Birational geometry of moduli spaces of sheaves and Bridgeland stability
aa r X i v : . [ m a t h . AG ] J un BIRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES ANDBRIDGELAND STABILITY
JACK HUIZENGA
Abstract.
Moduli spaces of sheaves and Hilbert schemes of points have experienced a recentresurgence in interest in the past several years, due largely to new techniques arising from Bridgelandstability conditions and derived category methods. In particular, classical questions about thebirational geometry of these spaces can be answered by using new tools such as the positivitylemma of Bayer and Macr`ı. In this article we first survey classical results on moduli spaces ofsheaves and their birational geometry. We then discuss the relationship between these classicalresults and the new techniques coming from Bridgeland stability, and discuss how cones of ampledivisors on these spaces can be computed with these new methods. This survey expands uponthe author’s talk at the 2015 Bootcamp in Algebraic Geometry preceding the 2015 AMS SummerResearch Institute on Algebraic Geometry at the University of Utah.
Contents
1. Introduction 12. Moduli spaces of sheaves 23. Properties of moduli spaces 94. Divisors and classical birational geometry 155. Bridgeland stability 246. Examples on P Introduction
The topic of vector bundles in algebraic geometry is a broad field with a rich history. In the70’s and 80’s, one of the main questions of interest was the study of low rank vector bundles onprojective spaces P r . One particularly challenging conjecture in this subject is the following. Conjecture 1.1 (Hartshorne [Har74]) . If r ≥ then any rank bundle on P r C splits as a directsum of line bundles. The Hartshorne conjecture is connected to the study of subvarieties of projective space of smallcodimension. In particular, the above statement implies that if X ⊂ P r is a codimension 2 smoothsubvariety and K X is a multiple of the hyperplane class then X is a complete intersection. Thus,early intersect in the study of vector bundles was born out of classical questions in projectivegeometry. Date : June 24, 2016.2010
Mathematics Subject Classification.
Primary: 14J60. Secondary: 14E30, 14J29, 14C05.
Key words and phrases.
Moduli spaces of sheaves, Hilbert schemes of points, ample cone, Bridgeland stability.During the preparation of this article the author was partially supported by a National Science FoundationMathematical Sciences Postdoctoral Research Fellowship.
Study of these types of questions led naturally to the study of moduli spaces of (semistable)vector bundles, parameterizing the isomorphism classes of (semistable) vector bundles with givennumerical invariants on a projective variety X (we will define semistable later—for now, view itas a necessary condition to get a good moduli space). As often happens in mathematics, thesespaces have become interesting in their own right, and their study has become an entire industry.Beginning in the 80’s and 90’s, and continuing to today, people have studied the basic questionsof the geometry of these spaces. Are they smooth? Irreducible? What do their singularities looklike? When is the moduli space nonempty? What are divisors on the moduli space? Especiallywhen X is a curve or surface, satisfactory answers to these questions can often be given. We willsurvey several foundational results of this type in § birational geometry of moduli spaces of various geometric objects. Loosely speaking, the goal of such a program is tounderstand alternate birational models, or compactifications , of a moduli space as themselves beingmoduli spaces for slightly different geometric objects. For instance, the Hassett-Keel program[HH13] studies alternate compactifications of the Deligne-Mumford compactification M g of themoduli space of stable curves. Different compactifications can be obtained by studying (potentiallyunstable) curves with different types of singularities. In addition to being interesting in their ownright, moduli spaces provide explicit examples of higher dimensional varieties which can frequentlybe understood in great detail. We survey the birational geometry of moduli spaces of sheaves froma classical viewpoint in § stabilitycondition [Bri07, Bri08]. Very roughly, there is a complex manifold Stab( X ), the stability manifold ,parameterizing stability conditions σ on X . There is a moduli space corresponding to each condition σ , and the stability manifold decomposes into chambers where the corresponding moduli spacedoes not change as σ varies in the chamber. For one of these chambers, the Gieseker chamber ,the corresponding moduli space is the ordinary moduli space of semistable sheaves. The modulispaces corresponding to other chambers often happen to be the alternate birational models of theordinary moduli space. In this way, the birational geometry of a moduli space of sheaves can beviewed in terms of a variation of the moduli problem. In § P in §
6. Finally, we close the paper in § Acknowledgements.
I would especially like to thank Izzet Coskun and Benjamin Schmidt formany discussions on Bridgeland stability and related topics. In addition, I would like to thankthe referee of this article for many valuable comments, as well as Barbara Bolognese, Yinbang Lin,Eric Riedl, Matthew Woolf, and Xialoei Zhao. Finally, I would like to thank the organizers of the2015 Bootcamp in Algebraic Geometry and the 2015 AMS Summer Research Institute on AlgebraicGeometry, as well as the funding organizations for these wonderful events.2.
Moduli spaces of sheaves
The definition of a Bridgeland stability condition is motivated by the classical theory of semistablesheaves. In this section we review the basics of the theory of moduli spaces of sheaves, particularlyfocusing on the case of a surface. The standard references for this material are Huybrechts-Lehn[HL10] and Le Potier [LeP97].2.1.
The moduli property.
First we state an idealized version of the moduli problem. Let X bea smooth projective variety with polarization H , and fix a set of discrete numerical invariants of a IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 3 coherent sheaf E on X . This can be accomplished by fixing the Hilbert polynomial P E ( m ) = χ ( E ⊗ O X ( mH ))of the sheaf.A family of sheaves on X over S is a (coherent) sheaf E on X × S which is S -flat. For a point s ∈ S , we write E s for the sheaf E| X ×{ s } . We say E is a family of semistable sheaves of Hilbertpolynomial P if E s is semistable with Hilbert polynomial P for each s ∈ S (see § moduli functor M ′ ( P ) : Sch o → Setby defining M ′ ( P )( S ) to be the set of isomorphism classes of families of semistable sheaves on X with Hilbert polynomial P . We will sometimes just write M ′ for the moduli functor when thepolynomial P is understood.Let p : X × S → S be the projection. If E is a family of semistable sheaves on X with Hilbertpolynomial P and L is a line bundle on S , then E ⊗ p ∗ L is again such a family. The sheaves E s and( E ⊗ p ∗ L ) | X ×{ s } parameterized by any point s ∈ S are isomorphic, although E and E ⊗ p ∗ L neednot be isomorphic. We call two families of sheaves on X equivalent if they differ by tensoring by aline bundle pulled back from the base, and define a refined moduli functor M by modding out bythis equivalence relation: M = M ′ / ∼ .The basic question is whether or not M can be represented by some nice object, e.g. by aprojective variety or a scheme. We recall the following definitions. Definition 2.1.
A functor F : Sch o → Set is represented by a scheme X if there is an isomorphismof functors F ∼ = Mor
Sch ( − , X ).A functor F : Sch o → Set is corepresented by a scheme X if there is a natural transformation α : F →
Mor
Sch ( − , X ) with the following universal property: if X ′ is a scheme and β : F →
Mor
Sch ( − , X ′ ) a natural transformation, then there is a unique morphism π : X → X ′ such that β is the composition of α with the transformation Mor Sch ( − , X ) → Mor
Sch ( − , X ′ ) induced by π . Remark 2.2.
Note that if F is represented by X then it is also corepresented by X .If F is represented by X , then F (Spec C ) ∼ = Mor Sch (Spec C , X ). That is, the points of X are inbijective correspondence with F (Spec C ). This need not be true if F is only corepresented by X .If F is corepresented by X , then X is unique up to a unique isomorphism.We now come to the basic definition of moduli space of sheaves. Definition 2.3.
A scheme M ( P ) is a moduli space of semistable sheaves with Hilbert polynomial P if M ( P ) corepresents M ( P ). It is a fine moduli space if it represents M ( P ).The most immediate consequence of M being a moduli space is the existence of the moduli map .Suppose E is a family of semistable sheaves on X parameterized by S . Then we obtain a morphism S → M which intuitively sends s ∈ S to the isomorphism class of the sheaf E s .In the special case when the base { s } is a point, a family over { s } is the isomorphism classof a single sheaf, and the moduli map { s } → M sends that class to a corresponding point. Thecompatibilities in the definition of a natural transformation ensure that in the case of a family E parameterized by a base S the image in M of a point s ∈ S depends only on the isomorphism classof the sheaf E s parameterized by s .In the ideal case where the moduli functor M has a fine moduli space, there is a universal sheaf U on X parameterized by M . We have an isomorphism M ( M ) ∼ = Mor Sch ( M, M )and the distinguished identity morphism M → M corresponds to a family U of sheaves parameter-ized by M (strictly speaking, U is only well-defined up to tensoring by a line bundle pulled back J. HUIZENGA from M ). This universal sheaf has the property that if E is a family of semistable sheaves on X parameterized by S and f : S → M is the moduli map, then E and (id X × f ) ∗ U are equivalent.2.2. Issues with a naive moduli functor.
In this subsection we give some examples to illustratethe importance of the as-yet-undefined semistability hypothesis in the definition of the modulifunctor. Let M n be the naive moduli functor of (flat) families of coherent sheaves with Hilbertpolynomial P on X , omitting any semistability hypothesis. We might hope that this functor is(co)representable by a scheme M n with some nice properties, such as the following.(1) M n is a scheme of finite type.(2) The points of M n are in bijective correspondence with isomorphism classes of coherentsheaves on X with Hilbert polynomial P .(3) A family of sheaves over a smooth punctured curve C − { pt } can be uniquely completed toa family of sheaves over C .However, unless some restrictions are imposed on the types of sheaves which are allowed, allthree hopes will fail. Properties (2) and (3) will also typically fail for semistable sheaves, but thisfailure occurs in a well-controlled way. Example 2.4.
Consider X = P , and let P = P O ⊕ P = 2 m + 2 be the Hilbert polynomial of therank 2 trivial bundle. Then for any n ≥
0, the bundle O P ( n ) ⊕ O P ( − n )also has Hilbert polynomial 2 m + 2, and h ( O P ( n ) ⊕ O P ( − n )) = n + 1. If there is a moduli scheme M n parameterizing all sheaves on P of Hilbert polynomial P , then M n cannot be of finite type.Indeed, the loci W n = { E : h ( E ) ≥ n } ⊂ M ( P )would then form an infinite decreasing chain of closed subschemes of M ( P ). Example 2.5.
Again consider X = P and P = 2 m + 2. Let S = Ext ( O P (1) , O P ( − C . For s ∈ S , let E s be the sheaf 0 → O P ( − → E s → O P (1) → s . One checks that if s = 0 then E s ∼ = O ⊕ P , but the extension issplit for s = 0. It follows that the moduli map S → M n must be constant, so O P ⊕ O P and O P (1) ⊕ O P ( −
1) are identified in the moduli space M n . Example 2.6.
Suppose X is a smooth variety and F is a coherent sheaf with dim Ext ( F, F ) ≥ S ⊂ Ext ( F, F ) be a 1-dimensional subspace, and for any s ∈ S let E s be the correspondingextension of F by F . Then if s, s ′ ∈ S are both not zero, we have E s ∼ = E s ′ = E = F ⊕ F. As in the previous example, we see that F ⊕ F and a nontrivial extension of F by F must beidentified in M n . Therefore any two extensions of F by F must also be identified in M n .If F is semistable, then Example 2.6 is an example of a nontrivial family of S -equivalent sheaves.A major theme of this survey is that S -equivalence is the main source of interesting birational mapsbetween moduli spaces of sheaves.2.3. Semistability.
Let E be a coherent sheaf on X . We say that E is pure of dimension d if thesupport of E is d -dimensional and every nonzero subsheaf of E has d -dimensional support. Remark 2.7.
If dim X = n , then E is pure of dimension n if and only if E is torsion-free. IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 5 If E is pure of dimension d then the Hilbert polynomial P E ( m ) has degree d . We write it in theform P E ( m ) = α d ( E ) m d d ! + · · · , and define the reduced Hilbert polynomial by p E ( m ) = P E ( m ) α d ( E ) . In the principal case of interest where d = n = dim X , Riemann-Roch gives α n ( E ) = r ( E ) H n where r ( E ) is the rank, and p E ( m ) = P E ( m ) r ( E ) H n . Definition 2.8.
A sheaf E is (semi)stable if it is pure of dimension d and any proper subsheaf F ⊂ E has p F < ( − ) p E , where polynomials are compared at large values. That is, p F < p E means that p F ( m ) < p E ( m ) forall m ≫ Theorem 2.9 ([HL10, Theorem 4.3.4]) . Let ( X, H ) be a smooth, polarized projective variety, andfix a Hilbert polynomial P . There is a projective moduli scheme of semistable sheaves on X ofHilbert polynomial P . While the definition of Gieseker stability is compact, it is frequently useful to use the Riemann-Roch theorem to make it more explicit. We spell this out in the case of a curve or surface. Wedefine the slope of a coherent sheaf E of positive rank on an n -dimensional variety by µ ( E ) = c ( E ) .H n − r ( E ) H n . Example 2.10 (Stability on a curve) . Suppose C is a smooth curve of genus g . The Riemann-Rochtheorem asserts that if E is a coherent sheaf on C then χ ( E ) = c ( E ) + r ( E )(1 − g ) . Polarizing C with H = p a point, we find P E ( m ) = χ ( E ( m )) = c ( E ( m )) + r ( E )(1 − g ) = r ( E ) m + ( c ( E ) + r ( E )(1 − g )) , and so p E ( m ) = m + c ( E ) r ( E ) + (1 − g ) . We conclude that if F ⊂ E then p F < ( − ) p E if and only if µ ( F ) < ( − ) µ ( E ). Example 2.11 (Stability on a surface) . Let X be a smooth surface with polarization H , and let E be a sheaf of positive rank. We define the total slope and discriminant by ν ( E ) = c ( E ) r ( E ) ∈ H ( X, Q ) and ∆( E ) = 12 ν ( E ) − ch ( E ) r ∈ Q . With this notation, the Riemann-Roch theorem takes the particularly simple form χ ( E ) = r ( E )( P ( ν ( E )) − ∆( E )) , J. HUIZENGA where P ( ν ) = χ ( O X ) + ν ( ν − K X ) (see [LeP97]). The total slope and discriminant behave wellwith respect to tensor products: if E and F are locally free then ν ( E ⊗ F ) = ν ( E ) + ν ( F )∆( E ⊗ F ) = ∆( E ) + ∆( F ) . Furthermore, ∆( L ) = 0 for a line bundle L ; equivalently, in the case of a line bundle the Riemann-Roch formula is χ ( L ) = P ( c ( L )). Then we compute χ ( E ( m )) = r ( E )( P ( ν ( E ) + mH ) − ∆( E ))= r ( E )( χ ( O X ) + 12 ( ν ( E ) + mH )( ν ( E ) + mH − K X ) − ∆( E ))= r ( E )( P ( ν ( E )) + 12 ( mH ) + mH. ( ν ( E ) + 12 K X ) − ∆( E ))= r ( E ) H m + r ( E ) H. ( ν ( E ) + 12 K X ) m + χ ( E ) , so p E ( m ) = 12 m + H. ( ν ( E ) + K X ) H m + χ ( E ) r ( E ) H Now if F ⊂ E , we compare the coefficients of p F and p E lexicographically to determine when p F < ( − ) p E . We see that p F < ( − ) p E if and only if either µ ( F ) < µ ( E ), or µ ( F ) = µ ( E ) and χ ( F ) r ( F ) H < ( − ) χ ( E ) r ( E ) H . Example 2.12 (Slope stability) . The notion of slope semistability has also been studied extensivelyand frequently arises in the study of Gieseker stability. We say that a torsion-free sheaf E on avariety X with polarization H is µ -(semi)stable if every subsheaf F ⊂ E of strictly smaller rankhas µ ( F ) < ( − ) µ ( E ). As we have seen in the curve and surface case, the coefficient of m n − in thereduced Hilbert polynomial p E ( m ) is just µ ( E ) up to adding a constant depending only on ( X, H ).This observation gives the following chain of implications: µ -stable ⇒ stable ⇒ semistable ⇒ µ -semistable . While Gieseker (semi)stability gives the best moduli theory and is therefore the most common towork with, it is often necessary to consider these various other forms of stability to study ordinarystability.
