aa r X i v : . [ m a t h . AG ] D ec THE MOVABLE CONE VIA INTERSECTIONS
BRIAN LEHMANN
Abstract.
We characterize the movable cone of divisors using inter-sections against curves on birational models. Introduction
Cones of divisors play an essential role in describing the birational geom-etry of a smooth complex projective variety X . A key feature of these conesis their interplay with cones of curves via duality statements. The dual ofthe nef cone and the pseudo-effective cone of divisors were determined by[Kle66] and [BDPP04] respectively. We consider the third cone commonlyused in birational geometry: the movable cone of divisors. Definition 1.1.
Let X be a smooth projective variety over C . The movablecone Mov ( X ) ⊂ N ( X ) is the closure of the cone generated by classes ofeffective Cartier divisors L such that the base locus of | L | has codimensionat least 2. We say a divisor is movable if its numerical class lies in Mov ( X ). Definition 1.2.
Let X be a smooth projective variety over C . We say thatan irreducible curve C on X is movable in codimension 1, or a mov -curve,if it deforms to cover a codimension 1 subset of X .It is natural to guess that a divisor L is movable if and only if it has non-negative intersection with every mov -curve. This is false, as demonstratedby [Pay06] Example 1. Nevertheless, Debarre and Lazarsfeld have askedwhether one can formulate a duality statement for movable divisors andmov -curves. This has been accomplished for toric varieties in [Pay06] andfor Mori Dream Spaces in [Cho10] by taking other birational models of X into account. Our main theorem proves an analogous statement for allsmooth varieties.Before stating this theorem, we need to analyze the behavior of the mov-able cone under birational transformations. Suppose that φ : Y → X isa birational map of smooth projective varieties and that L is a movabledivisor on X . It is possible that φ ∗ L is not movable – for example, some φ -exceptional centers could be contained in the base locus of L . The followingdefinition from [Nak04] allows us to quantify the loss in movability. The author is supported by NSF Award 1004363.
Definition 1.3.
Let X be a smooth projective variety over C and let L bea pseudo-effective R -divisor on X . Fix an ample divisor A on X . For anyprime divisor Γ on X we define σ Γ ( L ) = inf { mult Γ ( L ′ ) | L ′ ≥ L ′ ∼ R L + ǫA for some ǫ > } where ∼ R denotes R -linear equivalence. As demonstrated by [Nak04] III.1.5Lemma, σ Γ is independent of the choice of A .Suppose that E is an exceptional divisor for a birational map φ : Y → X . The R -divisor σ E ( φ ∗ L ) E represents the “extra contribution” from E to the non-movability of φ ∗ L . By subtracting these contributions, we canunderstand the geometry of the original divisor L . Definition 1.4.
Let X be a smooth projective variety over C and let L bea pseudo-effective R -divisor on X . Suppose that φ : Y → X is a birationalmap from a smooth variety Y . The movable transform of L on Y is definedto be φ − ( L ) := φ ∗ L − X E φ − exceptional σ E ( φ ∗ L ) E. Note that the movable transform is not linear and is only defined forpseudo-effective divisors. We can now state our main theorem.
Theorem 1.5.
Let X be a smooth projective variety over C and let L be apseudo-effective R -divisor. L is not movable if and only if there is a mov -curve C on X and a birational morphism φ : Y → X from a smooth variety Y such that φ − mov ( L ) · e C < where e C is the strict transform of a generic deformation of C . There does not seem to be an easy way to translate Theorem 1.5 into astatement involving only intersections on X . This is a symptom of the factthat the natural operation on movable divisors is the push-forward and notthe pull-back.The proof of Theorem 1.5 is accomplished by reinterpreting the orthogo-nality theorem of [BDPP04] and [BFJ09] using the techniques of [Leh10]. Example 1.6.
For surfaces Theorem 1.5 reduces to the usual duality of thenef and pseudo-effective cones.
Example 1.7.
