The jumping coefficients of non-Q-Gorenstein multiplier ideals
aa r X i v : . [ m a t h . AG ] J a n THE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIERIDEALS PATRICK GRAF
Abstract.
Let a ⊂ O X be a coherent ideal sheaf on a normal complex variety X , and let c ≥ X, a c ) which coincides with the usual notion whenever the canonical divisor K X is Q -Cartier.We investigate the properties of the jumping numbers associated to these multiplier ideals.We show that the set of jumping numbers of a pair is unbounded, countable and satisfies acertain periodicity property. We then prove that the jumping numbers form a discrete set ofreal numbers if the locus where K X fails to be Q -Cartier is zero-dimensional. It follows thatdiscreteness holds whenever X is a threefold with rational singularities.Furthermore, we show that the jumping numbers are rational and discrete if one removesfrom X a closed subset W ⊂ X of codimension at least three, which does not depend on a . Wealso obtain that outside of W , the multiplier ideal reduces to the test ideal modulo sufficientlylarge primes p ≫ Introduction
The theory of multiplier ideal sheaves has become an important part of complex algebraic andcomplex analytic geometry. To a coherent ideal sheaf a ⊂ O X on a smooth (or more generallynormal and Q -Gorenstein) complex variety X and a real exponent c ≥
0, it associates an idealsheaf J ( X, a c ) satisfying strong vanishing theorems. If the subscheme Z ⊂ X correspondingto a is a divisor D , the multiplier ideal can be thought of as measuring the failure of the pair( X, cD ) to be klt.In particular, multiplier ideals are able to detect the log canonical threshold of D , which isdefined as the smallest number t such that ( X, tD ) is log canonical but not klt, and whichis an important invariant of the pair (
X, D ). However, using multiplier ideals one sees thatthe log canonical threshold is merely the first member of an infinite sequence of numbers, the jumping coefficients (or jumping numbers) attached to (
X, D ). Intuitively, the multiplier ideals J ( X, tD ) get smaller as t increases, and the jumping coefficients are those values of t where J ( X, tD ) jumps. These numbers first appeared implicitly in the work of Libgober [Lib83] andLoeser and Vaqui´e [LV90]. They were studied systematically by Ein, Lazarsfeld, Smith, andVarolin [ELSV04].On the other hand, in positive characteristic there is the theory of test ideals, which isunderstood to be the analogue of the theory of multiplier ideals (see e.g. [Smi00]). Test idealscan be defined for any variety, that is, no Q -Gorenstein assumption is required. It is thusnatural to wonder whether also in characteristic zero, the theory of multiplier ideals can beextended to arbitrary (say normal) varieties. Such an extension was developed by de Fernexand Hacon [dFH09] and elaborated on by Boucksom, de Fernex, and Favre [BdFF12]. Multiplier Date : September 18, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Singularities of pairs, multiplier ideals, jumping numbers, test ideals.The author was supported in full by a research grant of the Deutsche Forschungsgemeinschaft (DFG). ideals in this generality are still poorly understood, partly due to the asymptotic nature of theirdefinition involving infinitely many resolutions of singularities.The aim of this paper is to study jumping numbers in the non- Q -Gorenstein case. Let us fixour definitions. Definition 1.1 (Pairs and jumping numbers) . A pair ( X, Z ) consists of a normal complexvariety X and a proper closed subscheme Z ( X . A positive real number ξ > jumping number of the pair ( X, Z ) if J ( X, ξZ ) ( J ( X, λZ ) for all 0 ≤ λ < ξ. The set of jumping numbers of (
X, Z ) is denoted by Ξ(
X, Z ) ⊂ R + .For the reader’s convenience, in Sections 3 and 4 of this paper we recall the definition of themultiplier ideals J ( X, cZ ) according to [dFH09]. We then begin by establishing some basicproperties of jumping numbers, which give a first idea what the set of jumping numbers of apair looks like.
Proposition 1.2 (Basic properties of jumping numbers) . Let ( X, Z ) be a pair.(1.2.1) (Nonemptiness) If Z = ∅ , then Ξ( X, Z ) = ∅ .(1.2.2) (Unboundedness) If ξ ∈ Ξ( X, Z ) , then also ξ + 1 ∈ Ξ( X, Z ) . In particular, if Z = ∅ then the set Ξ( X, Z ) is unbounded above.(1.2.3) (DCC property) The set Ξ( X, Z ) satisfies the descending chain condition, i.e. anydecreasing subsequence of Ξ( X, Z ) becomes stationary. In particular, if ξ ≥ is anyreal number, then ( ξ, ξ + ε ] ∩ Ξ( X, Z ) = ∅ for sufficiently small ε > , depending on ξ .(1.2.4) (Countability) The set Ξ( X, Z ) is countable.(1.2.5) (Periodicity) If ξ > dim X − , then ξ ∈ Ξ( X, Z ) if and only if ξ + 1 ∈ Ξ( X, Z ) . In the Q -Gorenstein case, it is elementary to see that for any pair ( X, Z ) the jumping numbersΞ(
X, Z ) form a discrete set of rational numbers. De Fernex and Hacon [dFH09, Rem. 4.10]asked whether this still holds true in general.
Question 1.3.
Let ( X, Z ) be a pair. Then is the set Ξ( X, Z ) of jumping numbers a discreteset of rational numbers? A positive answer at least to the discreteness part of the question is known in several cases:if X is projective with at most log terminal or isolated singularities [Urb12a, Thm. 5.2], if X is a toric variety [Urb12b, Sec. 5], or if K X is numerically Q -Cartier [BdFFU14, Thm. 1.3].For the definition of numerically Q -Cartier divisors and numerically Q -factorial varieties, seeSection 3.C below.While it is expected that the set Ξ( X, Z ) is always discrete, at the moment we are un-able to prove this. Therefore we have to content ourselves with considering special classes ofsingularities. Our two main results are the following.
Theorem 1.4 (Discreteness for isolated non- Q -Gorenstein loci) . Let ( X, Z ) be a pair such thatthe non- Q -Cartier locus of K X is zero-dimensional. Then Ξ( X, Z ) is a discrete subset of R . Recall that the non- Q -Cartier locus of a Weil divisor D on a normal variety X is definedas the closed subset of X consisting of those points where all positive multiples mD fail to beCartier. in the sense of [dFH09], i.e. without assuming that K X is Q -Cartier HE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIER IDEALS 3 Theorem 1.5 (Discreteness in dimension three) . Let ( X, Z ) be a pair, where X is a nor-mal threefold (not necessarily projective) whose locus of non-rational singularities is zero-dimensional. Then Ξ( X, Z ) is discrete.Remark . By Proposition 1.2.3, the conclusion about discreteness can be rephrased as sayingthat Ξ(
X, Z ) satisfies ACC (the ascending chain condition) for bounded subsequences.Theorem 1.5 follows from Theorem 1.4 combined with the following result also proved in thisarticle.