Example 2.13 (Elementary modifications) . As an example where µ -stability is useful, suppose X is a smooth surface and E is a torsion-free sheaf on X . Let p ∈ X be a point where X is locallyfree, and consider sheaves E ′ defined as kernels of maps E → O p , where O p is a skyscraper sheaf:0 → E ′ → E → O p → . Intuitively, E ′ is just E with an additional simple singularity imposed at p . Such a sheaf E ′ iscalled an elementary modification of E . We have µ ( E ) = µ ( E ′ ) and χ ( E ′ ) = χ ( E ) −
1, which makeselementary modifications a useful tool for studying sheaves by induction on the Euler characteristic.Suppose E satisfies one of the four types of stability discussed in Example 2.12. If E is µ -(semi)stable, then it follows that E ′ is µ -(semi)stable as well. Indeed, if F ⊂ E ′ with r ( F ) < r ( E ′ ),then also F ⊂ E , so µ ( F ) < ( − ) µ ( E ). But µ ( E ) = µ ( E ′ ), so µ ( F ) < ( − ) µ ( E ′ ) and E ′ is µ -(semi)stable.On the other hand, elementary modifications do not behave as well with respect to Gieseker(semi)stability. For example, take X = P . Then E = O P ⊕ O P is semistable, but any anyelementary modification E ′ of O P ⊕ O P at a point p ∈ P is isomorphic to I p ⊕ O P , where I p isthe ideal sheaf of p . Thus E ′ is not semistable. IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 7
It is also possible to give an example of a stable sheaf E such that some elementary modificationis not stable. Let p, q, r ∈ P be distinct points. Then ext ( I r , I { p,q } ) = 2. If E is any non-splitextension 0 → I { p,q } → E → I r → E is clearly µ -semistable. In fact, E is stable: the only stable sheaves F of rank 1 and slope0 with p F ≤ p E are O P and I s for s ∈ P a point, but Hom( I s , E ) = 0 for any s ∈ P since thesequence is not split. Now if s ∈ P is a point distinct from p, q, r and E → O s is a map such thatthe composition I { p,q } → E → O s is zero, then the corresponding elementary modification0 → E ′ → E → O s → I { p,q } ⊂ E ′ . We have p I { p,q } = p E ′ , so E ′ is strictly semistable. Example 2.14 (Chern classes) . Let K ( X ) be the Grothendieck group of X , generated by classes[ E ] of locally free sheaves, modulo relations [ E ] = [ F ] + [ G ] for every exact sequence0 → F → E → G → . There is a symmetric bilinear
Euler pairing on K ( X ) such that ([ E ] , [ F ]) = χ ( E ⊗ F ) whenever E, F are locally free sheaves. The numerical Grothendieck group K num ( X ) is the quotient of K ( X )by the kernel of the Euler pairing, so that the Euler pairing descends to a nondegenerate pairingon K num ( X ).It is often preferable to fix the Chern classes of a sheaf instead of the Hilbert polynomial. Thisis accomplished by fixing a class v ∈ K num ( X ). Any class v determines a Hilbert polynomial P v = ( v , [ O X ( m )]). In general, a polynomial P can arise as the Hilbert polynomial of severalclasses v ∈ K num ( X ). In any family E of sheaves parameterized by a connected base S the sheaves E s all have the same class in K num ( X ). Therefore, the moduli space M ( P ) splits into connectedcomponents corresponding to the different vectors v with P v = P . We write M ( v ) for the connectedcomponent of M ( P ) corresponding to v .2.4. Filtrations.
In addition to controlling subsheaves, stability also restricts the types of mapsthat can occur between sheaves.
Proposition 2.15. (1) (See-saw property) In any exact sequence of pure sheaves → F → E → Q → of the same dimension d , we have p F < ( − ) p E if and only if p E < ( − ) p Q .(2) If F, E are semistable sheaves of the same dimension d and p F > p E , then Hom(
F, E ) = 0 .(3) If
F, E are stable sheaves and p F = p E , then any nonzero homomorphism F → E is anisomorphism.(4) Stable sheaves E are simple: Hom( E, E ) = C .Proof. (1) We have P E = P F + P Q , so α d ( E ) = α d ( F ) + α d ( Q ) and p E = P E α d ( E ) = P F + P Q α d ( E ) = α d ( F ) p F + α d ( Q ) p Q α d ( E ) . Thus p E is a weighted mean of p F and p Q , and the result follows.(2) Let f : F → E be a homomorphism, and put C = Im f and K = ker f . Then C is pure ofdimension d since E is, and K is pure of dimension d since F is. By (1) and the semistability of F , we have p C ≥ p F > p E . This contradicts the semistability of E since C ⊂ E .(3) Since p F = p E , F and E have the same dimension. With the same notation as in (2), weinstead find p C ≥ p F = p E , and the stability of E gives p C = p E and C = E . If f is not anisomorphism then p K = p F , contradicting stability of F . Therefore f is an isomorphism. J. HUIZENGA (4) Suppose f : E → E is any homomorphism. Pick some point x ∈ X . The linear transformation f x : E x → E x has an eigenvalue λ ∈ C . Then f − λ id E is not an isomorphism, so it must be zero.Therefore f = λ id E . (cid:3) Harder-Narasimhan filtrations enable us to study arbitrary pure sheaves in terms of semistablesheaves. Proposition 2.15 is one of the important ingredients in the proof of the next theorem.
Theorem and Definition 2.16 ([HL10]) . Let E be a pure sheaf of dimension d . Then there is aunique filtration E ⊂ E ⊂ · · · ⊂ E ℓ = E called the Harder-Narasimhan filtration such that the quotients gr i = E i /E i − are semistable ofdimension d and reduced Hilbert polynomial p i , where p > p > · · · > p ℓ . In order to construct (semi)stable sheaves it is frequently necessary to also work with sheavesthat are not semistable. The next example outlines one method for constructing semistable vectorbundles. This general method was used by Dr´ezet and Le Potier to classify the possible Hilbertpolynomials of semistable sheaves on P [LeP97, DLP85]. Example 2.17.
Let (
X, H ) be a smooth polarized projective variety. Suppose A and B arevector bundles on X and that the sheaf H om ( A, B ) is globally generated. For simplicity assume r ( B ) − r ( A ) ≥ dim X . Let S ⊂ Hom(
A, B ) be the open subset parameterizing injective sheaf maps;this is set is nonempty since H om ( A, B ) is globally generated. Consider the family E of sheaves on X parameterized by S where the sheaf E s parameterized by s ∈ S is the cokernel0 → A s → B → E s → . Then for general s ∈ S , the sheaf E s is a vector bundle [Hui16, Proposition 2.6] with Hilbertpolynomial P := P B − P A . In other words, restricting to a dense open subset S ′ ⊂ S , we get afamily of locally free sheaves parameterized by S ′ .Next, semistability is an open condition in families. Thus there is a (possibly empty) opensubset S ′′ ⊂ S ′ parameterizing semistable sheaves. Let ℓ > P , . . . , P ℓ such that P + · · · + P ℓ = P and the corresponding reduced polynomials p , . . . , p ℓ have p > · · · > p ℓ . Then there is a locally closed subset S P ,...,P ℓ ⊂ S ′ parameterizing sheaves with aHarder-Narasimhan filtration of length ℓ with factors of Hilbert polynomial P , . . . , P ℓ . Such lociare called Shatz strata in the base S ′ of the family.Finally, to show that S ′′ is nonempty, it suffices to show that the Shatz stratum S P correspondingto semistable sheaves is dense. One approach to this problem is to show that every Shatz stratum S P ,...,P ℓ with ℓ ≥ P .Just as the Harder-Narasimhan filtration allows us to use semistable sheaves to build up arbitrarypure sheaves, Jordan-H¨older filtrations decompose semistable sheaves in terms of stable sheaves. Theorem and Definition 2.18. [HL10]
Let E be a semistable sheaf of dimension d and reducedHilbert polynomial p . There is a filtration E ⊂ E ⊂ · · · ⊂ E ℓ = E called the Jordan-H¨older filtration such that the quotients gr i = E i /E i − are stable with reducedHilbert polynomial p . The filtration is not necessarily unique, but the list of stable factors is uniqueup to reordering. We can now precisely state the critical definition of S -equivalence. IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 9
Definition 2.19.
Semistable sheaves E and F are S -equivalent if they have the same list of Jordan-H¨older factors.We have already seen an example of an S -equivalent family of semistable sheaves in Example2.6, and we observed that all the parameterized sheaves must be represented by the same point inthe moduli space. In fact, the converse is also true, as the next theorem shows. Theorem 2.20.
Two semistable sheaves
E, F with Hilbert polynomial P are represented by thesame point in M ( P ) if and only if they are S -equivalent. Thus, the points of M ( P ) are in bijectivecorrespondence with S -equivalence classes of semistable sheaves with Hilbert polynomial P .In particular, if there are strictly semistable sheaves of Hilbert polynomial P , then M ( P ) is nota fine moduli space. Remark 2.21.
The question of when the open subset M s ( P ) parameterizing stable sheaves is afine moduli space for the moduli functor M s ( P ) of stable families is somewhat delicate; in this casethe points of M s ( P ) are in bijective correspondence with the isomorphism classes of stable sheaves,but there still need not be a universal family.One positive result in this direction is the following. Let v ∈ K num ( X ) be the numerical class ofa stable sheaf with Hilbert polynomial P (see Example 2.14). Consider the set of integers of theform ( v , [ F ]), where F is a coherent sheaf and ( − , − ) is the Euler pairing. If their greatest commondivisor is 1, then M s ( v ) carries a universal family. (Note that the number-theoretic requirementalso guarantees that there are no semistable sheaves of class v .) See [HL10, § Properties of moduli spaces
To study the birational geometry of moduli spaces of sheaves in depth it is typically necessaryto have some kind of control over the geometric properties of the space. For example, is the modulispace nonempty? Smooth? Irreducible? What are the divisor classes on the moduli space?Our original setup of studying a smooth projective polarized variety (
X, H ) of any dimensionis too general to get satisfactory answers to these questions. We first mention some results onsmoothness which hold with a good deal of generality, and then turn to more specific cases withfar more precise results.3.1.
Tangent spaces, smoothness, and dimension.
Let (
X, H ) be a smooth polarized variety,and let v ∈ K num ( X ). The tangent space to the moduli space M = M ( v ) is typically only well-behaved at points E ∈ M parameterizing stable sheaves E , due to the identification of S -equivalenceclasses of sheaves in M .Let D = Spec C [ ε ] / ( ε ) be the dual numbers, and let E be a stable sheaf. Then the tangentspace to M is the subset of Mor( D, M ) corresponding to maps sending the closed point of D tothe point E . By the moduli property, such a map corresponds to a sheaf E on X × D , flat over D ,such that E = E .Deformation theory identifies the set of sheaves E as above with the vector space Ext ( E, E ),so there is a natural isomorphism T E M ∼ = Ext ( E, E ). The obstruction to extending a first-orderdeformation is a class Ext ( E, E ), and if Ext ( E, E ) = 0 then M is smooth at E .For some varieties X it is helpful to improve the previous statement slightly, since the vanishingExt ( E, E ) = 0 can be rare, for example if K X is trivial. If E is a vector bundle, lettr : E nd ( E ) → O X be the trace map , acting fiberwise as the ordinary trace of an endomorphism. Then H i ( E nd ( E )) ∼ = Ext i ( E, E ) , so there are induced maps on cohomologytr i : Ext i ( E, E ) → H i ( O X ) . We write Ext i ( E, E ) ⊂ Ext i ( E, E ) for ker tr i , the subspace of traceless extensions . The subspacesExt i ( E, E ) can also be defined if E is just a coherent sheaf, but the construction is more delicateand we omit it. Theorem 3.1.
The tangent space to M at a stable sheaf E is canonically isomorphic to Ext ( E, E ) ,the space of first order deformations of E . If Ext ( E, E ) = 0 , then M is smooth at E of dimension ext ( E, E ) . We now examine several consequences of Theorem 3.1 in the case of curves and surfaces.
Example 3.2.
Suppose X is a smooth curve of genus g , and let M ( r, d ) be the moduli space ofsemistable sheaves of rank r and degree d on X . Then the vanishing Ext ( E, E ) = 0 holds for anysheaf E , so the moduli space M ( P ) is smooth at every point parameterizing a stable sheaf E . Sincestable sheaves are simple, the dimension at such a sheaf isext ( E, E ) = 1 − χ ( E, E ) = r ( g −
1) + 1 . Example 3.3.
Let (
X, H ) be a smooth variety, and let v = [ O X ] ∈ K num ( X ) be the numerical classof O X . The moduli space M ( v ) parameterizes line bundles numerically equivalent to O X ; it is theconnected component Pic X of the Picard scheme Pic X which contains O X . For any line bundle L ∈ M ( v ), we have E nd ( L ) ∼ = O X and the trace map E nd ( L ) → O X is an isomorphism. ThusExt ( L, L ) = 0, and M ( v ) is smooth of dimension ext ( L, L ) = h ( O X ) =: q ( X ), the irregularity of X . Example 3.4.
Suppose (
X, H ) is a smooth surface and E ∈ M s ( P ) is a stable vector bundle. Thesheaf map tr : E nd ( E ) → O X is surjective, so the induced map tr : Ext ( E, E ) → H ( O X ) is surjective since X is a surface.Therefore ext ( E, E ) = 0 if and only if ext ( E, E ) = h ( O X ). We conclude that if ext ( E, E ) = 0then M ( P ) is smooth at E of local dimensiondim E M ( P ) = ext ( E, E ) = 1 − χ ( E, E ) + ext ( E, E )= 1 − χ ( E, E ) + h ( O X )= 2 r ∆( E ) + χ ( O X )(1 − r ) + q ( X ) . Example 3.5.
If (
X, H ) is a smooth surface such that
H.K X <
0, then the vanishing Ext ( E, E ) =0 is automatic. Indeed, by Serre duality,Ext ( E, E ) ∼ = Hom( E, E ⊗ K X ) ∗ . Then µ ( E ⊗ K X ) = µ ( E ) + µ ( K X ) = µ ( E ) + H.K X < µ ( E ) , so Hom( E, E ⊗ K X ) = 0 by Proposition 2.15.The assumption H.K X < X is a del Pezzo or Hirzebruch surface.Thus the moduli spaces M ( v ) for these surfaces are smooth at points corresponding to stablesheaves. Example 3.6.
If (
X, H ) is a smooth surface and K X is trivial (e.g. X is a K3 or abelian surface),then the weaker vanishing Ext ( E, E ) = 0 holds. The trace map tr : H ( E nd ( E )) → H ( O X ) isSerre dual to an isomorphism H ( O X ) → H ( E nd ( E )) = Hom( E, E ) , so tr is an isomorphism and Ext ( E, E ) = 0. IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 11
Existence and irreducibility.
