for i = 1 and2 with property φ∗1(KX1

) ∼ φ∗2(KX2) on Y .

Theorem 5.2. (Kontsevich) Suppose thatX1 is K− equivalent to X2. Then the Hodgenumber hp,q(X1) = hp,q(X2) for all p andq.

In what follows we describe some applica-tions of these spaces to the singularities ofpairs.

Consider a divisorial valuation of the formvalE with center cX(E) in X (the center isthe image of E in X). The log discrepancyof the pair (X,λ · Y ) along E is

a(E,X, λ · Y ) = kE + 1− λ · valE(IY ),

where IY is the ideal of Y in X . The idea isto measure the singularities of the pair (X,λ ·

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Y ) along B using the log discrepancies alongdivisors with center contained in B.

Definition 5.3. Let B ⊂ X be a nonemptyclosed subset. The minimal log discrepancy of(X,λ · Y ) over B is defined by(2)mld(B;X,λ·Y ) := inf

cX(E)⊆Ba(E;X,λ·Y ).

Theorem 5.4. (Ein, Mustata, and Yasuda)(Inversionof Adjunction) Let D be a smooth divisoron the smooth variety X and let B be anonempty proper closed subset of D. If Y isa closed subscheme of X such that D 6⊆ Y ,and if λ ∈ R+, then

mld(B;X,D + λ · Y ) = mld(B;D,λ · Y |D).

Remark 5.5. The notion of minimal log dis-crepancy plays an important role in the Mini-mal Model Program. It can be defined underweak assumptions on the singularities of X :

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one requires only that X is normal and Q-Gorenstein. Kollar and Shokurov have con-jectured the statement of Theorem 5.4 withthe assumption that X and D are only nor-mal and Q-Gorenstein. It is easy to see thatthe inequality ”≤” holds in general, and theopposite inequality is known as Inversion ofAdjunction. Theorem 5.4 has been general-ized in [21] to the case when both X and Dare normal locally complete intersections.

We have natural projection maps inducedby truncation Xm+1 → Xm. Since X issmooth, this is locally trivial in the Zariskitopology, with fiber An. We similarly haveprojection maps X∞ → Xm. A subset C ofX∞ is called a cylinder if it is the inverse im-age of a constructible set S in some Xm. If Cis a closed cylinder that is the inverse image ofa closed subset S ⊂ X∞, its codimension inX∞ is equal to the codimension of S in Xm.

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Consider a nonzero ideal sheaf a ⊆ OXdefining a subscheme Y ⊂ X . Given a finitejet or an arc γ on X , the order of contact ofthe corresponding scheme Y — along γ is de-fined in the natural way. Pulling a back viaγ yields an ideal (te) in C[t]/(tm+1) or C[[t]],and one sets

ordγ(a) = ordγ(Y ) = e.

For a fixed integer p ≥ 0, we define the con-tact locus

Contp(Y ) = Contp(a) =def

γ ∈ X∞ | ordγ(a) = p

.

Note that this is a locally closed cylinder. Asubset of X∞ is called an irreducible closedcontact subvariety if it is the closure of anirreducible component of Contp(Y ) for somep and Y .

Suppose now that W is an arbitrary irre-ducible closed cylinder in X∞. We can natu-rally associate a valuation of the function fieldofX toW as follows. If f is a nonzero rational

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function of X , we put

valW (f ) = ordγ(f ) for a general γ ∈ W.

This valuation is not identically zero if andonly if W does not dominate X .

If µ : X ′ −→ X is a proper birationalmorphism, with X ′ smooth, and if E is an ir-reducible divisor on X ′, then we define a val-uation by

valE(f ) = the vanishing order of f µ along E.

A valuation on the function field of X is calleda divisorial valuation (with center on X) ifit is of the form m · valE for some positiveinteger m and some divisor E as above.