Theorem 1.7 (Generic numerical Q -factoriality) . Let X be a normal complex variety. Thenthere is a closed subset W ⊂ X of codimension at least three such that X \ W is numerically Q -factorial. Concerning rationality of jumping numbers, Urbinati [Urb12a, Thm. 3.6] gave an exampleof a normal projective threefold X with a single isolated singularity x ∈ X such that forthe reduced subscheme Z = { x } , the set Ξ( X, Z ) consists only of irrational numbers. FromTheorem 1.7 and [BdFFU14, Thm. 1.3], we see that quite generally this phenomenon can onlyhappen in codimension at least three.
Corollary 1.8 (Generic discreteness and rationality) . Let ( X, Z ) be a pair. Then there is adense open subset U ⊂ X , not depending on Z , whose complement has codimension at leastthree and such that Ξ( U, Z | U ) is a discrete set of rational numbers.Remark . In the example of Urbinati mentioned above, all the jumping numbers are alge-braic. What is more, all of them are contained in Q ( √ Q -Gorenstein case.Under a Q -Gorenstein assumption, multiplier ideals are known to reduce to the correspondingtest ideals in sufficiently large characteristic [Smi00, Har01, HY03, Tak04]. It has been askedwhether this is still true in general (see e.g. [Sch11, Rem. 6.2]). An affirmative partial answerwas given in [dFDTT15, Thm. 1]. Using that result, from Theorem 1.7 we deduce the followingstatement. Corollary 1.10 (Generic comparison to test ideals) . Let ( X, Z ) be a pair, where Z is aneffective Q -Cartier Weil divisor on X . Then there is a dense open subset U ⊂ X independentof Z , whose complement has codimension at least three, such that the following holds: Given amodel of ( U, Z | U ) over a finitely generated Z -subalgebra A of C , and a rational number c ≥ ,there is a dense open subset S ⊂ Spec A such that for all closed points s ∈ S , we have J ( U, cZ | U ) s = τ ( U s , c ( Z | U ) s ) . Here τ ( U s , c ( Z | U ) s ) denotes the big (non-finitistic) test ideal of the pair ( U s , c ( Z | U ) s ) . Outline of proof of Theorem 1.4.
The key point is the following conjecture of Urbinati[Urb12b, Conj. 4.6].
Conjecture 1.11 (Global generation conjecture) . Let X be a complex normal projective varietyand D a Weil divisor on X . Then there is an ample Cartier divisor H on X such that for any m ∈ N , the sheaf O X ( m ( D + H )) is globally generated. It was noted by Urbinati in [Urb12a, Sec. 5] that Conjecture 1.11 is closely related to thequestion of discreteness of jumping numbers. In fact, a proof of the conjecture would provide
PATRICK GRAF an unconditionally positive answer to that question. We establish a weak form of the conjec-ture that only deals with isolated points of the non- Q -Cartier locus of the Weil divisor D inquestion (Theorem 7.1). Turning to the proof of Theorem 1.4, assume first for simplicity that X is projective, and suppose by way of contradiction that we are given a strictly descendingchain of multiplier ideals(1.12) J ( X, t Z ) ) J ( X, t Z ) ) · · · , where the sequence ( t k ) converges to an accumulation point t of Ξ( X, Z ). Theorem 7.1 enablesus to find an ample line bundle L on X such that L ⊗ J ( X, t k Z ) is globally generated for all k ≥
1. Taking global sections in (1.12) then yields a contradiction.In general, if X is a quasi-projective variety such that the non- Q -Cartier locus of K X iszero-dimensional, to apply the above reasoning we need to compactify X to a projective variety X . The non- Q -Cartier locus of K X may then no longer be zero-dimensional. We thereforeconstruct a sequence of ideal sheaves I k ⊂ O X which on X restricts to the sequence (1.12) andwhich has the property that all L ⊗ I k are globally generated on the open set X ⊂ X , for someample line bundle L on X . Then we conclude as before. Acknowledgements.
I would like to thank Thomas Peternell and Stefano Urbinati for stim-ulating discussions on the subject. Stefano Urbinati kindly read a draft version of this paper.Furthermore I would like to thank Fabrizio Catanese for explaining Example 3.7 to me.2.
Notation and conventions
We work over the field of complex numbers C throughout. A variety is an integral separatedscheme of finite type over C . A pair ( X, Z ) consists of a normal variety X and a proper closedsubscheme Z ( X . Unless otherwise specified, by a divisor on a normal variety X we meana Weil divisor with integer coefficients. For k ∈ { Z , Q , R } , a k -divisor is a Weil divisor withcoefficients in k . The group of Z -divisors on X modulo linear equivalence is denoted Cl( X ). Boundaries. A boundary on a normal variety X is an effective Q -divisor ∆ such that K X + ∆is Q -Cartier. In this case we will say that ( X, ∆) is a log pair . Note that the coefficients of ∆may be larger than 1. If f : Y → X is a proper birational morphism from a normal variety Y ,we write K ∆ Y/X := K Y + f − ∗ ∆ − f ∗ ( K X + ∆)for the relative canonical divisor of ( X, ∆). Reflexive sheaves. If F is a coherent sheaf on a normal variety X , we denote its reflexivehull (i.e. its double dual) by F ∗∗ . The torsion subsheaf of F is denoted tor F . We often write F (cid:14) tor as a shorthand for F (cid:14) tor F . Log resolutions.
We will need to discuss log resolutions of different kinds of objects, all ofwhich are of course variations on the same theme. For the existence of such resolutions, werefer to [dFH09, Thm. 4.2]. Let X be a normal variety. A log resolution of X is a properbirational morphism f : e X → X , where e X is smooth and Exc( f ), the exceptional locus of f , isa divisor with simple normal crossings (snc, for short). A log resolution of a log pair ( X, ∆) isa log resolution of X such that Exc( f ) ∪ f − ∗ ∆ has snc support.The constant sheaf of rational functions on X is denoted by K X . A fractional ideal sheaf a ⊂ K X on X is a coherent O X -submodule of K X . A log resolution of a nonzero fractionalideal sheaf a ⊂ K X is a log resolution f : e X → X of X such that a · O e X = O e X ( E ) ⊂ K e X for some Cartier divisor E on e X , where Exc( f ) ∪ supp( E ) has simple normal crossings. Notethat a · O e X ∼ = f ∗ a (cid:14) tor, i.e. we do not take the reflexive hull. A log resolution of a subscheme HE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIER IDEALS 5 Z ( X is a log resolution of the ideal sheaf a ⊂ O X ⊂ K X of Z . That is, the scheme-theoreticinverse image f − ( Z ) ⊂ e X is required to be a Cartier divisor whose support has simple normalcrossings with Exc( f ).If we need to simultaneously resolve several objects of the types described above, we willtalk about joint log resolutions . E.g. using the above notation, a joint log resolution of ( X, ∆), a , and Z would be a log resolution of each of the three objects such that Exc( f ) ∪ f − ∗ ∆ ∪ supp( E ) ∪ f − ( Z ) has snc support. Remark . If Z ⊂ X is a Weil divisor such that K X + Z is Q -Cartier, then being a logresolution of the subscheme Z ⊂ X is a slightly stronger notion than being a log resolution ofthe log pair ( X, Z ). We trust that this will not lead to any confusion.