What are the possible numerical invariants v ∈ K num ( X ) ofa semistable sheaf on X ? When the moduli space is nonempty, is it irreducible? As usual, the caseof curves is simplest.3.2.1. Existence and irreducibility for curves.
Let M = M ( r, d ) be the moduli space of semistablesheaves of rank r and degree d on a smooth curve X of genus g ≥
1. Then M is nonempty andirreducible, and unless X is an elliptic curve and r, d are not coprime then the stable sheaves aredense in M . To show M ( r, d ) is nonempty one can follow the basic outline of Example 2.17. Formore details, see [LeP97, Chapter 8].Irreducibility of M ( r, d ) can be proved roughly as follows. We may as well assume r ≥ d ≥ rg by tensoring by a sufficiently ample line bundle. Let L denote a line bundle of degree d on X , and consider extensions of the form0 → O r − X → E → L → . As L and the extension class vary, we obtain a family of sheaves E parameterized by a vector bundle S over the component Pic d ( X ) of the Picard group.On the other hand, by the choice of d , any semistable E ∈ M ( r, d ) is generated by its globalsections. A general collection of r − E will be linearly independent at every x ∈ X , sothat the quotient of the corresponding inclusion O r − X → E is a line bundle. Thus every semistable E fits into an exact sequence as above. The (irreducible) open subset of S parameterizing semistablesheaves therefore maps onto M ( r, d ), and the moduli space is irreducible.3.2.2. Existence for surfaces.
For surfaces the existence question is quite subtle. The first generalresult in this direction is the Bogomolov inequality.
Theorem 3.7 (Bogomolov inequality) . If ( X, H ) is a smooth surface and E is a µ H -semistablesheaf on X then ∆( E ) ≥ . Remark 3.8.
Note that the discriminant ∆( E ) is independent of the particular polarization H ,so the inequality holds for any sheaf which is slope-semistable with respect to some choice ofpolarization.Recall that line bundles L have ∆( L ) = 0, so in a sense the Bogomolov inequality is sharp.However, there are certainly Chern characters v with ∆( v ) ≥ v . A refined Bogomolov inequality should bound ∆( E ) from below in terms ofthe other numerical invariants of E . Solutions to the existence problem for semistable sheaves ona surface can often be viewed as such improvements of the Bogomolov inequality.3.2.3. Existence for P . On P , the classification of Chern characters v such that M ( v ) is nonemptyhas been carried out by Dr´ezet and Le Potier [DLP85, LeP97]. A (semi)exceptional bundle is arigid (semi)stable bundle, i.e. a (semi)stable bundle with Ext ( E, E ) = 0. Examples of exceptionalbundles include line bundles, the tangent bundle T P , and infinitely more examples obtained by aprocess of mutation . The dimension formula for a moduli space of sheaves on P readsdim M ( v ) = r (2∆ −
1) + 1 , so an exceptional bundle has discriminant ∆ = − r < . The dimension formula suggests animmediate refinement of the Bogomolov inequality: if E is a non-exceptional stable bundle, then∆( E ) ≥ .However, exceptional bundles can provide even stronger Bogomolov inequalities for non-exceptional bundles. For example, suppose E is a semistable sheaf with 0 < µ ( E ) <
1. ThenHom( E, O X ) = 0 and Ext ( E, O X ) ∼ = Hom( O X , E ⊗ K X ) ∗ = 0 Μ D Figure 1.
The curve δ ( µ ) occurring in the classification of stable bundles on P .If ( r, µ, ∆) are the invariants of an integral Chern character, then there is a non-exceptional stable bundle E with these invariants if and only if ∆ ≥ δ ( µ ). Theinvariants of the first several exceptional bundles are also displayed.by semistability and Proposition 2.15. Thus χ ( E, O X ) ≤
0. By the Riemann-Roch theorem, thisinequality is equivalent to the inequality∆( E ) ≥ P ( − µ ( E ))where P ( x ) = x + x + 1; this inequality is stronger than the ordinary Bogomolov inequality forany µ ( E ) ∈ (0 , P into account in a similar manner, one defines afunction δ : R → R with the property that any non-semiexceptional semistable bundle E satisfies∆( E ) ≥ δ ( µ ( E )). The graph of δ is Figure 1. Dr´ezet and Le Potier prove the converse theorem:exceptional bundles are the only obstruction to the existence of stable bundles with given numericalinvariants. Theorem 3.9.
Let v be an integral Chern character on P . There is a non-exceptional stablevector bundle on P with Chern character v if and only if ∆( v ) ≥ δ ( µ ( v )) . The method of proof follows the outline indicated in Example 2.17.3.2.4.
Existence for other rational surfaces.
In the case of X = P × P , Rudakov [Rud89, Rud94]gives a solution to the existence problem that is similar to the Dr´ezet-Le Potier result for P .However, the geometry of exceptional bundles is more complicated than for P , and as a result theclassification is somewhat less explicity. To our knowledge a satisfactory answer to the existenceproblem has not yet been given for a Del Pezzo or Hirzebruch surface. IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 13
Irreducibility for rational surfaces.
For many rational surfaces X it is known that the modulispace M H ( v ) is irreducible. One common argument is to introduce a mild relaxation of the notionof semistability and show that the stack parameterizing such objects is irreducible and contains thesemistable sheaves as an open dense substack.For example, Hirschowitz and Laszlo [HiL93] introduce the notion of a prioritary sheaf on P .A torsion-free coherent sheaf E on P is prioritary ifExt ( E, E ( − . By Serre duality, any torsion-free sheaf whose Harder-Narasimhan factors have slopes that are “nottoo far apart” will be prioritary, so it is very easy to construct prioritary sheaves. For example,semistable sheaves are prioritary, and sheaves of the form O P ( a ) ⊕ k ⊕ O P ( a + 1) ⊕ l are prioritary.The class of prioritary sheaves is also closed under elementary modifications, which makes it possibleto study them by induction on the Euler characteristic as in Example 2.13.The Artin stack P ( v ) of prioritary sheaves with invariants v is smooth, essentially becauseExt ( E, E ) = 0 for any prioritary sheaf. There is a unique prioritary sheaf of a given slope andrank with minimal discriminant, given by a sheaf of the form O P ( a ) ⊕ k ⊕ O P ( a + 1) ⊕ l with theintegers a, k, l chosen appropriately. Hirschowitz and Laszlo show that any connected componentof P ( v ) contains a sheaf which is an elementary modification of another sheaf. By inductionon the Euler characteristic, they conclude that P ( v ) is connected, and therefore irreducible. Sincesemistability is an open property, the stack M ( v ) of semistable sheaves is an open substack of P ( v )and therefore dense and irreducible if it is nonempty. Thus the coarse space M ( v ) is irreducible aswell.Walter [Wal93] gives another argument establishing the irreducibility of the moduli spaces M H ( v )on a Hirzebruch surface whenever they are nonempty. The arguments make heavy use of theruling, and study the stack of sheaves which are prioritary with respect to the fiber class. In moregenerality, he also studies the question of irreducibility on a geometrically ruled surface, at leastunder a condition on the polarization which ensures that semistable sheaves are prioritary withrespect to the fiber class.3.2.6. Existence and irreducibility for K3’s.
By work of Yoshioka, Mukai, and others, the existenceproblem has a particularly simple and beautiful solution when (
X, H ) is a smooth K3 surface (see[Yos01], or [BM14a, BM14b] for a simple treatment). Define the
Mukai pairing h− , −i on K num ( X )by h v , w i = − χ ( v , w ); we can make sense of this formula by the same method as in Example2.14. Since X is a K3 surface, K X is trivial and the Mukai pairing is symmetric by Serre duality.By Example 3.4, if there is a stable sheaf E with invariants v then the moduli space M ( v ) hasdimension 2 + h v , v i at E . If E is a stable sheaf of class v with h v , v i = −
2, then E is called spherical and the moduli space M H ( v ) is a single reduced point.A class v ∈ K num ( X ) is called primitive if it is not a multiple of another class. If the polarization H of X is chosen suitably generically, then v being primitive ensures that there are no strictlysemistable sheaves of class v . Thus, for a generic polarization, a necessary condition for theexistence of a stable sheaf is that h v , v i ≥ − Definition 3.10.
A primitive class v = ( r, c, d ) ∈ K num ( X ) is called positive if h v , v i ≥ − r >
0, or(2) r = 0 and c is effective, or(3) r = 0, c = 0, and d > sheaf of class v , so they are very mild. Theorem 3.11.
Let ( X, H ) be a smooth K3 surface. Let v ∈ K num ( X ) , and write v = m v , where v is primitive and m is a positive integer.If v is positive, then the moduli space M H ( v ) is nonempty. If furthermore m = 1 and thepolarization H is sufficiently generic, then M H ( v ) is a smooth, irreducible, holomorphic symplecticvariety.If M H ( v ) is nonempty and the polarization is sufficiently generic, then v is positive. The Mukai pairing can be made particularly simple from a computational standpoint by studyingit in terms of a different coordinate system. Let H ∗ alg ( X ) = H ( X, Z ) ⊕ NS( X ) ⊕ H ( X, Z ) . Then there is an isomorphism v : K num ( X ) → H ∗ alg ( X, Z ) defined by v ( v ) = v · p td( X ) . The vector v ( v ) is called a Mukai vector . The Todd class td( X ) ∈ H ∗ alg ( X ) is (1 , , p td( X ) = (1 , , v ( v ) = (ch ( v ) , ch ( v ) , ch ( v ) + ch ( v )) = ( r, c , r + c − c ) . Suppose v , w ∈ K num ( X ) have Mukai vectors v ( v ) = ( r, c, s ), v ( w ) = ( r ′ , c ′ , s ′ ). Since p td( X ) isself-dual, the Hirzebruch-Riemann-Roch theorem gives h v , w i = − χ ( v , w ) = − Z X v ∗ · w · td( X ) = − Z X ( r, − c, s ) · ( r ′ , c ′ , s ′ ) = cc ′ − rs ′ − r ′ s. It is worth pointing out that Theorem 3.11 can also be stated as a strong Bogomolov inequality,as in the Dr´ezet-Le Potier result for P . Let v be a primitive vector which is the vector of acoherent sheaf. The irregularity of X is q ( X ) = 0 and χ ( O X ) = 2, so as in Example 3.4 h v , v i = 2 r ∆( v ) + 2(1 − r ) − r (∆( v ) − . Therefore, v is positive and non-spherical if and only if ∆( v ) ≥ General surfaces.
On an arbitrary smooth surface (
X, H ) the basic geometry of the modulispace is less understood. To obtain good results, it is necessary to impose some kind of additionalhypotheses on the Chern character v .For one possibility, we can take v to be the character of an ideal sheaf I Z of a zero-dimensionalscheme Z ⊂ X of length n . Then the moduli space of sheaves of class v with determinant O X isthe Hilbert scheme of n points on X , written X [ n ] . It parameterizes ideal sheaves of subschemes Z ⊂ X of length n . Remark 3.12.
Note that any rank 1 torsion-free sheaf E with determinant O X admits an inclusion E → E ∗∗ := det E = O X , so that E is actually an ideal sheaf. Unless X has irregularity q ( X ) = 0,the Hilbert scheme X [ n ] and moduli space M ( v ) will differ, since the latter space also containssheaves of the form L ⊗ I Z , where L is a line bundle numerically equivalent to O X . In fact, M ( v ) ∼ = X [ n ] × Pic ( X ).Classical results of Fogarty show that Hilbert schemes of points on a surface are very well-behaved. Theorem 3.13 ([Fog68]) . The Hilbert scheme of points X [ n ] on a smooth surface X is smooth andirreducible. It is a fine moduli space, and carries a universal ideal sheaf. At the other extreme, if the rank is arbitrary then there are
O’Grady-type results which showthat the moduli space has many good properties if we require the discriminant of our sheaves tobe sufficiently large.
IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 15
Theorem 3.14 ([HL10, O’G96]) . There is a constant C depending on X, H , and r , such that if v has rank r and ∆( v ) ≥ C then the moduli space M H ( v ) is nonempty, irreducible, and normal.The µ -stable sheaves E such that ext ( E, E ) = 0 are dense in M H ( v ) , so M H ( v ) has the expecteddimension dim M H ( v ) = 2 r ∆( E ) + χ ( O X )(1 − r ) + q ( X ) . Divisors and classical birational geometry
In this section we introduce some of the primary objects of study in the birational geometry ofvarieties. We then study some simple examples of the birational geometry of moduli spaces fromthe classical point of view.4.1.
Cones of divisors.
Let X be a normal projective variety. Recall that X is factorial if everyWeil divisor on X is Cartier, and Q -factorial if every Weil divisor has a multiple that is Cartier.To make the discussion in this section easier we will assume that X is Q -factorial. This means thatdescribing a codimension 1 locus on X determines the class of a Q -Cartier divisor. Definition 4.1.
Two Cartier divisors D , D (or Q - or R -Cartier divisors) are numerically equiv-alent , written D ≡ D , if D · C = D · C for every curve C ⊂ X . The Neron-Severi space N ( X )is the real vector space Pic( X ) ⊗ R / ≡ .4.1.1. Ample and nef cones.
The first object of study in birational geometry is the ample cone
Amp( X ) of X . Roughly speaking, the ample cone parameterizes the various projective embeddingsof X . A Cartier divisor D on X is ample if the map to projective space determined by O X ( mD )is an embedding for sufficiently large m . The Nakai-Moishezon criterion for ampleness says that D is ample if and only if D dim V .V > V ⊂ X . In particular, ampleness onlydepends on the numerical equivalence class of D . A positive linear combination of ample divisorsis also ample, so it is natural to consider the cone spanned by ample classes. Definition 4.2.
The ample cone
Amp( X ) ⊂ N ( X ) is the open convex cone spanned by thenumerical classes of ample Cartier divisors.An R -Cartier divisor D is ample if its numerical class is in the ample cone.From a practical standpoint it is often easier to work with nef (i.e. numerically effective ) divisorsinstead of ample divisors. We say that a Cartier divisor D is nef if D.C ≥ C ⊂ X .This is clearly a numerical condition, so nefness extends easily to R -divisors and they span a coneNef( X ), the nef cone of X . By Kleiman’s theorem, the problems of studying ample or nef conesare essentially equivalent. Theorem 4.3 ([Deb01, Theorem 1.27]) . The nef cone is the closure of the ample cone, and theample cone is the interior of the nef cone:
Nef( X ) = Amp( X ) and Amp( X ) = Nef( X ) ◦ . Nef divisors are particularly important in birational geometry because they record the behaviorof the simplest nontrivial morphisms to other projective varieties, as the next example shows.
Example 4.4.
Suppose f : X → Y is any morphism of projective varieties. Let L be a very ampleline bundle on Y , and consider the line bundle f ∗ L . If C ⊂ X is any irreducible curve, we can findan effective divisor D ⊂ Y representing L such that the image of C is not contained entirely in D .This implies C. ( f ∗ L ) ≥
0, so f ∗ L is nef. Note that if f contracts some curve C ⊂ X to a point,then C. ( f ∗ L ) = 0, so f ∗ L is on the boundary of the nef cone.As a partial converse, suppose D is a nef divisor on X such that the linear series | mD | is basepoint free for some m >
0; such a divisor class is called semiample . Then for sufficiently large anddivisible m , the image of the map φ | mD | : X → | mD | ∗ is a projective variety Y m carrying an ample line bundle L such that φ ∗| mD | L = O X ( mD ) . See [Laz04, Theorem 2.1.27] for details and a moreprecise statement.