Suppose that X is a smooth Mori dream space and L is an R -divisor on X . By running the L -MMP as in [HK00], we obtain a smallmodification φ : X X ′ , a morphism f : X → Z , and an ample R -divisor A on Z such that φ − ∗ L ≡ f ∗ A where φ − ∗ denotes the strict transform.Let W be a smooth variety admitting birational maps ψ : W → X and ψ ′ : W → X ′ . Using [Nak04] III.5.5 Proposition, one easily verifies that ψ − mov ( L ) ≡ ψ ′∗ ( φ − ∗ L ) . HE MOVABLE CONE VIA INTERSECTIONS 3
Thus Theorem 1.5 implies the statements of [Pay06] and [Cho10]: for asmooth toric variety or Mori Dream Space X , a divisor class is movableiff its strict transform class on every Q -factorial small modification X ′ hasnon-negative intersection with every mov -curve on X ′ . Example 1.8.
Suppose that X is a smooth projective variety with K X numerically trivial. [Cho10] explains how to apply techniques of the mini-mal model program to analyze Mov ( X ). Just as before, a divisor class ismovable if and only if its strict transform class on every Q -factorial smallmodification has non-negative intersection with every mov -curve. When X is hyperk¨ahler, [Huy03] and [Bou04] show that in fact it suffices to considersmall modifications that are also smooth hyperk¨ahler varieties.More generally, [Cho10] shows that small modifications can detect certainregions of Mov ( X ) by using the minimal model program.We will also prove a slightly stronger version of Theorem 1.5 that involvesthe non-nef locus B − ( L ) of L (which will be defined in Definition 2.2).Although the non-nef locus represents the “obstruction” to the nefness of L ,it is not true that B − ( L ) is covered by curves C with L · C <
0. However,Proposition 3.2 formulates a birational version of this negativity using themovable transform.Finally, we will use Proposition 3.2 to understand k -movability for k > k -movable cone of X to be the closure of the cone in N ( X )generated by effective Cartier divisors whose base locus has codimension atleast k −
1. We say that a divisor is k -movable if its numerical class lies inthe k -movable cone. Note that the 1-movable cone is just Mov ( X ).Debarre and Lazarsfeld have asked whether there is a duality betweenthe k -movable cone of divisors and the closure of the cone of irreduciblecurves that deform to cover a codimension k subset (for 0 < k < dim X ).Corollary 3.3 constructs a birational version of this duality. Again, thisgeneralizes results for toric varieties in [Pay06] and for Mori dream spacesin [Cho11]. 2. Background
Throughout X will denote a smooth projective variety over C . We use thenotations ∼ , ∼ Q , ∼ R , ≡ to denote respectively linear equivalence, Q -linearequivalence, R -linear equivalence, and numerical equivalence of R -divisors.The volume of an R -divisor L isvol X ( L ) = lim sup m →∞ h ( X, ⌊ mL ⌋ ) m dim X . Divisorial Zariski decomposition.
Let L be a pseudo-effective R -divisor on a smooth projective variety X . Recall that for a prime divisor Γon X we have defined σ Γ ( L ) = inf { mult Γ ( L ′ ) | L ′ ≥ L ′ ∼ R L + ǫA for some ǫ > } . BRIAN LEHMANN where A is any fixed ample divisor. [Nak04] III.1.11 Corollary shows thatthere are only finitely many prime divisors Γ on X with σ Γ ( L ) >
0, allowingus to make the following definition.
Definition 2.1 ([Nak04] III.1.16 Definition) . Let L be a pseudo-effective R -divisor on X . Define N σ ( L ) = X σ E ( L ) E P σ ( L ) = L − N σ ( L )The decomposition L = N σ ( L ) + P σ ( L ) is called the divisorial Zariski de-composition of L .Note that for a birational morphism φ : Y → X we have φ − mov ( L ) = P σ ( φ ∗ L ) + φ − ∗ N σ ( L ) where φ − ∗ denotes the strict transform. The divisorialZariski decomposition is closely related to the non-nef locus of L . Definition 2.2.