Global generation of sheaves. If F is a coherent sheaf on a variety X and x ∈ X is a point,we say that F is globally generated at x if the natural morphism H ( X, F ) ⊗ C O X −→ F is surjective at x . We say that F is globally generated on an open subset U ⊂ X if F isglobally generated at every point x ∈ U . Relative N´eron–Severi spaces.
Let f : Y → X be a proper morphism between normalvarieties, where Y is Q -factorial. We denote by N ( Y /X ) Q the vector space of Q -divisors on Y modulo f -numerical equivalence (that is, numerical equivalence on all curves contracted by f ).We denote by N ( Y /X ) Q the vector space spanned by all curves contracted by f , modulonumerical equivalence. These two vector spaces are finite-dimensional, and dual to each othervia the intersection pairing N ( Y /X ) Q × N ( Y /X ) Q −→ Q . Their common dimension is called the relative Picard number ρ ( Y /X ) of f .3. Pullbacks of Weil divisors and numerical Q -factoriality In order to define multiplier ideals in the absence of any Q -Gorenstein assumption, it isnecessary to dispose of a notion of pullback for arbitrary Weil divisors. Such a pullback wasintroduced in [dFH09] and rephrased using the language of nef envelopes in [BdFF12]. For acertain class of Weil divisors, called numerically Q -Cartier divisors, this pullback is particularlywell-behaved. Numerically Q -Cartier divisors were introduced in [BdFF12] from a b -divisorialpoint of view and reinterpreted in a somewhat more down-to-earth fashion in [BdFFU14]. Forease of reference, we recall in this section the relevant facts and definitions. For more detailsand full proofs, we refer the reader to the original papers cited above. Remark . Closely related notions of pullback and numerical Q -Cartierness were alreadydiscussed by Nakayama [Nak04, Ch. II, Lem. 2.12 and Ch. III, Cor. 5.11].3.A. Pulling back Weil divisors.
Consider a proper birational morphism f : Y → X betweennormal varieties, and let D be a Weil divisor on X . The natural pullback f ♮ D of D along f isdefined by O Y ( − f ♮ D ) = ( O X ( − D ) · O Y ) ∗∗ , where we consider O X ( − D ) ⊂ K X as a fractional ideal sheaf on X . We define the pullback f ∗ D of D along f by setting f ∗ D := lim k →∞ f ♮ ( kD ) k = inf k ≥ f ♮ ( kD ) k , PATRICK GRAF where the limit and the infimum are to be understood coefficient-wise. The limits actually existin R , hence f ∗ D is a well-defined R -divisor. If D is Q -Cartier, then this definition agrees withthe usual notion of pullback. We have f ∗ ( mD ) = m · f ∗ D for any integer m ≥
0. Thus we canextend the map f ∗ to arbitrary Q -divisors by clearing denominators. Furthermore, we have f ∗ ( − D ) ≥ − f ∗ D . However, this inequality may be strict – see Section 3.C below.3.B. Relative canonical divisors.
Let f : Y → X be as before, and fix canonical divisors K X and K Y on X and on Y , respectively, that satisfy K X = f ∗ K Y . For any m ≥
1, we definethe m -th limiting relative canonical divisor of Y over X by K Y/X,m := K Y − m f ♮ ( mK X )and the relative canonical divisor of Y over X by K − Y/X := K Y − f ∗ K X = lim m →∞ K Y/X,m . For any m , we have the inequality K Y/X,m ≤ K − Y/X . If K X is Q -Cartier, then K − Y/X coincideswith the usual relative canonical divisor K Y/X . Note, however, that here we have not definedthe symbol K Y/X in general.3.C.
Numerically Q -Cartier divisors. We next address the question of when the equality f ∗ ( − D ) = − f ∗ D holds. Proposition 3.2 (Numerically Q -Cartier divisors) . Let X be a normal variety and D a Q -divisor on X . Then the following conditions are equivalent:(3.2.1) There is a proper birational morphism f : Y → X , where Y is Q -factorial, and an f -numerically trivial Q -divisor D ′ on Y such that f ∗ D ′ = D .(3.2.2) For any proper birational morphism f : Y → X where Y is Q -factorial, there is an f -numerically trivial Q -divisor D ′ on Y such that f ∗ D ′ = D .(3.2.3) For any proper birational morphism f : Y → X , we have f ∗ ( − D ) = − f ∗ D .Proof. See [BdFFU14, Props. 5.3, 5.9]. (cid:3)
Definition 3.3 (Numerical Q -factoriality) . A Q -divisor D on a normal variety X is called numerically Q -Cartier if the equivalent conditions of Proposition 3.2 are satisfied. The vectorspace of Q -divisors on X modulo the subspace of numerically Q -Cartier divisors is denoted byCl num ( X ) Q . The variety X is called numerically Q -factorial if Cl num ( X ) Q = 0, i.e. if every Q -divisor on X is numerically Q -Cartier. Remark . In (3.2.1) and (3.2.2), the divisor D ′ is unique, and D ′ = f ∗ D . Hence for anumerically Q -Cartier divisor D , the R -divisor f ∗ D is in fact a Q -divisor. In particular, if K X is numerically Q -Cartier, then for any normal modification Y → X the relative canonicaldivisor K − Y/X is a Q -divisor. Example . Any normal surface is numerically Q -factorial. This follows from theexistence of Mumford’s pullback for divisors on surfaces [KM98, Ch. 4.1]. Example . Let Y be a smooth projective variety and L an ample divisor on Y . Weconsider the affine cone X = C a ( Y, L ) over Y with respect to L , defined as the spectrum of thesection ring R ( X, L ) := M n ≥ H ( Y, nL ) . This is a generalization of the classical affine cone over a variety embedded in projective space.
HE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIER IDEALS 7 For any prime divisor D on Y , the cone C ( D ) = C a ( D, L | D ) is a divisor on X . The map D C ( D ) extends linearly to give an isomorphismCl( X ) ∼ = Pic( Y ) . Z · L. The divisor C ( D ) is Q -Cartier if and only if D and L are Q -linearly proportional, while C ( D ) isnumerically Q -Cartier if and only if D and L are numerically proportional [BdFF12, Ex. 2.31,Lem. 2.32]. The canonical divisor of X is given by K X = C ( K Y ).It follows that X is Q -factorial if and only if on Y , numerical equivalence coincides with Q -linear equivalence, and the Picard number ρ ( Y ) = 1. The first condition is well-known to beequivalent to H ( Y, O Y ) = 0. On the other hand, X is numerically Q -factorial if and only if ρ ( Y ) = 1. Example . This example is a continuation of the previous one. We make thefollowing claim.