Example 4.5.
Classically, to compute the nef (and hence ample) cone of a variety X one typi-cally first constructs a subcone Λ ⊂ Nef( X ) by finding divisors D on the boundary arising frominteresting contractions X → Y as in Example 4.4. One then dually constructs interesting curves C on X to span a cone Nef( X ) ⊂ Λ ′ given as the divisors intersecting the curves nonnegatively. Ifenough divisors and curves are constructed so that Λ = Λ ′ , then they equal the nef cone.One of the main features of the positivity lemma of Bayer and Macr`ı will be that it produces nefdivisors on moduli spaces of sheaves M without having to worry about finding a map M → Y toa projective variety giving rise to the divisor. A priori these nef divisors may not be semiample orhave sections at all, so it may or may not be possible to construct these divisors and prove theirnefness via more classical constructions. See § Example 4.6.
For an easy example of the procedure in Example 4.5, consider the blowup X =Bl p P of P at a point p . Then Pic X ∼ = Z H ⊕ Z E , where H is the pullback of a line under themap π : X → P and E is the exceptional divisor. The Neron-Severi space N ( X ) is the two-dimensional real vector space spanned by H and E . Convex cones in N ( X ) are spanned by twoextremal classes.Since π contracts E , the class H is an extremal nef divisor. We also have a fibration f : X → P ,where the fibers are the proper transforms of lines through p . The pullback of a point in P is ofclass H − E , so H − E is an extremal nef divisor. Therefore Nef( X ) is spanned by H and H − E .4.1.2. (Pseudo)effective and big cones. The easiest interesting space of divisors to define is perhapsthe effective cone
Eff( X ) ⊂ N ( X ), defined as the subspace spanned by numerical classes ofeffective divisors. Unlike nefness and ampleness, however, effectiveness is not a numerical property:for instance, on an elliptic curve C , a line bundle of degree 0 has an effective multiple if and onlyif it is torsion.The effective cone is in general neither open nor closed. Its closure Eff( X ) is less subtle, andcalled the pseudo-effective cone . The interior of the effective cone is the big cone Big( X ), spannedby divisors D such that the linear series | mD | defines a map φ | mD | whose image has the samedimension as X . Thus, big divisors are the natural analog of birational maps. By Kodaira’sLemma [Laz04, Proposition 2.2.6], bigness is a numerical property. Example 4.7.
The strategy for computing pseudoeffective cones is typically similar to that forcomputing nef cones. On the one hand, one constructs effective divisors to span a cone Λ ⊂ Eff( X ).A moving curve is a numerical curve class [ C ] such that irreducible representatives of the class passthrough a general point of X . Thus if D is an effective divisor we must have D.C ≥
0; otherwise D would have to contain every irreducible curve of class C . Thus the moving curve classes duallydetermine a cone Eff( X ) ⊂ Λ ′ , and if Λ = Λ ′ then they equal the pseudoeffective cone. Thisapproach is justified by the seminal work of Boucksom-Demailly-P˘aun-Peternell, which establishesa duality between the pseudoeffective cone and the cone of moving curves [BDPP13]. Example 4.8. On X = Bl p P , the curve class H is moving and H.E = 0. Thus E spans anextremal edge of Eff( X ). The curve class H − E is also moving, and ( H − E ) = 0. Therefore H − E spans the other edge of Eff( X ), and Eff( X ) is spanned by H − E and E .4.1.3. Stable base locus decomposition.
The nef cone Nef( X ) is one chamber in a decomposition ofthe entire pseudoeffective cone Eff( X ). By the base locus Bs( D ) of a divisor D we mean the baselocus of the complete linear series | D | , regarded as a subset (i.e. not as a subscheme) of X . Byconvention, Bs( D ) = X if | D | is empty. The stable base locus of D is the subset B s( D ) = \ m> Bs( D ) IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 17 of X . One can show that B s( D ) coincides with the base locus Bs( mD ) of sufficiently large anddivisible multiples mD. Example 4.9.
The base locus and stable base locus of D depend on the class of D in Pic( X ), notjust on the numerical class of D . For example, if L is a degree 0 line bundle on an elliptic curve X , then Bs( L ) = X unless L is trivial, and B s( L ) = X unless L is torsion in Pic( X ).Since (stable) base loci do not behave well with respect to numerical equivalence, for the restof this subsection we assume q ( X ) = 0 so that linear and numerical equivalence coincide and N ( X ) Q = Pic( X ) ⊗ Q . Then the pseudoeffective cone Eff( X ) has a wall-and-chamber decomposi-tion where the stable base locus remains constant on the open chambers. These various chamberscontrol the birational maps from X to other projective varieties. For example, if f : X Y isthe rational map given by a sufficiently divisible multiple | mD | , then the indeterminacy locus ofthe map is contained in the stable base locus. Example 4.10.
Stable base loci decompositions are typically computed as follows. First, oneconstructs effective divisors in a multiple | mD | and takes their intersection to get a variety Y with B s( D ) ⊂ Y . In the other direction, one looks for curves C on X such that C.D <
0. Then anydivisor of class mD must contain C , so B s( D ) contains every curve numerically equivalent to C .When the Picard rank of X is two, the chamber decompositions can often be made very explicit.In this case it is notationally conventient to write, for example, ( D , D ] to denote the cone ofdivisors of the form a D + a D with a > a ≥ Example 4.11.
Let X = Bl p P . The nef cone is [ H, H − E ], and both H, H − E are basepointfree. Thus the stable base locus is empty in the closed chamber [ H, H − E ]. If D ∈ ( H, E ] is aneffective divisor, then
D.E <
0, so D contains E as a component. The stable base locus of divisorsin the chamber ( H, E ] is E .We now begin to investigate the birational geometry of some of the simplest moduli spaces ofsheaves on surfaces from a classical point of view.4.2. Birational geometry of Hilbert schemes of points.
Let X be a smooth surface withirregularity q ( X ) = 0, and let v be the Chern character of an ideal sheaf I Z of a collection Z of n points. Then M ( v ) is the Hilbert scheme X [ n ] of n points on X , parameterizing zero-dimensionalschemes of length n . See § Divisor classes.
Divisor classes on the Hilbert scheme X [ n ] can be understood entirely interms of the birational Hilbert-Chow morphism h : X [ n ] → X ( n ) to the symmetric product X ( n ) =Sym n X . Informally, this map sends the ideal sheaf of Z to the sum of the points in Z , withmultiplicities given by the length of the scheme at each point. Remark 4.12.
The symmetric product X ( n ) can itself be viewed as the moduli space of 0-dimensional sheaves with Hilbert polynomial P ( m ) = n . Suppose E is a zero-dimensional sheafwith constant Hilbert polynomial ℓ and that E is supported at a single point p . Then E admits alength ℓ filtration where all the quotients are isomorphic to O p . Thus, E is S -equivalent to O ⊕ ℓp .Since S -equivalent sheaves are identified in the moduli space, the moduli space M ( P ) is just X ( n ) .The Hilbert-Chow morphism h : X [ n ] → X ( n ) can now be seen to come from the moduli propertyfor X ( n ) . Let I be the universal ideal sheaf on X × X [ n ] . The quotient of the inclusion I → O X × X [ n ] is then a family of zero-dimensional sheaves of length n . This family induces a map X [ n ] → X ( n ) ,which is just the Hilbert-Chow morphism.The exceptional locus of the Hilbert-Chow morphism is a divisor class B on the Hilbert scheme X [ n ] . Alternately, B is the locus of nonreduced schemes. It is swept out by curves contained in fibers of the Hilbert-Chow morphism. A simple example of such a curve is given by fixing n − X and allowing a length 2 scheme Spec C [ ε ] / ( ε ) to “spin” at one additional point. Remark 4.13.
The divisor class B/ Z ⊂ X × X [ n ] denote the universal subscheme of length n , and let p : Z → X and q : Z → X [ n ] be the projections. Then the tautological bundle q ∗ p ∗ O X is a rank n vector bundlewith determinant of class − B/ L on X induces a line bundle L ( n ) on the symmetric product. Pulling backthis line bundle by the Hilbert-Chow morphism gives a line bundle L [ n ] := h ∗ L ( n ) . This gives aninclusion Pic( X ) → Pic( X [ n ] ). If L can be represented by a reduced effective divisor D , then L [ n ] can be represented by the locus D [ n ] := { Z ∈ X [ n ] : Z ∩ D = ∅} . Fogarty proves that the divisors mentioned so far generate the Picard group.
Theorem 4.14 (Fogarty [Fog73]) . Let X be a smooth surface with q ( X ) = 0 . Then Pic( X [ n ] ) ∼ = Pic( X ) ⊕ Z ( B/ . Thus, tensoring by R , N ( X [ n ] ) ∼ = N ( X ) ⊕ R B. There is another interesting way to use a line bundle on X to construct effective divisor classes.In examples, many extremal effective divisors can be realized in this way. Example 4.15.
Suppose L is a line bundle on X with m := h ( L ) > n . If Z ⊂ X is a generalsubscheme of length n , then H ( L ⊗ I Z ) ⊂ H ( L ) is a subspace of codimension n . Thus we get arational map φ : X [ n ] G := Gr( m − n, m )to the Grassmannian G of codimension n subspaces of H ( L ). The line bundle e L [ n ] := φ ∗ O G (1)(which is well-defined since the indeterminacy locus of φ has codimension at least 2) can be rep-resented by an effective divisor as follows. Let W ⊂ H ( L ) be a sufficiently general subspace ofdimension n ; one frequently takes W to be the subspace of sections of L passing through m − n general points. Then the locus e D [ n ] = { Z ∈ X [ n ] : H ( L ⊗ I Z ) ∩ W = { }} is an effective divisor representing φ ∗ O G (1).4.2.2. Curve classes.
Let C ⊂ X be an irreducible curve. There are two immediate ways that wecan induce a curve class on X [ n ] . Example 4.16.
Fix n − p , . . . , p n − on X which are not in C . Allowing an n th point p n to travel along C gives a curve e C [ n ] ⊂ X [ n ] . Example 4.17.
Suppose C admits a g n . If the g n is base-point free, then we get a degree n map C → P . The fibers of this map induce a rational curve P → X [ n ] , and we write C [ n ] for the classof the image. If the g n is not base-point free, we can first remove the basepoints to get a map P → X [ m ] for some m < n , and then glue the basepoints back on to get a map P → X [ n ] . Theclass C [ n ] doesn’t depend on the particular g n used to construct the curve (see for example [Hui12,Proposition 3.5] in the case of P ). IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 19
Remark 4.18.
Typically the curve classes C [ n ] are more interesting than e C [ n ] and they frequentlyshow up as extremal curves in the cone of curves. However, the class C [ n ] is only defined if C [ n ] carries an interesting linear series of degree n , while e C [ n ] always makes sense; thus curves of class e C [ n ] are also sometimes used.Both curve classes e C [ n ] and C [ n ] have the useful property that the intersection pairing withdivisors is preserved, in the sense that if D ⊂ X is a divisor then D [ n ] . e C [ n ] = D [ n ] .C [ n ] = D.C ;indeed, it suffices to check the equalities when D and C intersect transversely, and in that case D [ n ] and C [ n ] (resp. e C [ n ] ) intersect transversely in D.C points.The intersection with B is more interesting. Clearly e C [ n ] .B = 0 . On the other hand, the nonreduced schemes parameterized by a curve of class C [ n ] correspond toramification points of the degree n map C → P . The Riemann-Hurwitz formula then implies C [ n ] .B = 2 g ( C ) − n. One further curve class is useful; we write C for the class of a curve contracted by the Hilbert-Chowmorphism.4.2.3. The intersection pairing.
At this point we have collected enough curve and divisor classes tofully determine the intersection pairing between curves and divisors and find relations between thevarious classes. The classes C and C [ n ] for C any irreducible curve span N ( X ), so to completelycompute the intersection pairing we are only missing the intersection number C .B . However, sincethis intersection number is negative, we use the additional curve and divisor classes e C [ n ] and e D [ n ] to compute this number. To this end, we compute the intersection numbers of e D [ n ] with our curveclasses. Example 4.19.
To compute e D [ n ] .C , let m = h ( O X ( D )), fix m − n general points p , . . . , p m − n in X , and represent e D [ n ] as the set of schemes Z such that there is a curve on X of class D passingthrough p , . . . , p m − n and Z . Schemes parameterized by C are supported at n − q , . . . , q n − , with a spinning tangent vector at q n − . There is a unique curve D ′ of class D passingthrough p , . . . , p m − n , q , . . . , q n − , and it is smooth at q n − , so there is a single point of intersectionbetween C and e D [ n ] , occurring when the tangent vector at q n − is tangent to D ′ . Thus e D [ n ] .C = 1. Example 4.20.
Next we compute e C [ n ] . e D [ n ] . Represent e D [ n ] as in Example 4.19. The curve class e C [ n ] is represented by fixing n − q , . . . , q n − and letting q n travel along C . There is aunique curve D ′ of class D passing through p , . . . , p m − n , q , . . . , q n − , so e C [ n ] meets e D [ n ] when q n ∈ C ∩ D ′ . Thus e C [ n ] . e D [ n ] = C.D . Example 4.21.
For an irreducible curve C ⊂ X , write b C [ n ] for the curve class on X [ n ] obtainedby fixing n − X , fixing one point on C , and letting one point travel along C andcollide with the point fixed on C . It follows immediately that b C [ n ] .D [ n ] = C.D b C [ n ] . e D [ n ] = C.D − . Less immediately, we find b C [ n ] .B = 2: while the curve meets B set-theoretically in one point, atangent space calculation shows this intersection has multiplicity 2. We now collect our known intersection numbers. D [ n ] e D [ n ] BC [ n ] C.D g ( C ) − n e C [ n ] C.D C.D b C [ n ] C.D C.D − C e D [ n ] .C = 0, the divisors e D [ n ] are all not in the codimension one subspace N ( X ) ⊂ N ( X [ n ] ).Therefore the divisor classes of type D [ n ] and e D [ n ] together span N ( X ). It now follows that C + b C [ n ] = e C [ n ] since both sides pair the same with divisors D [ n ] and e D [ n ] , and thus C .B = −
2. We then also findrelations C [ n ] = e C [ n ] − ( g ( C ) − n ) C and e D [ n ] = D [ n ] − B. In particular, the divisors of type e D [ n ] are all in the half-space of divisors with negative coefficientof B in terms of the Fogarty isomorphism N ( X [ n ] ) ∼ = N ( X ) ⊕ R B . We can also complete ourintersection table. D [ n ] e D [ n ] BC [ n ] C.D C.D − ( g ( C ) − n ) 2 g ( C ) − n e C [ n ] C.D C.D b C [ n ] C.D C.D − C − Some nef divisors.
Part of the nef cone of X [ n ] now follows from our knowledge of theintersection pairing. First observe that since C .D [ n ] = 0 and C .B <
0, the nef cone is containedin the half-space of divisors with nonpositive B -coefficient in terms of the Fogarty isomorphism.If D is an ample divisor on X , then the divisor D ( n ) on the symmetric product is also ample, so D [ n ] is nef. Since a limit of nef divisors is nef, it follows that if D is nef on X then D [ n ] is nef on X [ n ] . Furtermore, if D is on the boundary of the nef cone of X then D [ n ] is on the boundary of thenef cone of X [ n ] . Indeed, if C.D = 0 then e C [ n ] .D [ n ] = 0 as well. This provesNef( X [ n ] ) ∩ N ( X ) = Nef( X ) , where by abuse of notation we embed N ( X ) in N ( X [ n ] ) by D D [ n ] .Boundary nef divisors which are not contained in the hyperplane N ( X ) are more interestingand more challenging to compute. Bridgeland stability and the positivity lemma will give us a toolfor computing and describing these classes.4.2.5. Examples.