A key invariant associated to a divisorialvaluation v is its log discrepancy. If E is adivisor as above, Suppose that kE is the co-efficient of E in the relative canonical divisorKX ′/X . Note that kE depends only on valE(it does not depend on the model X ′). Given

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an arbitrary divisorial valuation m · valE, wedefine its log discrepancy as m(kE + 1).

Consider a divisor E on X ′ as above. IfCm(E) is the closure of µ∞(Contm(E)), thenit is not hard to see that Cm(E) is an irre-ducible closed contact subvariety of X∞ suchthat valCm(E) = m·valE. The following result

of Ein, Lazarsfeld and Mustata [18] describesin general the connection between cylindersand divisorial valuations.

Theorem 5.6. Let X be a smooth variety.

(i) If W is an irreducible, closed cylinderin X∞ that does not dominate X, thenthe valuation valW is divisorial.

(ii) For every divisorial valuation m · valE,there is a unique maximal irreducibleclosed cylinder W such that valW = m ·valE: this is W = Cm(E).

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(iii) The map that sends m · valE to Cm(E)gives a bijection between divisorial val-uations of C(X) with center on X andthe set of irreducible closed contact sub-varieties of X∞.

The applicability of this result to the studyof singularities is due the following descriptionof log discrepancy of a divisorial valuation interms of the codimension of a certain set ofarcs.

Theorem 5.7. Given a divisorial valuationv = m · valE with center on X, if Cm(E)is its associated irreducible closed contactsubvariety in X∞, then the log discrepancyof v is equal to codim(Cm(E), X∞).

Combining the statements of the above the-orems, we deduce a lower bound for the codi-mension of an arbitrary cylinder in terms ofthe discrepancy of the corresponding divisor.

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Corollary 5.8. If W is a closed, irreduciblecylinder in X∞ that does not dominate X,then codim(W,X∞) is bounded below by thelog discrepancy of valW .

Remark 5.9. The above two theorems alsohold for singular varieties after some minormodifications using Nash’s blow-up and Mather’scanonical class.

As an application of Theorems 5.6 and 5.7,one gives in [18] a simple proof of the follow-ing result of Mustata [37] describing the logcanonical threshold in terms of the geometryof the space of jets.

Theorem 5.10. Let X be a smooth com-plex variety and Y be a proper closed sub-scheme of X. Let Xm and Ym be the spacesof mth order jets of X and Y , respectively.If c = lc(X, Y ), then

(i) For every m we have codim(Ym, Xm) ≥c · (m + 1).

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(ii) If m + 1 is sufficiently divisible, thencodim(Ym, Xm) = c · (m + 1).

The above results relating divisorial valu-ations with the space of arcs can be used tostudy more subtle invariants of singularitiesof pairs. Let Y be a closed subscheme of thesmooth variety X , and let λ be a positive realnumber. We associate a numerical invariantto the pair (X,λ · Y ) and to an arbitrarynonempty closed subset B ⊆ X , as follows.

We end with a result that translates prop-erties of the minimal log discrepancy over thesingular locus of a locally complete intersec-tion variety into geometric properties of itsspaces of jets.

Theorem 5.11. Let X be a normal locallycomplete intersection variety of dimensionn.

(i)Xm is irreducible for every m (and inthis case it is also reduced) if and only

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if mld(Xsing;X, ∅) ≥ 1 (this says that Xcanonical singularities).

(ii)Xm is normal for every m if and onlyif mld(Xsing;X, ∅) > 1 (this says that Xhas terminal singularities).

(iii) In general, we have codim((Xm)sing, Xm) ≥mld(Xsing;X, ∅) for every m.

Remark 5.12. The description in (ii) abovewas first proved in [38]. Note that since X isin particular Gorenstein, it is known that Xhas canonical singularities if and only if it hasrational singularities. All the statements inthe above theorem were obtained in [22] and[21] combining the description of minimal logdiscrepancies in terms of spaces of arcs andInversion of Adjunction.

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