In every dimension n ≥ , there exist isolated singularities that are numerically Q -factorial,but not Q -factorial (more precisely, the canonical divisor is not Q -Cartier). Namely, if Y is smooth projective of dimension n − K Y is ample,and H ( Y, O Y ) = 0, let B ∈ Pic ( Y ) be a numerically trivial non-torsion divisor on Y . Thenthe cone X = C a ( Y, K Y + B ) over Y has the required properties.Concerning the existence of Y , if n = 2 we may simply take Y to be a curve of genus g ≥
2. If n ≥
3, let A be an abelian n -fold of Picard number one [Laz04a, Ex. 1.2.26], andtake Y ⊂ A to be a smooth ample divisor. Then K Y is ample by the adjunction formula and H ( Y, O Y ) = H ( A, O A ) = 0 by the weak Lefschetz theorem [Laz04a, Ex. 3.1.24]. If n ≥ A → Pic Y is an isomorphism [Laz04a, Ex. 3.1.25], hence ρ ( Y ) = ρ ( A ) = 1. Incase n = 3, at least if Y is very general in a sufficiently ample linear system the infinitesimalNoether–Lefschetz theorem implies that H , ( A ) ∩ H ( A, Q ) −→ H , ( Y ) ∩ H ( Y, Q )is surjective [CGGH83, Cor. on p. 179]. By the Lefschetz (1 , ρ ( Y ) ≤ ρ ( A ) = 1.We end this section with some facts about numerically Q -Cartier divisors which we will uselater. Proposition 3.8 (Short exact sequence) . Let f : Y → X be a proper birational morphism,where Y is Q -factorial, and let E , . . . , E ℓ be the divisorial components of the exceptional locusof f . Then we have a short exact sequence of Q -vector spaces −→ ℓ M i =1 Q · [ E i ] −→ N ( Y /X ) Q −→ Cl num ( X ) Q −→ induced by the pushforward of divisors along f .Proof. See [BdFFU14, Cor. 5.4.ii)]. (cid:3)
Theorem 3.9 (Rational singularities) . If X has rational singularities, then the notions of Q -Cartier and numerically Q -Cartier divisors coincide, i.e. any numerically Q -Cartier divisor isalready Q -Cartier.Proof. See [BdFFU14, Thm. 5.11]. (cid:3)
PATRICK GRAF
Remark . In Theorem 3.9 it is sufficient to assume that X has ,meaning that R f ∗ O e X = 0 for some (equivalently, any) resolution f : e X → X . This can be seenfrom the proof of [BdFFU14, Thm. 5.11].4. Non- Q -Gorenstein multiplier ideals Following [dFH09], we recall in this section how to associate a multiplier ideal J ( X, cZ ) toa pair (
X, Z ) and a real number c ≥
0. If a ⊂ O X is the ideal sheaf of the subscheme Z , themultiplier ideal may also be denoted by J ( X, a c ).4.A. The classical case.
Classically, one defines multiplier ideals under some Q -Gorensteincondition. More precisely, assume that a boundary ∆ on X is given, i.e. the divisor K X + ∆ is Q -Cartier. Let f : Y → X be a joint log resolution of ( X, ∆) and a , and write a · O Y = O Y ( − D )with D an effective Cartier divisor. Then one defines J (( X, ∆) , cZ ) := f ∗ O Y ( ⌈ K Y − f ∗ ( K X + ∆) − cD ⌉ ) . One checks that this definition is independent of the choice of log resolution. See [Laz04b,Def. 9.3.60] for more details.4.B.
The general case.
Let (
X, Z ) be a pair and c ≥ K X on X . For any natural number m ≥
1, consider a joint log resolution f : Y → X of Z ⊂ X and O X ( − mK X ) ⊂ K X . Note that f may heavily depend on m . We define the m -th limitingmultiplier ideal sheaf of ( X, cZ ) as J m ( X, cZ ) := f ∗ O Y ( ⌈ K Y/X,m − cD ⌉ ) ⊂ O X , where D = f − ( Z ). This coherent ideal sheaf is independent of the choice of K X and f . Forany k, m >
0, one has an inclusion J m ( X, cZ ) ⊂ J km ( X, cZ ). By the Noetherian propertyof O X , it follows that the set {J m ( X, cZ ) } m ≥ has a unique maximal element. We define the multiplier ideal sheaf J ( X, cZ ) to be this maximal element. We have J ( X, cZ ) = J m ( X, cZ )for m sufficiently divisible. If K X is Q -Cartier, then this definition of multiplier ideal agreeswith the classical one. More precisely, we have J (( X, , cZ ) = J m ( X, cZ ) as soon as mK X isCartier.By [dFH09, Prop. 4.9], we have that J ( X, cZ ) ⊂ J ( X, c ′ Z ) for c ′ < c and that J ( X, cZ ) = J ( X, ( c + ε ) Z ) for ε > c . Hence the following definitionmakes sense. Definition 4.1 (Jumping numbers) . A positive real number ξ > jumping number of the pair (
X, Z ) if J ( X, ξZ ) ( J ( X, λZ ) for all 0 ≤ λ < ξ. We denote the set of jumping numbers of (
X, Z ) by Ξ(
X, Z ) ⊂ R + .4.C. Compatible boundaries.
The following definition [dFH09, Def. 5.1] serves to relate thegeneral case just described to the classical one.
Definition 4.2 (Compatible boundary) . Let (
X, Z ) be a pair, and fix an integer m ≥
2. Aboundary ∆ on X is said to be m -compatible for ( X, Z ) if there exists a canonical divisor K X on X and a joint log resolution f : Y → X of ( X, ∆), Z ⊂ X and O X ( − mK X ) such that:(4.2.1) the divisor m ∆ is integral, and ⌊ ∆ ⌋ = 0,(4.2.2) no component of ∆ is contained in the support of Z , and(4.2.3) the equality K ∆ Y/X = K Y/X,m holds.The point of this definition is the following observation.
HE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIER IDEALS 9 Proposition 4.3 (Realizing multiplier ideals as classical ones) . Let ( X, Z ) be a pair and c ≥ a real number. Choose an integer m such that J ( X, cZ ) = J m ( X, cZ ) , and let ∆ be an m -compatible boundary for ( X, Z ) . Then we have J ( X, cZ ) = J (( X, ∆) , cZ ) . Proof.
See [dFH09, Prop. 5.2]. (cid:3)
Of course, the usefulness of this notion depends on the existence of compatible boundaries.This was established in [dFH09, Thm. 5.4]. The proof given there is constructive and yieldsthe following more precise result, which we record for later use.