Let X be a smooth projective variety and let L be a pseudo-effective R -divisor on X . We define the R -stable base locus of L to be thesubset of X given by B R ( L ) = [ { Supp( L ′ ) | L ′ ≥ L ′ ∼ R L } . The non-nef locus of L is then defined to be B − ( L ) = [ A ample R -divisor B R ( L + A ) . The following proposition records the basic properties of the divisorialZariski decomposition.
Proposition 2.3 ([Nak04] III.1.14 Proposition, III.2.5 Lemma, V.1.3 The-orem) . Let X be a smooth projective variety and let L be a pseudo-effective R -divisor. (1) P σ ( L ) is a movable R -divisor. In particular for any prime divisor E the restriction P σ ( L ) | E is pseudo-effective. (2) If φ : Y → X is a birational morphism of smooth varieties and Γ is a prime divisor on Y that is not φ -exceptional, then σ Γ ( φ ∗ L ) = σ φ (Γ) ( L ) . (3) The union of the codimension components of B − ( L ) coincides with Supp( N σ ( L )) . Numerical dimension and orthogonality.
Given a pseudo-effectivedivisor L , the numerical dimension ν ( L ) of [Nak04] and [BDPP04] is a nu-merical measure of the “positivity” of L . There is also a restricted variant ν X | V ( L ) introduced in [BFJ09]; since the definition is somewhat involved,we will only refer to a special subcase using an alternate characterizationfrom [Leh10]. Definition 2.4.
Let L be a pseudo-effective divisor on X . Fix a primedivisor E on X and choose L ′ ≡ L whose support does not contain E . We HE MOVABLE CONE VIA INTERSECTIONS 5 say ν X | E ( L ) = 0 if lim inf φ vol e E ( P σ ( φ ∗ L ′ ) | e E ) = 0where φ : e X → X varies over all birational maps and e E denotes the stricttransform of E .The connection with geometry is given by the following version of theorthogonality theorem of [BDPP04] and [BFJ09]. Theorem 2.5 ([BFJ09], Theorem 4.15) . Let L be a pseudo-effective divisor.If a prime divisor E ⊂ X is contained in Supp( N σ ( L )) then ν X | E ( L ) = 0 .Proof. Fix an ample divisor A . Choose an ǫ > N σ ( L )) = Supp( N σ ( L + ǫA )) and apply [BFJ09] Theorem 4.15. Thecomparison between the numerical dimension of [BFJ09] and Definition 2.4is given by [Leh10] Theorem 7.1. (cid:3) Proof
Proof of Theorem 1.5:
Suppose that L is not movable. Denote by E a fixeddivisorial component of N σ ( L ).Fix a sufficiently general ample divisor A on X and choose ǫ small enoughso that E is a component of N σ ( L + ǫA ). Applying the orthogonality theoremof [BDPP04], we see there is a birational map φ : Y → X so that(1) e E is smooth.(2) vol e E ( P σ ( φ ∗ ( L + ǫA )) | e E ) < vol E ( A | E ) = vol e E ( φ ∗ A | e E ).(3) The strict transform of every component of N σ ( L ) is disjoint.There is a unique expression P σ ( φ ∗ ( L + ǫA )) = P σ ( φ ∗ L ) + φ ∗ A + α ( ǫ ) e E + F where e E is the strict transform of E , F is an effective divisor with F ≤ N σ ( φ ∗ L ) and the support of F does not contain E , and α ( ǫ ) is positive andgoes to 0 as ǫ goes to 0. By shrinking ǫ we may ensure that α ( ǫ ) < σ E ( L ).Condition (2) above, along with Lemma 3.1, show that the restriction( P σ ( φ ∗ L ) + α ( ǫ ) e E ) | e E is not pseudo-effective for any ǫ >
0. Since α ( ǫ ) <σ E ( L ), we also have that ( P σ ( φ ∗ L ) + σ E ( L ) e E ) | e E is not pseudo-effective. Asthe strict transform of components of N σ ( L ) are disjoint, the restriction of P σ ( φ ∗ L ) + φ − ∗ N σ ( L ) to e E is still not pseudo-effective.By [BDPP04, 0.2 Theorem] there is a curve e C whose deformations cover e E such that ( P σ ( φ ∗ L ) + φ − ∗ N σ ( L )) · e C < . Since e E is not φ -exceptional, C = φ ( e C ) is a mov -curve.Conversely, if L is movable, then φ − mov ( L ) = P σ ( φ ∗ L ) is also movable forevery φ . Thus every movable transform has non-negative intersection withthe strict transform of every mov -curve general in its family. (cid:3) BRIAN LEHMANN
Lemma 3.1.