We close our initial discussion of the birational geometry of Hilbert schemes ofpoints by considering several examples from this classical point of view.
Example 4.22 ( P n ] ) . The Neron-Severi space N ( P n ] ) of the Hilbert scheme of n points in P is spanned by H [ n ] and B , where H is the class of a line in P . Any divisior in the cone ( H, B ]is negative on C , so the locus B swept out by curves of class C is contained in the stable baselocus of any divisor in this chamber. Since B. e H [ n ] = 0 and e H [ n ] is the class of a moving curve, thedivisor B is an extremal effective divisor. IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 21
The divisor H [ n ] is an extremal nef divisor by § O P ( n −
1) is n -very ample , meaning that if Z ⊂ P is any zero-dimensional subscheme of length n , then H ( I Z ( n − n in H ( O P ( n − G is the Grassmannian of codimension- n planes in H ( O P ( n − φ : P n ] → G is a morphism . Thus φ ∗ O G (1) is nef. In our notation for divisors,putting D = ( n − H we conclude that e D [ n ] = ( n − H [ n ] − B is nef.Furthermore, e D [ n ] is not ample. Numerically, simply observe that e D [ n ] .H [ n ] = 0. More ge-ometrically, if two length n schemes Z, Z ′ are contained in the same line L then the subspaces H ( I Z ( n − H ( I Z ′ ( n − φ identifies Z and Z ′ . Note that if Z and Z ′ areboth contained in a single line L then their ideal sheaves can be written as extensions0 → O P ( − → I Z → O L ( − n ) → → O P ( − → I Z ′ → O L ( − n ) → . This suggests that if we have some new notion of semistability where I Z is strictly semistable withJordan-H¨older factors O P ( −
1) and O L ( − n ) then the ideal sheaves I Z and I Z ′ will be S -equivalent.Thus, in the moduli space of such objects, I Z and I Z ′ will be represented by the same point of themoduli space. Example 4.23 ( P ) . The divisor e H [2] = H [2] − B spanning an edge of the nef cone is also anextremal effective divisor on P . Indeed, the orthogonal curve class H [2] is a moving curve on P . Thus there two chambers in the stable base locus decomposition of Eff( P ). Example 4.24 ( P ) . By Example 4.22, on P the divisor 2 H [3] − B is an extremal nef divisor.The open chambers of the stable base locus decomposition are( H [3] , B ) , (2 H [3] − B, H [3] ) , and ( H [3] − B, H [3] − B ) . To establish this, first observe that H [3] − B is the class of the locus D of collinear schemes, since D.C = 1 and D. e H [3] = 1 . The divisor 2 H [3] − B is orthogonal to curves of class H [3] , so thelocus of collinear schemes swept out by these curves lies in the stable base locus of any divisor in( H [3] − B, H [3] − B ). In the other direction, any divisor in ( H [3] − B, H [3] − B ) is the sumof a divisor on the ray spanned by D and an ample divisor. It follows that the stable base locus inthis chamber is exactly D .For many more examples of the stable base locus decomposition of P n ] , see [ABCH13] forexplicit examples with n ≤
9, [CH14a] for a discussion of the chambers where monomial schemesare in the base locus, and [Hui16, CHW16] for the effective cone. Alternately, see [CH14b] for adeeper survey. Also, see the work of Li and Zhao [LZ16] for more recent developments unifyingseveral of these topics.4.3.
Birational geometry of moduli spaces of sheaves.
We now discuss some of the basicaspects of the birational geometry of moduli spaces of sheaves. Many of the concepts are mildgeneralizations of the picture for Hilbert schemes of points.
Line bundles.
The main method of constructing line bundles on a moduli space of sheavesis by a determinantal construction. First suppose E /S is a family of sheaves on X parameterizedby S . Let p : S × X → S and q : S × X → X be the projections. The Donaldson homomorphism is a map λ E : K ( X ) → Pic( S ) defined by the composition λ E : K ( X ) q ∗ → K ( S × X ) · [ E ] → K ( S × X ) p ! → K ( S ) det → Pic( S )Here p ! = P i ( − i R i p ∗ . Informally, we pull back a sheaf on X to the product, twist by the family E , push forward to S , and take the determinant line bundle. Thus we obtain from any class in K ( X ) a line bundle on the base S of the family E . The above discussion is sufficient to define linebundles on a moduli space M ( v ) of sheaves if there is a universal family E on M ( v ): there is thena map λ E : K ( X ) → Pic( M ( v )), and the image typically consists of many interesting line bundleson the moduli space.Things are slightly more delicate in the general case where there is no universal family. Asmotivation, given a class w ∈ K ( X ), we would like to define a line bundle L on M ( v ) with thefollowing property. Suppose E /S is a family of sheaves of character v and that φ : S → M ( v )is the moduli map. Then we would like there to be an isomorphism φ ∗ L ∼ = λ E ( w ) , so that thedeterminantal line bundle λ E ( w ) on S is the pullback of a line bundle on the moduli space M ( v ).In order for this to be possible, observe that the line bundle λ E ( w ) must be unchanged when itis replaced by E ⊗ p ∗ N for some line bundle N ∈ Pic( S ). Indeed, the moduli map φ : S → M ( v ) isnot changed when we replace E by E ⊗ p ∗ N , so φ ∗ L is unchanged as well. However, a computationshows that λ E⊗ p ∗ N ( w ) = λ E ( w ) ⊗ N ⊗ χ ( v ⊗ w ) . Thus, in order for there to be a chance of defining a line bundle L on M ( v ) with the desiredproperty we need to assume that χ ( v ⊗ w ) = 0.In fact, if χ ( v ⊗ w ) = 0, then there is a line bundle L as above on the stable locus M s ( v ),denoted by λ s ( w ). To handle things rigorously, it is necessary to go back to the construction ofthe moduli space via GIT. See [HL10, § M ( v ). Theorem 4.25 ([HL10, Theorem8.1.5]) . Let v ⊥ ⊂ K ( X ) denote the orthogonal complement of v with respect to the Euler pairing χ ( − ⊗ − ) . Then there is a natural homomorphism λ s : v ⊥ → Pic( M s ( v )) . In general it is a difficult question to completely determine the Picard group of the moduli space.One of the best results in this direction is the following theorem of Jun Li.
Theorem 4.26 ([Li94]) . Let X be a regular surface, and let v ∈ K ( X ) with rk v = 2 and ∆( v ) ≫ .Then the map λ s : v ⊥ ⊗ Q → Pic( M s ( v )) ⊗ Q is a surjection. More precise results are somewhat rare. We discuss a few of the main such examples here.
Example 4.27 (Picard group of moduli spaces of sheaves on P ) . Let M ( v ) be a moduli spaceof sheaves on P . The Picard group of this space was determined by Dr´ezet [Dre88]. The answerdepends on the δ -function introduced in the classification of semistable characters in § v is the character of an exceptional bundle then M ( v ) is a point and there is nothing to discuss. If δ ( µ ( v )) = ∆( v ), then M ( v ) is a moduli space of so-called height zero bundles and the Picard groupis isomorphic to Z . Finally, if δ ( µ ( v )) > ∆( v ) then the Picard group is isomorphic to Z ⊕ Z . Ineach case, the Donaldson morphism is surjective. IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 23
Example 4.28 (Picard group of moduli spaces of sheaves on P × P ) . Let M ( v ) be a modulispace of sheaves on P × P . Already in this case the Picard group does not appear to be known inevery case. See [Yos96] for some partial results, as well as results on ruled surfaces in general. Example 4.29 (Picard group of moduli spaces of sheaves on a K . Let X be a K v ∈ K num ( X ) be a primitive positive vector (see § H be a polarization whichis generic with respect to v . In this case the story is similar to the computation for P , with theBeauville-Bogomolov form playing the role of the δ function. If h v , v i = − M H ( v ) is a point.If h v , v i = 0, then the Donaldson morphism λ : v ⊥ ⊗ R → N ( M H ( v )) is surjective with kernelspanned by v , and N ( M H ( v )) is isomorphic to v ⊥ / v . Finally, if h v , v i > Example 4.30 (Brill-Noether divisors) . For birational geometry it is important to be able toconstruct sections of of line bundles. The determinantal line bundles introduced above frequentlyhave special sections vanishing on
Brill-Noether divisors . Let (
X, H ) be a smooth surface, and let v and w be an orthogonal pair of Chern characters, i.e. suppose that χ ( v ⊗ w ) = 0, and supposethat there is a reasonable, e.g. irreducible, moduli space M H ( v ) of semistable sheaves. Suppose F is a vector bundle with ch F = w , and consider the locus D F = { E ∈ M H ( v ) : H ( E ⊗ F ) = 0 } . If we assume that H ( E ⊗ F ) = 0 for every E ∈ M H ( v ) and that H ( E ⊗ F ) = 0 for a general E ∈ M H ( v ) then the locus D F will be an effective divisor. Furthermore, its class is λ ( w ∗ ).The assumption that H ( E ⊗ F ) = 0 often follows easily from stability and Serre duality. Forinstance, if µ H ( v ) , µ H ( w ) , µ H ( K X ) > F is a semistable vector bundle then H ( E ⊗ F ) = Ext ( F ∗ , E ) = Hom( E, F ∗ ( K X )) ∗ = 0by stability. On the other hand, it can be quite challenging to verify that H ( E ⊗ F ) = 0 for ageneral E ∈ M H ( v ). These types of questions have been studied in [CHW16] in the case of P and[Rya16] in the case of P × P . Interesting effective divisors arising in the birational geometry ofmoduli spaces frequently arise in this way.4.3.2. The Donaldson-Uhlenbeck-Yau compactification.
For Hilbert schemes of points X [ n ] , the sym-metric product X ( n ) offered an alternate compactification, with the map h : X [ n ] → X ( n ) being theHilbert-Chow morphism. Recall that from a moduli perspective the Hilbert-Chow morphism sendsthe ideal sheaf I Z to (the S -equivalence class of) the structure sheaf O Z . Thinking of O X as thedouble-dual of I Z , the sheaf O Z is the cokernel in the sequence0 → I Z → O X → O Z → . The Donaldson-Uhlenbeck-Yau compactification can be viewed as analogous to the compactificationof the Hilbert scheme by the symmetric product.Let (
X, H ) be a smooth surface, and let v be the Chern character of a semistable sheaf of positiverank. Set-theoretically, the Donaldson-Uhlenbeck-Yau compactification M DUYH ( v ) of the modulispace M H ( v ) can be defined as follows. Recall that the double dual of any torsion-free sheaf E on X is locally free, and there is a canonical inclusion E → E ∗∗ . (Note, however, that the double-dualof a Gieseker semistable sheaf is in general only µ H -semistable). Define T E as the cokernel0 → E → E ∗∗ → T E → , so that T E is a skyscraper sheaf supported on the singularities of E . In the Donaldson-Uhlenbeck-Yau compactification of M H ( v ), a sheaf E is replaced by the pair ( E ∗∗ , T E ) consisting of the µ H -semistable sheaf E ∗∗ and the S -equivalence class of T E , i.e. an element of some symmetricproduct X ( n ) . In particular, two sheaves which have isomorphic double duals and have singu-larities supported at the same points (counting multiplicity) are identified in M DUYH ( v ), even if the particular singularities are different. The Jun Li morphism j : M H ( v ) → M DUYH ( v ) inducingthe Donaldson-Uhlenbeck-Yau compactification arises from the line bundle λ ( w ) associated to thecharacter w of a 1-dimensional torsion sheaf supported on a curve whose class is a multiple of H .See [HL10, § Change of polarization.
Classically, one of the main interesting sources of birational mapsbetween moduli spaces of sheaves is provided by varying the polarization. Suppose that { H t } (0 ≤ t ≤
1) is a continuous family of ample divisors on X . Let E be a sheaf which is µ H -stable. Itmay happen for some time t > E is not µ H t -stable. In this case, there is a smallest time t where E is not µ H t -stable, and then E is strictly µ H t -semistable. There is then an exact sequence0 → F → E → G → µ H t -semistable sheaves with the same µ H t -slope. For t < t , we have µ H t ( F ) < µ H t ( E ) < µ H t ( G ) . On the other hand, in typical examples the inequalities will be reversed for t > t : µ H t ( F ) > µ H t ( E ) > µ H t ( G ) . While E is certainly not µ H t -semistable for t > t , if there are sheaves E ′ fitting as extensions insequences 0 → G → E ′ → F → E ′ is µ H t -stable for t > t (although they are certainly not µ H t -semistablefor t < t ).Thus, the set of H t -semistable sheaves changes as t crosses t , and the moduli space M H t ( v )changes accordingly. It frequently happens that only some very special sheaves become destabilizedas t crosses t , in which case the expectation would be that the moduli spaces for t < t and t > t are birational.To clarify the dependence between the geometry of the moduli space M H ( v ) and the choice ofpolarization H , we partition the cone Amp( X ) of ample divisors on X into chambers where themoduli space remains constant. Let v be a primitive vector, and suppose E has ch( E ) = v andis strictly H -semistable for some polarization H . Let F ⊂ E be an H -semistable subsheaf with µ H ( F ) = µ H ( E ). Then the locus Λ ⊂ Amp( X ) of polarizations H ′ such that µ H ′ ( F ) = µ H ′ ( E )is a hyperplane in the ample cone, called a wall . The collection of all walls obtained in this waygives the ample cone a locally finite wall-and-chamber decomposition. As H varies within an openchamber, the moduli space M H ( v ) remains unchanged. On the other hand, if H crosses a wall thenthe moduli spaces on either side may be related in interesting ways.Notice that if say X has Picard rank 1 or we are considering Hilbert schemes of points thenno interesting geometry can be obtained by varying the polarization. Recall that in Example 4.24we saw that even P has nontrivial alternate birational models. One of the goals of Bridgelandstability will be to view these alternate models as a variation of the stability condition. Variationof polarization is one of the simplest examples of how a stability condition can be modified in acontinuous way, and Bridgeland stability will give us additional “degrees of freedom” with whichto vary our stability condition. 5. Bridgeland stability
The definition of a Bridgeland stability condition needs somewhat more machinery than theprevious sections. However, we will primarily work with explicit stability conditions where theabstract nature of the definition becomes very concrete. While it would be a good idea to reviewthe basics of derived categories of coherent sheaves, triangulated categories, t-structures, and torsiontheories, it is also possible to first develop an appreciation for stability conditions and then go back
IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 25 and fill in the missing details. Good references for background on these topics include [GM03] and[Huy06].5.1.
Stability conditions in general.