Theorem 4.4 (Existence of compatible boundaries) . Let ( X, Z ) be a pair, and let m ≥ be anatural number. Choose an effective Weil divisor D on X such that K X − D is Cartier, andlet L ∈ Pic X be a line bundle such that L ( − mD ) := L ⊗ O X ( − mD ) is globally generated.Pick a finite-dimensional subspace V ⊂ H ( X, L ( − mD )) that generates L ( − mD ) , and let M be the divisor of a general element of V . Then ∆ := 1 m M is an m -compatible boundary for ( X, Z ) . (cid:3) It follows that for any pair (
X, Z ) and c ≥
0, the set of ideal sheaves (cid:8) J (( X, ∆) , cZ ) (cid:12)(cid:12) ∆ a boundary on X in the sense of Section 2 (cid:9) has a unique maximal element, namely J ( X, cZ ). This reduces some, but by no means all,questions about multiplier ideals to the classical case. Roughly speaking, as long as one isdealing with only finitely many values of (
X, Z ) and c , one may choose the number m sufficientlydivisible such that J = J m for all those pairs, and then one picks an m -compatible boundary.In general, however, it is hard to tell where the sequence of limiting multiplier ideals stabilizes.Even if the pair ( X, Z ) is fixed, given a collection of infinitely many values of c , the required m ’smight become arbitrarily large. One then needs to consider infinitely many different compatibleboundaries. If instead one tries to work directly with the definitions, one needs to considerinfinitely many resolutions of X .However, if K X is numerically Q -Cartier, we can circumvent these difficulties thanks to thefollowing theorem. Theorem 4.5 (Multiplier ideals on numerically Q -Gorenstein varieties) . Let ( X, Z ) be a pair,where K X is numerically Q -Cartier. Choose a log resolution f : e X → X of ( X, Z ) . Then forany t > we have J ( X, tZ ) = f ∗ O e X (cid:0) ⌈ K − e X/X − t · f − ( Z ) ⌉ (cid:1) . Proof.
See [BdFFU14, Thm. 1.3]. (cid:3) Elementary properties of jumping numbers
The aim of the present section is to prove Proposition 1.2. So let (
X, Z ) be a pair. We denotethe ideal sheaf of Z by a ⊂ O X . Nonemptiness.
Pick a canonical divisor K X on X and a number m such that J ( X, ∅ ) = J m ( X, ∅ ). If f : Y → X is a joint log resolution of ( X, Z ) and O X ( − mK X ), then J ( X, ∅ ) = f ∗ O Y ( ⌈ K Y/X,m ⌉ ) . As Z = ∅ , we have a · O Y = O Y ( − D ), where D is a nonzero effective Cartier divisor on Y . Pick x ∈ supp( Z ) arbitrarily, and let E ⊂ supp( D ) be a prime divisor with x ∈ f ( E ). Take anynonzero function germ h ∈ J ( X, ∅ ) x . Then for ξ ≫
0, the coefficient of E in −⌈ K − Y/X − ξD ⌉ will be larger than the order of vanishing of f ∗ h along E . Fix such a number ξ . Let ℓ be suchthat J ( X, ξZ ) = J ℓ ( X, ξZ ). Let g : e Y → X be a joint log resolution of ( X, Z ) and O X ( − ℓK X )that factors through f . Then J ( X, ξZ ) = g ∗ O e Y ( ⌈ K e Y /X,ℓ − ξD ⌉ ) . As K e Y /X,ℓ ≤ K − e Y /X , we see that g ∗ h is not contained in O e Y ( ⌈ K e Y /X,ℓ − ξD ⌉ ) at the genericpoint of the strict transform of E on e Y . Hence h is not contained in J ( X, ξZ ) x . It follows thatthe inclusion J ( X, ξZ ) ( J ( X, ∅ ) is strict, and so there must be a jumping number of the pair( X, Z ) in the interval (0 , ξ ].5.B.
Unboundedness.
Let ξ ∈ Ξ( X, Z ) be a jumping number. Spelled out explicitly, thismeans that for any 0 < ε ≤ ξ there is a point x ∈ X and a function h ∈ O X,x , both dependingon ε , such that h ∈ J ( X, ( ξ − ε ) Z ) x , but h
6∈ J ( X, ξZ ) x .Fix an arbitrary 0 < ε ≤ ξ . We regard a germ g ∈ a x as a regular function on a smallneighborhood U of x . For any proper birational morphism f : Y → X such that Y is normaland a · O Y = O Y ( − D ) is invertible, we may then writediv( f ∗ g ) = D | V + M g , where V = f − ( U ) and M g is an effective divisor on V . Here we take the divisor of f ∗ g as afunction, not as a section of O Y ( − D ).Choose a finite set of generators for the ideal a x = h g , . . . , g r i in the noetherian ring O X,x . Then after possibly shrinking U , the sheaf O Y ( − D ) | V will be generated by the sec-tions f ∗ g , . . . , f ∗ g r . It follows that if g = P ri =1 λ i g i ∈ a x is a general C -linear combination ofthe chosen generators, then M g and D | V do not have any common components.Now fix such a general g ∈ a x . As h ∈ J m ( X, ( ξ − ε ) Z ) x for some m , we get g · h ∈ J m ( X, ( ξ + 1 − ε ) Z ) x ⊂ J ( X, ( ξ + 1 − ε ) Z ) x from the definitions. On the other hand, we will show that g · h
6∈ J ( X, ( ξ + 1) Z ) x . Proceedingby contradiction, assume that g · h ∈ J m ( X, ( ξ + 1) Z ) x for some m . Since M g and D | V do notshare any common components, it follows that h ∈ J m ( X, ξZ ) x ⊂ J ( X, ξZ ) x , contradicting the definition of h . So J ( X, ( ξ + 1) Z ) ( J ( X, ( ξ + 1 − ε ) Z ). As ε was chosenarbitarily, this proves that ξ + 1 is a jumping number.The second statement of (1.2.2) follows from what we have just proved, combined with (1.2.1).5.C. DCC property.
Any decreasing sequence ξ > ξ > · · · contained in Ξ( X, Z ) would giverise to a strictly ascending chain of coherent ideal sheaves J ( X, ξ Z ) ( J ( X, ξ Z ) ( · · · , which is impossible by the Noetherian property of O X . For the second statement, assuming itwere false we can easily construct a decreasing sequence as above. HE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIER IDEALS 11 Countability.
Consider the following subset of R : A := (cid:8) a ∈ R (cid:12)(cid:12) Ξ( X, Z ) ∩ [0 , a ] is countable (cid:9) . As 0 ∈ A , this set is nonempty. If ( a n ) ⊂ A is a sequence converging to some number a , it isimmediate that also a ∈ A . Hence A is closed. By the second part of (1.2.3), the set A is open.It follows that A = R , and then Ξ( X, Z ) = S n ∈ N (cid:0) Ξ( X, Z ) ∩ [0 , n ] (cid:1) is countable. Remark . Of course, the proof just given has nothing to do with jumping numbers, and itshows quite generally that any DCC set is countable.5.E.
Periodicity.
By (1.2.2), it suffices to show that if ξ > dim X − ξ + 1 is not a jumping number. So we assume that J ( X, ( ξ − ε ) Z ) = J ( X, ξZ ) forsufficiently small ε >
0, and we need to show J ( X, ( ξ + 1 − ε ) Z ) = J ( X, ( ξ + 1) Z ) for small ε > Claim . For any c ≥ dim X −
1, we have J ( X, ( c + 1) Z ) = a · J ( X, cZ ) . Proof of Claim 5.2.