Let X be a smooth projective variety and let L and L ′ bepseudo-effective divisors on X . Then vol X ( L + L ′ ) ≥ vol X ( L ) .Proof. We may assume L is big since otherwise the inequality is automatic.Then for any sufficiently small ǫ > X ( L + L ′ ) = vol X ((1 − ǫ ) L + ( ǫL + L ′ )) ≥ (1 − ǫ ) dim X vol X ( L )since ǫL + L ′ is big. (cid:3) We now give an alternate formulation of Theorem 1.5.
Proposition 3.2.
Let X be a smooth projective variety and let L be apseudo-effective R -divisor. Suppose that V is an irreducible subvariety of X contained in B − ( L ) and let ψ : X ′ → X be a smooth birational model re-solving the ideal sheaf of V . Then there is a birational morphism φ : Y → X ′ from a smooth variety Y and an irreducible curve e C on Y such that φ − mov ( ψ ∗ L ) · e C < and ψ ◦ φ ( e C ) deforms to cover V .Proof. Let E be the ψ -exceptional divisor dominating V . Since we have E ⊂ Supp( N σ ( ψ ∗ L )), we may argue as in the proof of Theorem 1.5 for ψ ∗ L and E to find a birational map φ such that φ − mov ( ψ ∗ L ) | e E is not pseudo-effective.[BDPP04, 2.4 Theorem] shows that there is some curve e C on e E with φ − mov ( ψ ∗ L ) · e C < e C deforms to cover e E and is not contractedby any morphism from e E to a variety of positive dimension. Choosing e C on e E to satisfy this stronger property, we obtain the statement of Proposition3.2. (cid:3) Proposition 3.2 shows that the non-nef locus is covered by L -negativecurves in a birational sense. Alternatively, one can rephrase this resultusing k -movability. Corollary 3.3.
Let X be a smooth projective variety and let L be a pseudo-effective R -divisor. Then L is not k -movable if and only if there is a bi-rational morphism ψ : X ′ → X from a smooth variety X ′ , a birationalmorphism φ : Y → X ′ from a smooth variety Y , and an irreducible curve e C on Y such that φ − mov ( ψ ∗ L ) · e C < and ψ ◦ φ ( e C ) deforms to cover a k -dimensional subset of V .Proof. To say that L is not k -movable is equivalent to saying that B − ( L )has a component of dimension at least k . Apply Proposition 3.2 to obtainthe forward implication. The converse is immediate. (cid:3) Remark 3.4.
It is unclear whether Corollary 3.3 is the best formulationpossible for the duality of k -movable divisors. For Mori Dream Spaces va-rieties and for 2 < k < dim X , [Pay06] Theorem 1 and [Cho11] Corollary 3 HE MOVABLE CONE VIA INTERSECTIONS 7 prove a slightly stronger statement. The essential difference is that one doesnot need to blow-up along top-dimensional components of B − ( L ). Moreprecisely, if L is not k -movable, one may find a Q -factorial small modifi-cation f : X X ′ that is regular at the generic point of a component V ⊂ B − ( L ) of codimension at most k and a family of curves covering thestrict transform of V with f ∗ L · C <
0. In contrast, Corollary 3.3 mayproduce a birational map that is not regular at any point of V . References [BDPP04] S. Boucksom, J.P. Demailly, M. Pˇaun, and T. Peternell,
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