Let X be a smooth projective variety. We write D b ( X )for the bounded derived category of coherent sheaves on X . We also write K num ( X ) for theGrothendieck group of X modulo numerical equivalence. Following [Bri07], we make the followingdefinition. Definition 5.1. A Bridgeland stability condition on X is a pair σ = ( Z, A ) consisting of an R -linearmap Z : K num ( X ) ⊗ R → C (called the central charge ) and the heart A ⊂ D b ( X ) of a boundedt-structure (which is an abelian category). Additionally, we require that the following propertiesbe satisfied.(1) (Positivity) If 0 = E ∈ A , then Z ( E ) ∈ H := { re iθ : 0 < θ ≤ π and r > } ⊂ C . We define functions r ( E ) = ℑ Z ( E ) and d ( E ) = −ℜ Z ( E ), so that r ( E ) ≥ d ( E ) > r ( E ) = 0. Thus r and d are generalizations of the classical rank and degreefunctions. The (Bridgeland) σ -slope is defined by µ σ ( E ) = d ( E ) r ( E ) = − ℜ Z ( E ) ℑ Z ( E ) . (2) (Harder-Narasimhan filtrations) An object E ∈ A is called (Bridgeland) σ -(semi)stable if µ σ ( F ) < ( − ) µ σ ( E )whenever F ⊂ E is a subobject of E in A . We require that every object of A has a finiteHarder-Narasimhan filtration in A . That is, there is a unique filtration0 = E ⊂ E ⊂ · · · ⊂ E ℓ = E of objects E i ∈ A such that the quotients F i = E i /E i − are σ -semistable with decreasingslopes µ σ ( F ) > · · · > µ σ ( F ℓ ).(3) (Support property) The support property is one final more technical condition which mustbe satisfied. Fix a norm k · k on K num ( X ) ⊗ R . Then there must exist a constant C > k E k ≤ C k Z ( E ) k for all semistable objects E ∈ A . Remark 5.2.
Let (
X, H ) be a smooth surface. The subcategory coh X ⊂ D b ( X ) of sheaves withcohomology supported in degree 0 is the heart of the standard t-structure. We can then try todefine a central charge Z ( E ) = − c ( E ) .H + i rk( E ) H , and the corresponding slope function is the ordinary slope µ H . However, this does not give aBridgeland stability condition, since Z ( E ) = 0 for any finite length torsion sheaf. Thus it is notimmediately clear in what way Bridgeland stability generalizes ordinary slope- or Gieseker stability.Nonetheless, for any fixed polarization H and character v there are Bridgeland stability conditions σ where the σ -(semi)stable objects of character v are precisely the H -Gieseker (semi)stable sheavesof character v . See § Remark 5.3.
To work with the definition of a stability condition, it is crucial to understand whatit means for a map F → E between objects of the heart A to be injective. The following exerciseis a good test of the definitions involved. Exercise 5.4.
Let
A ⊂ D b ( X ) be the heart of a bounded t-structure, and let φ : F → E be a mapof objects of A . Show that φ is injective if and only if the mapping cone cone( φ ) of φ is also in A .In this case, there is an exact sequence0 → F → E → cone( φ ) → A .One of the most important features of Bridgeland stability is that the space of all stabilityconditions on X is a complex manifold in a natural way. In particular, we are able to continuouslyvary stability conditions and study how the set (or moduli space) of semistable objects varies withthe stability condition. Let Stab( X ) denote the space of stability conditions on X . Then Bridgelandproves that there is a natural topology on Stab( X ) such that the forgetful mapStab( X ) → Hom R ( K num ( X ) ⊗ R , C )( Z, A ) Z is a local homeomorphism. Thus if σ = ( Z, A ) is a stability condition and the linear map Z isdeformed by a small amount, there is a unique way to deform the category A to get a new stabilitycondition.5.1.1. Moduli spaces.
Let σ be a stability condition and fix a vector v ∈ K num ( X ). There is anotion of a flat family E /S of σ -semistable objects parameterized by an algebraic space S [BM14a].Correspondingly, there is a moduli stack M σ ( v ) parameterizing flat families of σ -semistable objectof character v . In full generality there are many open questions about the geometry of thesemoduli spaces. In particular, when is there a projective coarse moduli space M σ ( v ) parameterizing S -equivalence classes of σ -semistable objects of character v ?Several authors have addressed this question for various surfaces, at least when the stabilitycondition σ does not lie on a wall for v (see § M σ ( v ) when X is P [ABCH13], P × P or F [AM16], an abelian surface [MYY14], a K Stability conditions on surfaces.
Bridgeland [Bri08] and Arcara-Bertram [AB13] explainhow to construct stability conditions on a smooth surface. The construction is very explicit, andthese are the only kinds of stability conditions we will consider in this survey. Before beginning weintroduce some notation to make the definitions more succinct.Let X be a smooth surface and let H, D ∈ Pic( X ) ⊗ R be an ample divisor and an arbitrary twisting divisor, respectively. We formally define the twisted Chern character ch D = e − D ch.Explicitly expanding this definition, this means thatch D = ch ch D = ch − D ch ch D = ch − D ch + D . We can also define twisted slopes and discriminants by the formulas µ H,D = H. ch D H ch D ∆ H,D = 12 µ H,D − ch D H ch D . IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 27
For reasons that will become clear in § K X / D = ch D + K X µ H,D = µ H,D + K X ∆ H,D = ∆
H,D + K X . Remark 5.5.
Note that the twisted slopes µ H,D and µ H,D are primarily just a notational conve-nience; they only differ from the ordinary slope by a constant (depending on H and D ). On theother hand, twisted discriminants ∆ H,D and ∆
H,D do not obey such a simple formula, and aregenuinely useful.
Remark 5.6 (Twisted Gieseker stability) . We have already encountered H -Gieseker (semi)stabilityand the associated moduli spaces M H ( v ) of H -Gieseker semistable sheaves. There is a mild gener-alization of this notion called ( H, D ) -twisted Gieseker (semi)stability . A torsion-free coherent sheaf E is ( H, D )-twisted Gieseker (semi)stable if whenever F ( E we have(1) µ H,D ( F ) ≤ µ H,D ( E ) and(2) whenever µ H,D ( F ) = µ H,D ( E ), we have ∆ H,D ( F ) > ( − ) ∆ H,D ( E ).Compare with Example 2.11, which is the case D = 0. When H, D are Q -divisors, Matsukiand Wentworth [MW97] construct projective moduli spaces M H,D ( v ) of ( H, D )-twisted Giesekersemistable sheaves. Note that any µ H -stable sheaf is both H -Gieseker stable and ( H, D )-twistedGieseker stable, so that the spaces M H ( v ) and M H,D ( v ) are often either isomorphic or birational. Exercise 5.7.
Use the Hodge Index Theorem and the ordinary Bogomolov inequality (Theorem3.7) to show that if E is µ H -semistable then∆ H,D ( E ) ≥ . We now define a half-plane (or slice ) of stability conditions on X corresponding to a choice ofdivisors H, D ∈ Pic( X ) ⊗ R as above. First fix a number β ∈ R . We define two full subcategoriesof the category coh X of coherent sheaves by T β = { E ∈ coh X : µ H,D ( G ) > β for every quotient G of E }F β = { E ∈ coh X : µ H,D ( F ) ≤ β for every subsheaf F of E } . Note that by convention the (twisted) Mumford slope of a torsion sheaf is ∞ , so that T β containsall the torsion sheaves on X . On the other hand, sheaves in F β have no torsion subsheaf and soare torsion-free.For any β ∈ R , the pair of categories ( T β , F β ) form what is called a torsion pair . Briefly, thismeans that Hom( T, F ) = 0 for any T ∈ T β and F ∈ F β , and any E ∈ coh X can be expressednaturally as an extension 0 → F → E → T → T ∈ T β by a sheaf F ∈ F β . Then there is an associated t-structure with heart A β = { E • : H − ( E • ) ∈ F β , H ( E • ) ∈ T β , and H i ( E • ) = 0 for i = − , } ⊂ D b ( X ) , where we use a Roman H i ( E • ) to denote cohomology sheaves.Some objects of A β are the sheaves T in T β (viewed as complexes sitting in degree 0) and shifts F [1] where F ∈ F β , sitting in degree −
1. More generally, every object E • ∈ A β is an extension0 → H − ( E • )[1] → E • → H ( E • ) → , where the sequence is exact in the heart A β .To define stability conditions we now need to define central charges compatible with the hearts A β . Let α ∈ R > be an arbitrary positive real number. We define Z β,α = − ch D + βH + α H D + βH + iH ch D + βH , and put σ β,α = ( Z β,α , A β ). Note that if E is an object of nonzero rank with twisted slope µ H,D and discriminant ∆
H,D then the corresponding Bridgeland slope is µ σ β,α = − ℜ Z β,α ℑ Z β,α = ( µ H,D − β ) − α − H,D µ H,D − β . Theorem 5.8 ([AB13]) . Let X be a smooth surface, and let H, D ∈ Pic( X ) ⊗ R with H ample.If β, α ∈ R with α > , then the pair σ β,α = ( Z β,α , A β ) defined above is a Bridgeland stabilitycondition. The most interesting part of the theorem is the verification of the Positivity axiom 1 in theDefinition 5.1 of a stability condition, which we now sketch. The other parts are quite formal.
Sketch proof of positivity.
Note that Z := Z β,α is an R -linear map. Since the upper half-plane H = { re iθ : 0 < θ ≤ π and r > } is closed under addition, the exact sequence0 → H − ( E • )[1] → E • → H ( E • ) → Z ( T ) ∈ H and Z ( F [1]) ∈ H whenever T ∈ T β and F ∈ F β .If T ∈ T β is not torsion, then µ H,D ( T ) > β is finite. Expanding the definitions immediatelygives H. ch D + βH ( T ) >
0, so Z ( T ) ∈ H . If T is torsion with positive-dimensional support, thenagain H. ch D + βH ( T ) > Z ( T ) ∈ H . Finally, if T = 0 has zero-dimensional support then − ch D + βH ( T ) = − ch ( T ) > Z ( T ) ∈ H .Suppose 0 = F ∈ F β . If actually µ H,D ( F ) < β , then H. ch D + βH ( F ) < Z ( F [1]) ∈ H againfollows. So suppose that µ H,D ( F ) = β , which gives ℑ Z ( F ) = 0. By the definition of F β , thesheaf F is torsion-free and µ H,D + βH -semistable of µ H,D + βH slope 0. By Exercise 5.7 we find that∆ H,D + βH ( F ) ≥
0. The formula for the twisted discriminant and the fact that α > ℜ Z ( F ) <
0, so ℜ Z ( F [1]) > (cid:3) To summarize, if we let Π = { ( β, α ) : β, α ∈ R , α > } , the choice of a pair of divisors H, D ∈ Pic( X ) ⊗ R with H ample defines an embeddingΠ → Stab( X )( β, α ) σ β,α . This half-plane of stability conditions is called the (
H, D ) -slice of the stability manifold. We willsometimes abuse notation and write σ ∈ Π for a stability condition σ parameterized by the slice.While the stability manifold can be rather large and unwieldy in general (having complex dimensiondim R K num ( X ) ⊗ R ), much of the interesting geometry can be studied by inspecting the differentslices of the manifold.5.3. Walls.
Fix a class v ∈ K ( X ). The stability manifold Stab( X ) of X admits a locally finitewall-and-chamber decomposition such that the set of σ -semistable objects of class v does not varyas σ varies within an open chamber. This is analogous to the wall-and-chamber decomposition ofthe ample cone Amp( X ) for classical stability, see § v is primitive, then a stability condition σ lies on a wall if and only if there is a strictly σ -semistable object of character v .For computations, the entire stability manifold can be rather unwieldy to work with. Onecommonly restricts attention to stability conditions in some easily parameterized subset of thestability manifold. Here we focus on the ( H, D )-slice { σ β,α : β, α ∈ R , α > } of stability conditionson a smooth surface X determined by a choice of divisors H, D ∈ Pic( X ) ⊗ R with H ample. Definition 5.9.
Let X be a smooth surface, and fix divisors H, D ∈ Pic( X ) ⊗ R with H ample.Let v , w ∈ K num ( X ) be two classes which have different µ σ β,α -slopes for some ( β, α ) with α > IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 29 (1) The numerical wall for v determined by w is the subset W ( v , w ) = { ( β, α ) : µ σ β,α ( v ) = µ σ β,α ( w ) } ⊂ Π . (2) The numerical wall for v determined by w is a wall if there is some ( β, α ) ∈ W ( v , w ) andan exact sequence 0 → F → E → G → σ β,α -semistable objects with ch F = w and ch E = v .5.3.1. Geometry of numerical walls.
The geometry of the numerical walls in a slice of the stabilitymanifold is particularly easy to describe. Verifying the following properties is a good exercise inthe algebra of Chern classes and the Bridgeland slope function.(1) First suppose v has nonzero rank and that the Bogomolov inequality ∆ H,D ( v ) ≥ β = µ H,D ( v ) is a numerical wall. The other numerical walls formtwo nested families of semicircles on either side of the vertical wall. These semicircles havecenters on the β -axis, and their apexes lie along the hyperbola ℜ Z β,α ( v ) = 0 in Π. The twofamilies of semicircles accumulate at the points( µ H,D ( v ) ± q H,D ( v ) , ℜ Z β,α ( v ) = 0 with the β -axis. See Figure 2 for an approximate illustration. µ H,D − q H,D µ H,D + q H,D µ H,D
Figure 2.
Schematic diagram of numerical walls in the (
H, D )-slice for a nonzerorank character v with slope µ H,D and discriminant ∆
H,D .(2) If instead v has rank zero but c ( v ) = 0, then the curve ℜ Z β,α ( v ) = 0 in Π degenerates tothe vertical line β = ch D ( v )ch D ( v ) .H . The numerical walls for v are all semicircles with center (ch D ( v ) / (ch D ( v ) .H ) ,
0) and arbi-trary radius.
Exercise 5.10. In v , w have nonzero rank and different slopes, the numerical semicircular wall W ( v , w ) has center ( s W ,
0) and radius ρ W satisfying s W = µ H,D ( v ) + µ H,D ( w )2 − ∆ H,D ( v ) − ∆ H,D ( w ) µ H,D ( v ) − µ H,D ( w ) ρ W = ( s W − µ H,D ( v )) − H,D ( v ) . If ( s W − µ H,D ( v )) ≤ H,D ( v ), then the wall is empty. Remark 5.11.
Let v be a character of nonzero rank. It follows from the above discussion that if W, W ′ are numerical walls for v both lying left of the vertical wall β = µ H,D ( v ) then W is nestedinside W ′ if and only if s W > s W ′ , where the center of W (resp. W ′ ) is ( s W ,
0) (resp. ( s W ′ , Walls and destabilizing sequences.
In the definition of a wall W := W ( v , w ) for v determinedby a character w we required that there is some point ( β, α ) ∈ W and a destabilizing exact sequence0 → F → E → G → σ β,α -semistable objects, where ch( E ) = v and ch( F ) = w . Note that since ( β, α ) ∈ W wein particular have µ σ β,α ( F ) = µ σ β,α ( E ) = µ σ β,α ( G ). The above sequence is an exact sequenceof objects of the categories A β . By the geometry of the numerical walls, the wall W separatesthe slice Π into two open regions Ω , Ω ′ . Relabeling the regions if necessary, for σ ∈ Ω we have µ σ ( F ) > µ σ ( E ). Therefore E is not σ -semistable for any σ ∈ Ω. On the other hand, E may be σ -semistable for σ ∈ Ω; at least the subobject F ⊂ E does not violate the semistability of E .Our definition of a wall is perhaps somewhat unsatisfactory due to the dependence on pickingsome point ( β, α ) ∈ W where there is a destabilizing exact sequence as above. The next resultshows that this definition is equivalent to an a priori stronger definition which appears more natural.Roughly speaking, destabilizing sequences “persist” along the entire wall. Proposition 5.12 ([ABCH13, Lemma 6.3] for P , [Mac14] in general) . Suppose that → F → E → G → is an exact sequence of σ β,α -semistable objects of the same σ β,α -slope. Put ch F = w and ch E = v ,and suppose v and w do not have the same slope everywhere in the ( H, D ) -slice. Let W = W ( v , w ) be the wall defined by these characters. If ( β ′ , α ′ ) ∈ W is any point on the wall, then the aboveexact sequence is an exact sequence of σ β ′ ,α ′ -semistable objects of the same σ β ′ ,α ′ -slope.In particular, each of the objects F, E, G appearing in the above sequence lie in the category A β ′ . Note that the first part of the proposition is essentially equivalent to the final statement byExercise 5.4.5.4.