Fix a number m such that J ( X, cZ ) = J m ( X, cZ ) and J ( X, ( c + 1) Z ) = J m ( X, ( c + 1) Z ), and choose an m -compatible boundary ∆ for the pair ( X, Z ). We then have J ( X, cZ ) = J (( X, ∆) , cZ ) and J ( X, ( c + 1) Z ) = J (( X, ∆) , ( c + 1) Z )by Proposition 4.3. The claim now follows from Skoda’s theorem [Laz04b, Thm. 9.6.21.ii)],asserting that J (( X, ∆) , ( c + 1) Z ) = a · J (( X, ∆) , cZ ) . Note that [Laz04b, Thm. 9.6.21] is only applicable if X is smooth, ∆ = 0, and c is rational.However, as explained in Remark 9.6.23 and after Variant 9.6.39 of [Laz04b], the statementremains true in the generality required here. Indeed, upon replacing the relative vanishingof [Laz04b, Variant 9.4.4] by Thm. 9.4.17 of that book, the proof of Skoda’s theorem goesthrough verbatim, even if c is not rational. (cid:3) Returning to the proof of (1.2.5), for ε > J ( X, ( ξ + 1 − ε ) Z ) = a · J ( X, ( ξ − ε ) Z ) = a · J ( X, ξZ ) = J ( X, ( ξ + 1) Z ) , finishing the proof of (1.2.5) and the whole Proposition 1.2. (cid:3) Resolutions of sheaves
As a preparation for the proof of Theorem 1.4, we recall here the notion of resolution mor-phism for a coherent sheaf (not to be confused with resolutions in the sense of homologicalalgebra) and a positivity property of such resolutions which will prove crucial for us.The construction presented here is essentially due to Rossi [Ros68, Thm. 3.5] in the moregeneral context of coherent analytic sheaves on complex spaces. However, as the positivityproperty mentioned above is not addressed in [Ros68], we have chosen to include here a fullaccount of the construction of sheaf resolutions (in the algebraic case). Our proof is somewhatsimpler than the original argument. Furthermore, it answers a question of Rossi [Ros68, p. 72]asking for a universal property of his construction.
Definition 6.1 (Resolution of a sheaf) . Let F be a coherent sheaf on a normal variety X . A resolution of F is a proper birational morphism f : Y → X such that Y is normal and f ∗ F (cid:14) toris locally free. A resolution f of F is called minimal if every other resolution of F factorsthrough f . Note that a resolution of a sheaf F on X will usually not be a resolution of singularities for X . Of course, the minimal resolution of F is unique if it exists. Theorem 6.2 (Existence of resolutions) . Let X be a normal variety and F a coherent sheafon X . Then the minimal resolution f : Y → X of F exists. Furthermore, if F has rank one,then f ∗ F (cid:14) tor is an f -ample invertible sheaf.Proof. First we construct the required resolution locally. After shrinking X , we may assumethat there is a surjection α : O ⊕ pX ։ F for some integer p . Let X ◦ be the open subset of X where F is locally free, say of rank r . Then α determines a morphism F : X ◦ → G := Gr( p, r )into the Grassmannian of r -dimensional quotients of C p . We may view F as a rational map X G .Let π : e X → X be a resolution of indeterminacy for F , i.e. a blowup such that F extends toa morphism ϕ : e X → G . e X π (cid:15) (cid:15) ϕ & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ X F / / ❴❴❴❴❴❴ G Let O ⊕ pG ։ E be the tautological quotient bundle on G . Then we have a surjection π ∗ F ։ ϕ ∗ E .It is an isomorphism on π − ( X ◦ ), hence it induces an isomorphism π ∗ F (cid:14) tor ∼ = ϕ ∗ E . Inparticular, π ∗ F (cid:14) tor is locally free. Conversely, if π : e X → X is a blowup such that π ∗ F (cid:14) toris locally free, then the surjection O ⊕ p e X π ∗ α −−→ π ∗ F −→ π ∗ F (cid:14) torprovides a morphism e X → G that extends F .Now let Y ′ ⊂ X × G be the closure of the graph of F , and let Y be the normalization of Y ′ . Y ν / / f ' ' Y ′ pr (cid:15) (cid:15) pr & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ X F / / ❴❴❴❴❴❴ G Then f = pr ◦ ν : Y → X is a resolution of indeterminacy for F , and every other such resolution π : e X → X with e X normal factors through f . By what we have observed above, this means that f is a minimal resolution of F . By uniqueness of the minimal resolution, f does not dependon the surjection α chosen in the beginning. In particular, the local constructions glue to givea globally defined minimal resolution of F . This proves the first half of the theorem.For the second statement, assume that F has rank one. Since the statement is local, we mayagain assume that we are in the local situation described above, and we continue to use thatnotation. Note that since F has rank one, G = Gr( p,
1) = P p − . Sopr ∗ F (cid:14) tor ∼ = pr ∗ O P p − (1) . It follows immediately that pr ∗ F (cid:14) tor is ample (even very ample) on the fibres of pr . Asampleness is preserved under finite pullbacks, pulling back everything to the normalization Y of Y ′ we get that f ∗ F (cid:14) tor is ample on the fibres of f . By [Laz04a, Thm. 1.7.8], this impliesthe f -ampleness of f ∗ F (cid:14) tor. (cid:3) Remark . If F has rank one (which is the only case we shall need), an alternative approachis as follows. Replacing F by F (cid:14) tor, we may assume that F is torsion-free. Then F isisomorphic to a fractional ideal sheaf a ⊂ K X . Let f : Y → X be the normalization of HE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIER IDEALS 13 the blowing-up of a , which can be defined in exactly the same fashion as the blowing-up of anordinary ideal sheaf [Har77, Ch. II, Sec. 7]. Then the assertions we need to prove follow directlyfrom the analogues of [Har77, Ch. II, Props. 7.14, 7.10]. Remark . If F has rank r ≥
2, then there is no resolution f of F such that f ∗ F (cid:14) toris an f -ample vector bundle. For the minimal resolution, this can be seen from the proof ofTheorem 6.2 and the fact that the tautological quotient bundle on the Grassmannian Gr( p, r )is not ample if r ≥ Remark . It is clear that both the proof of Theorem 6.2 as well as the alternative approachoutlined in Remark 6.3 work over an algebraically closed field of arbitrary characteristic.7.
Global generation of Weil divisorial sheaves
The purpose of the present section is to prove the following theorem, which is a special caseof Conjecture 1.11 and generalizes the previously known special case [Urb12a, Prop. 5.5], where X is required to have isolated singularities. Theorem 7.1 (Global generation for isolated non- Q -Cartier loci) . Let X be a normal projectivevariety and D a Weil divisor on X , with non- Q -Cartier locus W ⊂ X . Define the open set U ⊂ X as the complement of W union the isolated points of W . Then there is an ample Cartierdivisor H on X such that for m ∈ N sufficiently divisible, the sheaf O X ( m ( D + H )) is globallygenerated on U . The proof proceeds along the general lines of [Urb12a], except that instead of passing toa resolution of X and using Kawamata–Viehweg vanishing, we only resolve the sheaf O X ( D )(in the sense of Section 6) and use Fujita vanishing. In particular, all the key ingredientsof our proof (the others being relative Serre vanishing and Castelnuovo–Mumford regularity)remain valid in arbitrary characteristic. Hence we see that Theorem 7.1 is also true in positivecharacteristic, as one might expect. Remark . In Conjecture 1.11, the conclusion holds for all m ∈ N and not just for sufficientlydivisible m . This should also be true of Theorem 7.1. We have not paid attention to this, as itis not needed for our purposes and would only complicate the proof.Before starting the proof, we record one auxiliary lemma. Lemma 7.3 (Extensions of globally generated sheaves) . Let X be a projective variety, and let −→ F ′ −→ F −→ F ′′ −→ be a short exact sequence of coherent sheaves on X . Assume that F ′ is globally generated andthat H ( X, F ′ ) = 0 . If x ∈ X is any point such that F ′′ is globally generated at x , then so is F .Proof. We have the following diagram with exact rows:0 / / H ( F ′ ) ⊗ O X,x / / (cid:15) (cid:15) (cid:15) (cid:15) H ( F ) ⊗ O X,x / / (cid:15) (cid:15) H ( F ′′ ) ⊗ O X,x / / (cid:15) (cid:15) (cid:15) (cid:15) / / F ′ x / / F x / / F ′′ x / / . The outer vertical maps are surjective. By the four lemma, so is the middle one. (cid:3)
Proof of Theorem 7.1.
For convenience of the reader, the proof is divided into foursteps.
Step 1: Blowing up.
Let e D = N · D be the smallest positive multiple of D which is Cartieroutside of W , and let e H = N · H for some very ample Cartier divisor H on X . Let f : Y → X be the minimal resolution of O X ( e D ). Then f is an isomorphism outside of W , and O Y ( B ) := f ∗ O X ( e D ) . toris an f -ample invertible sheaf by Theorem 6.2. Note that we are free to replace D by D + N · e H for any N >
0. This does not change N and f , and it changes B to B + f ∗ ( N N · e H ). Henceby [KM98, Prop. 1.45], we may assume that the Cartier divisor B is globally ample on Y . Step 2: Vanishing theorems.
By Fujita vanishing [Laz04a, Thm. 1.4.35], we have H i ( Y, O Y ( mB + P )) = 0 for i > m ≫ P on Y .Furthermore, by relative Serre vanishing [Laz04a, Thm. 1.7.6] R i f ∗ O Y ( mB ) = 0 for i > m ≫ f and the sheaf O Y ( mB + ℓ · f ∗ e H ) that H i ( X, f ∗ O Y ( mB ) ⊗ O X ( ℓ e H )) = 0 for i > m ≫ ℓ ≥ m ≫ F m := f ∗ O Y ( mB ) ⊗ O X ( m e H )is globally generated and satisfies H ( X, F m ) = 0. Step 3: Pushing down.
Observe that F m is torsion-free and that its reflexive hull F ∗∗ m isisomorphic to O X ( m ( e D + e H )). Indeed, both sheaves are reflexive and they agree outside of W ,which has codimension at least two in X . Thus the natural map F m → F ∗∗ m yields a shortexact sequence 0 −→ F m −→ O X ( m ( e D + e H )) −→ Q m −→ , where Q m is supported on W . We are aiming to show that the middle term O X ( m ( e D + e H ))is globally generated on U . So let x ∈ U be arbitrary. Then either x W , whence the stalk Q m,x is zero, or x ∈ W is isolated, hence so is x ∈ supp Q m . In either case, Q m is globallygenerated at x . By Lemma 7.3, also O X ( m ( e D + e H )) is globally generated at x . Since x ∈ U was arbitrary, it follows that O X ( m ( e D + e H )) is globally generated on U . Step 4: End of proof.
We have shown that there is an m ∈ N such that for m ≥ m , the sheaf O X ( m ( e D + e H )) is globally generated on U . Since m ( e D + e H ) = mN · ( D + H ), this proves theclaim of the theorem if we take “ m sufficiently divisible” to mean “ m divisible by m N ”. (cid:3) Discreteness for isolated non- Q -Gorenstein loci This section is devoted to the proof of Theorem 1.4, repeated here for the reader’s conve-nience.
Theorem 8.1 (Discreteness for isolated non- Q -Gorenstein loci) . Let ( X, Z ) be a pair such thatthe non- Q -Cartier locus of K X is zero-dimensional. Then Ξ( X, Z ) is a discrete subset of R . HE JUMPING COEFFICIENTS OF NON- Q -GORENSTEIN MULTIPLIER IDEALS 15 Auxiliary results.
We begin with a few easy observations.
Lemma 8.2 (Descending chains of ideals) . Let X be a projective variety, and let J ⊃ J ⊃ · · · be a descending chain of coherent ideal sheaves on X . Let U ⊂ X be an open subset. Assumethat there exists a line bundle L ∈ Pic X such that for any k ≥ , the sheaf L ⊗ J k is globallygenerated on U . Then the sequence J | U ⊃ J | U ⊃ · · · stabilizes, i.e. we have J k | U = J k +1 | U for k ≫ .Proof. The chain of complex vector spaces H ( X, L ⊗ J ) ⊃ H ( X, L ⊗ J ) ⊃ · · · stabilizes for dimension reasons. So for k ≫ H ( X, L ⊗ J k ) ⊗ C O U (cid:15) (cid:15) (cid:15) (cid:15) H ( X, L ⊗ J k +1 ) ⊗ C O U (cid:15) (cid:15) (cid:15) (cid:15) ( L ⊗ J k ) | U ( L ⊗ J k +1 ) | U . ? _ o o This implies ( L ⊗ J k ) | U = ( L ⊗ J k +1 ) | U , and then J k | U = J k +1 | U . (cid:3) Proposition 8.3 (Global generation of twisted multiplier ideals) . Let ( X, Z ) be a pair, where X is projective, and let c ≥ be a real number. Choose a boundary ∆ on X and a Cartierdivisor B such that O X ( B ) ⊗ J Z is globally generated. Furthermore let L be a very ampleCartier divisor such that L − ( K X + ∆ + cB ) is big and nef. Then for n ≥ dim X + 1 , the sheaf O X ( nL ) ⊗ J (( X, ∆) , cZ ) is globally generated.Proof. This is a combination of Nadel vanishing in the form of [Laz04b, Prop. 9.4.18] andCastelnuovo–Mumford regularity. (cid:3)
Proof of Theorem 8.1.
Again, for the sake of readability the proof is divided into fivesteps.
Step 0: Setup of notation.
We need to show that Ξ(
X, Z ) is discrete. Arguing by contradiction,assume that t ∈ R is an accumulation point. Then there is a sequence ( t k ) ⊂ Ξ( X, Z ) \ { t } converging to t . By Proposition 1.2(3), we may assume that the sequence ( t k ) is strictlyincreasing. Step 1: Simplifying assumptions.
Cover X by finitely many affine open subsets U i . Then clearlyΞ( X, Z ) = [ i Ξ( U i , Z | U i ) . For some index i , the set Ξ( U i , Z | U i ) contains a subsequence of ( t k ). Hence we may assume that X = U i ⊂ A N is affine. Taking the closure of X in P N and normalizing yields the followingassumption. Additional Assumption . The variety X is embedded as an open set X ⊂ X in a normalprojective variety X .We denote by Z the closure of Z considered as a locally closed subscheme of X . Step 2: Constructing compatible boundaries.