Large volume limit.
As mentioned earlier, (twisted) Gieseker moduli spaces of sheaves onsurfaces can be recovered as certain moduli spaces of Bridgeland-semistable objects. We say thatan object E • ∈ A β is a sheaf if it is isomorphic to a sheaf sitting in degree 0. We continue to workin an ( H, D )-slice of stability conditions on a smooth surface X . Theorem 5.13 ([ABCH13, §
6] for P , [Mac14] in general) . Let v ∈ K num ( X ) be a character ofpositive rank with ∆ H,D ( v ) ≥ . Let β < µ H,D ( v ) , and suppose α ≫ (depending on v ). Then anobject E • ∈ A β is σ β,α -semistable if and only if it is an ( H, D ) -semistable sheaf.Proof. Since β < µ
H,D ( v ), the stability condition σ β,α lies left of the vertical wall β = µ H,D ( v ).The walls for v are locally finite. Considering a neighborhood of a stability condition on thevertical wall shows that there is some largest semicircular wall W left of the vertical wall. The setof σ -semistable objects is constant as σ varies in the chamber between W and the vertical wall.It is therefore enough to show the following two things. (1) If E • ∈ A β has ch E • = v and is σ β,α -semistable for α ≫ E • is an ( H, D )-semistable sheaf. (2) If E is an ( H, D )-semistablesheaf of character v , then E is σ β,α -semistable for α ≫
0. That is, we may pick α depending on E , and not just depending on v .(1) First suppose E • ∈ A β is σ β,α -semistable for α ≫ E • = v . If E • is not a sheaf, thenwe have an interesting exact sequence0 → H − ( E • )[1] → E • → H ( E • ) → IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 31 in A β . Since F := H − ( E • ) ∈ F β , the formula for the Bridgeland slope shows that µ σ β,α ( F [1]) = µ σ β,α ( F ) → ∞ as α → ∞ . On the other hand, since G := H ( E • ) ∈ T β we have µ σ β,α ( G ) → −∞ as α → ∞ ,noting that rk( G ) > v >
0. This is absurd since E • is σ β,α -semistable for α ≫
0, and weconclude that E := E • ∈ T β is a sheaf.Similar arguments show that E is ( H, D )-semistable. First suppose E has a µ H,D -stable subsheaf F with µ H,D ( F ) > µ H,D ( E ). Then the corresponding exact sequence of sheaves0 → F → E → G → T β . Indeed, any quotient of an object in T β is in T β , and F ∈ T β by construction. Thus this is actually an exact sequence in A β . The formula for the Bridgelandslope then shows that µ σ β,α ( F ) > µ σ β,α ( E ) for α ≫
0, violating the σ β,α -semistability of E . Weconclude that E is µ H,D -semistable. To see that E is ( H, D )-semistable, suppose there is a sequence0 → F → E → G → µ H,D -slope, but ∆
H,D ( F ) < ∆ H,D ( E ). Then the formula for the Bridgelandslope gives µ σ β,α ( F ) > µ σ β,α ( E ) for every α , again contradicting the σ β,α -semistability of E forlarge α .(2) Suppose E is ( H, D )-semistable of character v , and suppose F • is a subobject of E in A β .Taking the long exact sequence in cohomology sheaves of the exact sequence0 → F • → E → G • → A β gives an exact sequence of sheaves0 → H − ( F • ) → → H − ( G • ) → H ( F • ) → E → H ( G • ) → . Therefore H − ( F • ) = 0, i.e. F := F • is a sheaf in T β . The ( H, D )-semistability of E then gives µ H,D ( F ) ≤ µ H,D ( E ), with ∆ H,D ( F ) ≥ ∆ H,D ( E ) in case of equality. The formula for µ σ β,α thenshows that if α ≫ µ σ β,α ( F ) ≤ µ σ β,α ( E ). It follows from the finiteness of the walls that E is actually σ β,α -semistable for large α . (cid:3) In particular, if v ∈ K num ( X ) is the character of an ( H, D )-semistable sheaf of positive rank,then there is some largest wall W lying to the left of the vertical wall (or, possibly, there are nowalls left of the vertical wall). This wall is called the Gieseker wall . For stability conditions σ inthe open chamber C bounded by the Gieseker wall and the vertical wall, we have M σ ( v ) ∼ = M H,D ( v ) . Therefore, any moduli space of twisted semistable sheaves can be recovered as a moduli space ofBridgeland semistable objects. 6.
Examples on P In this subsection we investigate a couple of the first interesting examples of Bridgeland stabilityconditions and their relationship to birational geometry. We focus here on the characters of somesmall Hilbert schemes of points on P . In these cases the definitions simplify considerably, andthings can be understood explicitly. Notation.
Let X = P , and fix the standard polarization H . We take D = 0; in general, thechoice of twisting divisor is only interesting modulo the polarization, as adding a multiple of thepolarization to D only translates the ( H, D )-slice. The twisting divisor becomes more relevant inexamples of higher Picard rank. Additionally, since K P is parallel to H , we may as well work withthe ordinary slope and discriminant µ = ch r ∆ = 12 µ − ch r instead of the more complicated µ H, and ∆ H, . With these conventions, if v and w are charactersof positive rank then the wall W ( v , w ) has center ( s W ,
0) and radius ρ W given by s W = µ ( v ) + µ ( w )2 − ∆( v ) − ∆( w ) µ ( v ) − µ ( w ) ρ W = ( s W − µ ( v )) − v ) . If we further let v = ch I Z be the character of an ideal of a length n scheme Z ∈ P n ] , then theformulas further simplify to s W = µ ( w )2 + n − ∆( w ) µ ( w ) ρ W = s W − n. The main question to keep in mind is the following.
Question 6.1.
Let I Z be the ideal sheaf of Z ∈ P n ] . For which stability conditions σ in the slice is I Z a σ -semistable object? What does the destabilizing sequence of I Z look like along the wall whereit is destabilized? Note that since I Z is a Gieseker semistable sheaf, it is σ β,α -semistable if α ≫ β < µ ( I Z ).There will be some wall W left of the vertical wall where I Z is destabilized by some subobject F .For stability conditions σ below this wall, I Z is never σ -semistable. Thus the region in the slicewhere I Z is σ -semistable is bounded by the wall W and the vertical wall. It potentially consists ofseveral of the chambers in the wall-and-chamber decomposition of the slice.6.2. Types of walls.
There are two very different ways in which an ideal sheaf I Z of length n canbe destabilized along a wall. The simplest way I Z can be destabilized is if it is destabilized by anactual subsheaf, i.e. if there is an exact sequence of sheaves0 → I Y ( − k ) → I Z → T → Y of length ℓ . The character w =ch I W ( − k ) has ( r, µ, ∆) = (1 , − k, ℓ ), so this wall has center ( s W ,
0) with(1) s W = − k − n − ℓk . A wall obtained in this way is called a rank one wall .On the other hand, subobjects of I Z in the categories A β need not be subsheaves of I Z ! Inparticular, it is entirely possible that I Z is destabilized by a sequence0 → F → I Z → G → w = ch F has rk w ≥
2. Such destabilizing sequences, giving so-called higher rank walls , aresomewhat more troublesome to deal with. It will be helpful to bound their size, which we now do.As in the proof of Theorem 5.13, the long exact sequence of cohomology sheaves shows that anysubobject F ⊂ I Z in a category A β must actually be a sheaf (but not necessarily a subsheaf). Let IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 33 K and C be the kernel and cokernel, respectively, of the map of sheaves F → I Z , so that there isan exact sequence of sheaves 0 → K → F → I Z → C → . In order for G to be in the categories A β along the wall W = W ( w , v ) (which must be the case byProposition 5.12), it is necessary and sufficient that we have K ∈ F β and C ∈ T β for all β alongthe wall. Indeed, K and C are the cohomology sheaves of the mapping cone of the map F → I Z ,so this follows from Exercise 5.4. The sequence0 → F → I Z → G → F is additionally in T β for β along the wall. Thesebasic considerations lead to the following result. Lemma 6.2 ([ABCH13], or see [Bol +
16, Lemma 3.1 and Corollary 3.2] for a generalization toarbitrary surfaces) . If an ideal sheaf I Z of n points in P is destabilized along a wall W given by asubobject F of rank at least , then the radius ρ W of W satisfies ρ W ≤ n . Proof.
We use the notation from above. Since I Z is rank 1 and torsion-free, a nonzero map F → I Z has torsion cokernel. Therefore C is torsion, and it is no condition at all to have C ∈ T β along thewall. We further deduce that c ( C ) ≥
0, so c ( K ) ≥ c ( F ) and rk( K ) = rk( F ) −
1. Let ( s W , ρ W be the center and radius of W . Since F ∈ T β along the wall and K ∈ F β along the wall,we have2 ρ W ≤ µ ( F ) − µ ( K ) = c ( F )rk( F ) − c ( K )rk( K ) ≤ − c ( F )rk( F )(rk( F ) − ≤ − µ ( F ) ≤ − s W − ρ W , so 3 ρ W ≤ − s W . Squaring both sides, 9 ρ W ≤ s W = ρ W + 2 n by the formula for the radius. Theresult follows. (cid:3) Small examples.
We now consider the stability of ideal sheaves of small numbers of pointsin P in detail. Example 6.3 (Ideals of 2 points) . Let I Z be the ideal of a length 2 scheme Z ∈ P . Such anideal fits in an exact sequence 0 → O P ( − → I Z → O L ( − → L is the line spanned by Z . If W = W (ch O P ( − , I Z ) is the wall defined by this sequence,then Z is certainly not σ -semistable for stability conditions σ inside W . On the other hand, weclaim that I Z is σ -semistable for stability conditions σ on or above W .To see this, we rule out the possibility that I Z is destabilized along some wall W ′ which is largerthan W . The wall W has center ( s W ,
0) with s W = − / ρ W = 3 / W passes through the point ( − , W ′ is given by a rank 1 subobject I Y ( − k ) thenwe must have − k > − I Y ( − k ) to be in the categories A β along the wall W ′ . Thisthen forces k = 0, which means I Y does not define a semicircular wall. This is absurd.The other possibility is that W ′ is a higher-rank wall. But then by Lemma 6.2, W ′ has radius ρ W ′ satisfying ρ W ′ ≤ /
2. This contradicts that W ′ is larger than W .Note that in the above example, if σ is a stability condition on the wall W then I Z is strictly σ -semistable and S -equivalent to any ideal I Z ′ where Z ′ lies on the line spanned by Z . Thus theset of S -equivalence classes of σ -semistable objects is naturally identified with P ∗ . Example 6.4 (Ideals of 3 collinear points) . Let I Z be the ideal of a length 3 scheme Z ∈ P which is supported on a line. As in Example 6.3, we claim that I Z is destablized by the sequence0 → O P ( − → I Z → O L ( − → . That is, if W is the wall corresponding to the sequence, then I Z is σ -semistable for conditions σ on or above the wall. (From the existence of the sequence it is immediately clear that I Z is not σ -semistable below the wall.)We compute s W = − / ρ W = . As in Example 6.3, we conclude that there is no largerrank 1 wall. Any higher rank wall W ′ would have ρ W ′ ≤ /
2, so there can be no larger higher rankwall either. Therefore I Z is σ -semistable on and above W .For the next example we will need one additional useful fact. Proposition 6.5 ([ABCH13, Proposition 6.2]) . A line bundle O P ( − k ) or a shifted line bundle O P ( − k )[1] is σ β,α -stable whenever it is in the category A β . Thus, O P ( − k ) is σ β,α -stable if β < − k ,and O P ( − k )[1] is σ β,α -stable if β ≥ − k . In the next example we see our first example of an ideal sheaf destabilized by a higher ranksubobject.
Example 6.6 (Ideals of 3 general points) . Let I Z be the ideal of a length 3 scheme Z ∈ P whichis not supported on a line. In this case, the ideal I Z has a minimal resolution of the form0 → O P ( − → O P ( − → I Z → O P ( − → I Z → O P ( − [1] → · . Consider the wall W = W ( O P ( − , I Z ) defined by this sequence. It has center at ( s W ,
0) with s W = − /
2, and its radius is 1 /
2. By Proposition 6.5 and Exercise 5.4, the above triangle gives anexact sequence 0 → O P ( − → I Z → O P ( − [1] → A β along the wall. Then for any σ on the wall, I Z is an extension of σ -semistableobjects of the same slope, and hence is σ -semistable. It follows that I Z is destabilized preciselyalong W . Remark 6.7 (Correspondence between birational geometry and Bridgeland stability) . In[ABCH13, § σ -semistabilityof ideal sheaves I Z of up to 9 points are completely determined by similar methods. A remarkablecorrespondence between these regions of stability and the stable base locus decomposition was ob-served and conjectured to hold in general. The following result has since been proved by Li andZhao. Theorem 6.8 ([LZ16]) . Let Z ∈ P n ] . Let W be the Bridgeland wall where the ideal sheaf I Z isdestabilized. Also, let yH − B be the ray in the Mori cone past which the point Z ∈ P n ] entersthe stable base locus. Then s W = − y − . Therefore, computations in Bridgeland stability provide a dictionary between semistability andbirational geometry. Compare with Examples 4.23, 4.24, 6.3, 6.4, 6.6, which establish the cases n = 2 , P can be interpreted as a Bridgeland moduli space, and they match up the wallsin the Mori chamber decomposition of the effective cone with the walls in the wall-and-chamber IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 35 decomposition of the stability manifold. As a consequence, they are able to give new computationsof the effective, movable, and ample cones of divisors on these spaces. A crucial ingredient in thisprogram is the smoothness of these Bridgeland moduli spaces, as well as a Dr´ezet-Le Potier typeclassification of characters v for which Bridgeland moduli spaces are nonempty [LZ16, Theorems0.1 and 0.2].The next exercise computes the Gieseker wall for a Hilbert scheme of points on P . This is theeasiest case of the main problem we will discuss in the next section. Exercise 6.9.
Following Examples 6.3 and 6.4, show that the largest wall where some ideal sheaf I Z of n points is destabilized is the wall W (ch O P ( − , I Z ). Furthermore, an ideal I Z is destabilizedalong this wall if and only if Z lies on a line. Remark 6.10.
A similar program to the above has also been undertaken on some other rationalsurfaces such as Hirzebruch and del Pezzo surfaces. See [BC13].7.
The positivity lemma and nef cones
We close the survey by discussing the positivity lemma of Bayer and Macr`ı and recent applicationsof this tool to the computation of cones of nef divisors on Hilbert schemes of points and modulispaces of sheaves. This provides an example where Bridgeland stability provides insight that atpresent is not understood from a more classical point of view.7.1.
The positivity lemma.