Let D be an effective Weil divisor on X such that K X − D is Cartier, and let D be its closure in X . By our assumptions, the non- Q -Cartier locusof D is finite. Hence by Theorem 7.1, there exists an ample Cartier divisor H on X and apositive integer m such that O X ( m ( H − D )) is globally generated on X for all m ∈ m · N .For such m , define∆ m := 1 m M, where M ∈ | m ( H − D ) | is a general element.Then by Theorem 4.4, the divisor ∆ m := ∆ m | X is an m -compatible boundary for the pair( X, Z ). Note that the Q -linear equivalence class of ∆ m ∼ Q H − D does not depend on m . Notealso that ∆ m need not be the closure of ∆ m , as ∆ m might have components contained in X \ X . Step 3: Global generation.
Let B be an ample Cartier divisor on X such that O X ( B ) ⊗ J Z isglobally generated. As we have remarked above, the numerical equivalence class of K X + ∆ m is independent of m . Thus we can find a very ample Cartier divisor L on X such that L − ( K X + ∆ m + t B ) is big and nef for all m ∈ m · N .Then also L − ( K X + ∆ m + t k B ) is big and nef for k ≥
1. Fix n ≥ dim X + 1. By Proposition 8.3, O X ( nL ) ⊗ J (( X, ∆ m ) , t k Z ) is globally generated for all m ∈ m · N , k ≥ Claim . The sheaf O X ( nL ) ⊗ J ( X, t k Z ) is globally generated on X for all k ≥ Proof of Claim 8.5.
For m ∈ m · N sufficiently divisible, we have J ( X, t k Z ) = J m ( X, t k Z ).Fix such an m . Then, since ∆ m is m -compatible, J ( X, t k Z ) | X = J ( X, t k Z ) = J (( X, ∆ m ) , t k Z ) = J (( X, ∆ m ) , t k Z ) | X . We also have O X ( nL ) ⊗ J (( X, ∆ m ) , t k Z ) ⊂ O X ( nL ) ⊗ J ( X, t k Z )by [dFH09, Rem. 5.3]. As the left-hand side sheaf is globally generated, so is the right-handside one wherever they agree. In particular, this is the case on the open subset X ⊂ X . (cid:3) Step 4: End of proof.
Consider the descending chain J ( X, t Z ) ⊃ J ( X, t Z ) ⊃ · · · . By Claim 8.5 and Lemma 8.2, the restriction of this chain to X stabilizes. This restriction isnothing but J ( X, t Z ) ⊃ J ( X, t Z ) ⊃ · · · . However, by the definition of jumping numbers we have J ( X, t k Z ) ) J ( X, t k +1 Z ) for all k .This is a contradiction, showing that Ξ( X, Z ) is discrete and thus finishing the proof. (cid:3) Generic numerical Q -factoriality In this section, we prove Theorem 1.7 from the introduction:
Theorem 9.1 (Generic numerical Q -factoriality) . Let X be a normal variety. Then there is aclosed subset W ⊂ X of codimension at least three such that X \ W is numerically Q -factorial.Proof. Let f : e X → X be a log resolution of X , with exceptional locus E = E + · · · + E k .Re-indexing, we may assume that for some number ℓ , we have codim X f ( E i ) = 2 for 1 ≤ i ≤ ℓ ,while codim X f ( E i ) ≥ ℓ < i ≤ k . We may remove the closed set S ℓ
Fix an index i . The morphism f i : E i → f ( E i ) is smooth of relativedimension one. Let E i g i / / B i h i / / f ( E i )be its Stein factorization. Then g i is smooth of relative dimension one with connected fibres,and h i is finite ´etale, say of degree n i .It is then clear that for any x ∈ f ( E i ), the fibre f − i ( x ) is a disjoint union of n i many smoothcurves C ( j ) i . Each of these curves is a (scheme-theoretic) fibre of g i , hence they all represent thesame class Γ i ∈ N ( E i /B i ) Q = N ( E i /f ( E i )) Q , independent of the point x . If γ i is the image ofΓ i under the natural map N ( E i /f ( E i )) Q → N ( e X/X ) Q , then [ C ( j ) i ] = γ i for all indices j . (cid:3) Claim 9.3 implies that N ( e X/X ) Q is spanned by γ , . . . , γ ℓ . In particular, the relative Picardnumber of f is ρ ( e X/X ) = ℓ . Hence the dimension of N ( e X/X ) Q is likewise ℓ . We now makeuse of the short exact sequence of Q -vector spaces0 −→ ℓ M i =1 Q · [ E i ] −→ N ( e X/X ) Q −→ Cl num ( X ) Q −→ num ( X ) Q = 0. This means that X is numerically Q -factorial. (cid:3) Discreteness in dimension three
Recall that Theorem 1.5 states the following:
Theorem 10.1 (Discreteness in dimension three) . Let ( X, Z ) be a pair, where X is a nor-mal threefold whose locus of non-rational singularities is zero-dimensional. Then Ξ( X, Z ) is adiscrete subset of R .Proof of Theorem 10.1. By Theorem 1.7 and the assumption, there is a finite subset W ⊂ X such that X \ W is numerically Q -factorial and has rational singularities. It then follows fromTheorem 3.9 that X \ W is Q -factorial, i.e. every Weil divisor on X is Q -Cartier outside afinite set of points. In particular, the non- Q -Cartier locus of K X is zero-dimensional. NowTheorem 1.4 applies to show that Ξ( X, Z ) is discrete. (cid:3)
Proof of Corollaries
Proof of Corollary 1.8.
By Theorem 9.1, there is an open subset U ⊂ X such that X \ U has codimension at least three in X , and U is numerically Q -factorial. In particular, K U isnumerically Q -Cartier. Note that U does not depend on Z . Let π : e U → U be a log resolutionof Z | U ⊂ U , so that J Z | U · O e U = O e U ( − D ) for some effective Cartier divisor D on e U . We maywrite K − e U/U = k X i =1 a i E i and D = k X i =1 r i E i for some k ∈ N and suitable a i ∈ Q , r i ∈ N and prime divisors E i on e U .By Theorem 4.5, for any t > J ( U, tZ | U ) = π ∗ O e U (cid:16) ⌈ K − e U/U − tD ⌉ (cid:17) = π ∗ O e U k X i =1 ⌈ a i − t · r i ⌉ E i ! . This implies that if t is a jumping number of ( U, Z | U ), then a i − t · r i is a negative integer forsome index 1 ≤ i ≤ k such that r i = 0. HenceΞ( U, Z | U ) ⊂ (cid:26) a i + mr i (cid:12)(cid:12)(cid:12)(cid:12) ∃ ≤ i ≤ k such that r i = 0, and m ∈ N (cid:27) . The set on the right-hand side is clearly discrete and consists of rational numbers only. SoΞ(
U, Z | U ) enjoys the same properties. This ends the proof. Proof of Corollary 1.10.
Immediate from Theorem 1.7 and [dFDTT15, Thm. 1].
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