The positivity lemma is a tool for constructing nef divisors onmoduli spaces of Bridgeland-semistable objects. On a surface, (twisted) Gieseker moduli spacescan themselves be viewed as Bridgeland moduli spaces, so this will also allow us to construct nefdivisors on classical moduli spaces. As with the construction of divisors on Gieseker moduli spaces,the starting point is to define a divisor on the base of a family of objects. When the moduli spacecarries a universal family, the family can be used to define a divisor on the moduli space.In this direction, let σ = ( Z, A ) be a stability condition on X , and let E /S be a flat familyof σ -semistable objects of character v parameterized by a proper algebraic space S . We define anumerical divisor class D σ, E ∈ N ( S ) on S depending on E and σ by specifying the intersection D σ, E .C with every curve class C ⊂ S . Let Φ E : D b ( S ) → D b ( X ) be the Fourier-Mukai transformwith kernel E , defined by Φ E ( F ) = q ∗ ( p ∗ F ⊗ E ) , where p : S × X → S and q : S × X → X are the projections and all the functors are derived. Thenwe declare D σ, E .C = ℑ (cid:18) − Z (Φ E ( O C )) Z ( v ) (cid:19) . Remark 7.1.
Note that if Z ( v ) = − D σ, E .C = ℑ ( Z (Φ E ( O C ))) . If Φ E ( O C ) ∈ A , then D σ, E .C ≥ E ( O C ) ∈ A , this fact nonetheless plays an important role in the proofof the positivity lemma.The positivity lemma states that this assignment actually defines a nef divisor on S . Furthermore,there is a simple criterion to detect the curves C meeting the divisor orthogonally. Theorem 7.2 (Positivity lemma, Theorem 4.1 [BM14a]) . The above assignment defines a well-defined numerical divisor class D σ, E on S . This divisor is nef, and a complete, integral curve C ⊂ S satisfies D σ, E .C = 0 if and only if the objects parameterized by two general points of C are S -equivalent with respect to σ . If the moduli space M σ ( v ) carries a universal family E , then Theorem 7.2 constructs a nef divisor D σ, E on the moduli space. In fact, the divisor does not depend on the choice of E ; we will see thisin the next subsection. Remark 7.3.
If multiplies of D σ, E define a morphism from S to projective space, then the curves C contracted by this morphism are characterized as the curves with D σ, E .C = 0. Thus, in a sense,all the interesting birational geometry coming from such a nef divisor D σ, E is due to S -equivalence.Unfortunately, in general, a nef divisor does not necessarily give rise to a morphism—multiples ofthe divisor do not necessarily have any sections at all. However, in such cases the positivity lemmais especially interesting. Indeed, one of the easiest ways to construct nef divisors is to pull backample divisors by a morphism (recall Examples 4.4 and 4.5). The positivity lemma can potentiallyproduce nef divisors not corresponding to any any map at all, in which case nefness is classicallymore difficult to check.7.2. Computation of divisors.
It is interesting to relate the Bayer-Macr`ı divisors D σ, E with thedeterminantal divisors on a base S arising from a family E /S . Now would be a good time to review § λ E : v ⊥ → N ( S )depending on a choice of family E /S , where v ⊥ ⊂ K num ( X ) R . Reviewing the definition of λ E , thedefinition only actually depends on the class of E ∈ K ( S × X ), so it immediately extends to thecase where E is a family of σ -semistable objects.Since the Euler pairing ( − , − ) is nondegenerate on K num ( X ) R , any linear functional on K num ( X ) R vanishing on v can be represented by a vector in v ⊥ . In particular, there is a unique vector w Z ∈ v ⊥ such that ℑ (cid:18) − Z ( w ) Z ( v ) (cid:19) = ( w Z , w )holds for all w ∈ K num ( X ) R . Note that the definition of w Z is essentially purely linear-algebraic,and makes no reference to S or E . The next result shows that the Bayer-Macr`ı divisors are alldeterminantal. Proposition 7.4 ([BM14a, Proposition 4.4]) . We have D σ, E = λ E ( w Z ) . If N is any line bundle on S , then we have D σ, E⊗ p ∗ N = λ E⊗ p ∗ N ( w Z ) = λ E ( w Z ) = D σ, E . In particular, if S is a moduli space M σ ( v ) with a universal family E , then the divisor D σ := D σ, E does not depend on the choice of universal family. Remark 7.5.
See [BM14a, §
4] for less restrictive hypotheses under which a divisor can be definedon the moduli space.In explicit cases, it can be useful to compute the character w Z in more detail. The next resultdoes this in the case of an ( H, D )-slice of divisors on a smooth surface X (review § Lemma 7.6 ([Bol +
16, Proposition 3.8]) . Let X be a smooth surface and let H, D ∈ Pic( X ) ⊗ R ,with H ample. If σ is a stability condition in the ( H, D ) -slice with center ( s W , , then the character w Z is a multiple of ( − , − K X + s W H + D, m ) ∈ v ⊥ , where we write Chern characters as (ch , ch , ch ) . Here the number m is determined by the propertythat the character is in v ⊥ . IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 37
Gieseker walls and nef cones.
For the rest of the survey we let X be a smooth surfaceand fix an ( H, D )-slice Π of stability conditions. Let v ∈ K num ( v ) be the Chern character of an( H, D )-semistable sheaf of positive rank. Additionally assume for simplicity that M H,D ( v ) hasa universal family E , so that in particular every ( H, D )-semistable sheaf is (
H, D )-stable. Recallthat the
Gieseker wall W for v in the ( H, D )-slice is, by definition, the largest wall where an(
H, D )-semistable sheaf of character v is destabilized. For conditions σ on or above W , every( H, D )-semistable sheaf is σ -semistable. Therefore, for any such σ , the universal family E is afamily of σ -semistable objects parameterized by M H,D ( v ). Each condition σ on or above the walltherefore gives a nef divisor D σ = D σ, E on the moduli space. Corollary 7.7.
With notation as above, if s ≤ s W , then the divisor on M H,D ( v ) corresponding tothe class ( − , − K X + sH + D, m ) ∈ v ⊥ under the Donaldson homomorphism is nef. Now let σ be a stability condition on the Gieseker wall. It is natural to wonder whether the“final” nef divisor D σ produced by this method is a boundary nef divisor. This may or may not bethe case. By Theorem 7.2, the divisor D σ is on the boundary of Nef( M H,D ( v )) if and only if thereis a curve in M H,D ( v ) parameterizing sheaves which are generically S -equivalent with respect tothe stability condition σ . This happens if there is some sheaf E ∈ M H,D ( v ) destabilized along W by a sequence 0 → F → E → G → ( G, F ) to obtain non-isomorphic objects E ′ .This can be subtle, and typically requires further analysis.7.4. Nef cones of Hilbert schemes of points on surfaces.
In this section we survey the recentresults of [Bol +
16] computing nef divisors on the Hilbert scheme X [ n ] of points on a smooth surfaceof irregularity q ( X ) = 0. Let v = ch I Z , where Z ∈ X [ n ] . For each pair of divisors ( H, D ) on X ,we can interpret X [ n ] as the moduli space M H,D ( v ). A stability condition σ in the ( H, D )-slice ona wall W with center ( s W ,
0) induces a divisor D σ on X [ n ] with class a multiple of12 K [ n ] X − s W H [ n ] − D [ n ] − B. The ray spanned by this class tends to the ray spanned by H [ n ] as s W → −∞ . As s W varies inthe above expression we obtain a two-dimensional cone of divisors in N ( X [ n ] ) containing the rayspanned by the nef divisor H [ n ] . The positivity lemma allows us to study the nefness of divisors inthis cone by studying the Gieseker wall for v in the ( H, D )-slice. Changing the twisting divisor D changes which two-dimensional cone we look at, and the entire nef cone of X [ n ] can be studied bythe systematic variation of the twisting divisor.The main result we discuss in this section addresses the computation of the Gieseker wall inan ( H, D )-slice, at least assuming the number n of points is sufficiently large. We find that theGieseker wall, or more precisely the subobject computing it, “stabilizes” once n is sufficiently large. Theorem 7.8 ([Bol +
16, various results from § . There is a curve C ⊂ X (depending on H, D )such that if n ≫ then the Gieseker wall for v in the ( H, D ) -slice is computed by the rank 1subobject O X ( − C ) . The intersection number C.H is minimal among all effective curves C on X .The divisor D σ corresponding to a stability condition σ on the Gieseker wall is an extremal nefdivisor. Orthogonal curves to D σ can be obtained by letting n points move in a g n on C . Note that everything here has already been verified for X = P , and in fact n ≥ O P ( − Sketch proof.
Consider the character v as varying with n . Then µ H,D ( v ) is constant, and ∆ H,D ( v )is of the form n + const. Consider the wall W ′ given by a rank 1 object I Y ( − C ) with C an effectivecurve, and put w = ch I Y ( − C ). The wall W ′ has center at ( s W ′ ,
0) with s W ′ = µ H,D ( v ) + µ H,D ( w )2 − ∆ H,D ( v ) − ∆ H,D ( w ) µ H,D ( v ) − µ H,D ( w ) . As a function of n , this looks like(2) s W ′ = − nµ H ( v ) − µ H ( w ) + const = nµ H ( w ) + const = − nC.H + const , where the constant depends on w . Correspondingly, the radius ρ W ′ grows approximately linearlyin n .Note that the numerical wall given by O X ( − C ) is always at least as large as the numerical wallgiven by I Y ( − C ), by a discriminant calculation. Furthermore, if I Y ( − C ) gives an actual wall, i.e.if there is some I Z ∈ X [ n ] fitting in a sequence0 → I Y ( − C ) → I Z → T → , then O X ( − C ) also gives an actual wall. Thus, if the Gieseker wall is computed by a rank 1 sheafthen it is computed by a line bundle O X ( − C ).In fact, for n ≫ O X ( − C ) and not by somehigher rank subobject. This is because an analog of Lemma 6.2 for arbitrary surfaces shows thatany higher rank wall for v in the ( H, D )-slice has radius squared bounded by n times a constantdepending on H, D . On the other hand, as soon as we know there is some wall given by a rank 1subobject it follows that there are walls with radius which is linear in n , implying that the Giesekerwall is not a higher rank wall.To see that there is some rank 1 wall if n ≫
0, let C be any effective curve. For some Z ∈ X [ n ] ,there is an exact sequence of sheaves0 → O X ( − C ) → I Z → I Z ⊂ C → . We know the numerical wall W ′ corresponding to the subobject O X ( − C ) has radius which growslinearly with n . In particular, for n ≫ µ H,D ( O X ( − C ))and µ H,D ( I Z ) are constant with n but ∆ H,D ( I Z ) is unbounded with n , the sheaf O X ( − C ) iseventually in some of the categories along the wall W ′ . Thus the above exact sequence of sheavesis an exact sequence along the wall, and this wall is larger than any higher rank wall. We concludethat I Z is either destabilized along W ′ or destabilized along some possibly larger rank 1 wall. Eitherway, there is a rank 1 wall, and the Gieseker wall is a rank 1 wall, computed by some line bundle O X ( − C ) with C effective.More precisely, the curve C such that the subsheaf O X ( − C ) computes the Gieseker wall for n ≫ O X ( − C ), we find that C must be an effectivecurve of minimal H -degree. Furthermore, C must be chosen to minimize the constant whichappears in that formula (this depends additionally on D ). Any such curve C which asymptoticallyminimizes Formula (2) in this way computes the Gieseker wall for n ≫
0. Curves orthogonal tothe divisor D σ given by a stability condition on the Gieseker wall can now be obtained by varyingthe extension class in the sequence0 → O X ( − C ) → I Z → I Z ⊂ C → Z move in a pencil on C , which can certainly be done for n ≫ (cid:3) More care is taken in [Bol +
16] to determine the precise bounds on n which are necessary for thevarious steps of the proof. The general method is applied to compute nef cones of Hilbert schemes IRATIONAL GEOMETRY OF MODULI SPACES OF SHEAVES AND BRIDGELAND STABILITY 39 of sufficiently many points on very general surfaces in P , very general double covers of P , anddel Pezzo surfaces of degree 1. The last example provides an example of a surface of higher Picardrank, where the variation of the twisting divisor is exploited. See [Bol + § Problem 7.9.
Let X ⊂ P be a very general quintic surface, so that the Picard rank is by theNoether-Lefschetz theorem. Compute the nef cone of X [2] and X [3] . Once n ≥ +
16, Proposition 4.5].7.5.
Nef cones of moduli spaces of sheaves on surfaces.
We close our discussion with a surveyof the main result of [CH16b] on the cone of nef divisors on a moduli space of sheaves with largediscriminant on an arbitrary smooth surface. In the case of P , this result was first discovered in thepapers [CH16a, LZ16]. The picture for an arbitrary surface is a modest simultaneous generalizationof the P case as well as the Hilbert scheme case for an arbitrary surface (see § + X be a smooth surface and let H, D be divisors giving a slice of stability conditions.Let v be the character of an ( H, D )-semistable sheaf of positive rank. We assume the discriminant∆
H,D ( v ) ≫ M H,D ( v ) carries a (quasi-)universalfamily. The goal of [CH16b] is to compute the Gieseker wall for v in the ( H, D )-slice and to showthat the divisor D σ corresponding to a stability condition σ on the Gieseker wall is a boundary nefdivisor.The basic picture is similar to the case of a Hilbert scheme of points, and indeed Theorem 7.8will follow as a special case of this more general result. However, the asymptotics can easily bemade much more explicit in the Hilbert scheme case. The common thread between the two resultsis that as the discriminant ∆ H,D ( v ) is increased, the character w of a destabilizing subobject givingrise to the Gieseker wall stabilizes. It is furthermore easy to give properties which almost uniquelydefine the character w . Definition 7.10.
Fix an (
H, D )-slice. An extremal Chern character w for v is any charactersatisfying the following defining properties.(E1) We have 0 < r ( w ) ≤ r ( v ), and if r ( w ) = r ( v ), then c ( v ) − c ( w ) is effective.(E2) We have µ H ( w ) < µ H ( v ), and µ H ( w ) is as close to µ H ( v ) as possible subject to (E1).(E3) The moduli space M H,D ( w ) is nonempty.(E4) The discriminant ∆ H,D ( w ) is as small as possible, subject to (E1)-(E3).(E5) The rank r ( w ) is as large as possible, subject to (E1)-(E4).Note that properties (E1)-(E4) uniquely determine the slope µ H ( w ) and discriminant ∆ H,D ( w ),although c ( w ) is not necessarily uniquely determined. Condition (E5) uniquely specifies the rankof w . We then have the following theorem. Furthermore, notice that the definition does not dependon the discriminant ∆ H,D ( v ), so that w can be held constant as ∆ H,D ( v ) varies. Theorem 7.11 ([CH16b]) . Suppose ∆ H,D ( v ) ≫ . Then the Gieseker wall for v in the ( H, D ) -sliceis computed by a destabilizing subobject of character w , where w is an extremal Chern characterfor v . Furthermore, the divisor D σ corresponding to a stability condition σ on the Gieseker wall isa boundary nef divisor. The argument is largely similar to the proof of Theorem 7.8. First one shows that the destabilizingsubobject along the Gieseker wall must actually be a subsheaf, and not some higher rank object.This justifies restriction (E1) in the definition of w (note that if r ( w ) = r ( v ) then the only waythere can be an injection of sheaves F → E with ch F = w and ch E = v is if the induced mapdet F → det E is injective, forcing c ( v ) − c ( w ) to be effective. Next, one shows that the subsheaf defining the Gieseker wall must actually be an (
H, D )-semistable sheaf. Recalling the formula s W = µ H,D ( v ) + µ H,D ( w )2 − ∆ H,D ( v ) − ∆ H,D ( w ) µ H,D ( v ) − µ H,D ( w )for the center of a wall, conditions (E2)-(E4) then ensure that the numerical wall defined by w isas large as possible when ∆ H,D ( v ) ≫
0. Therefore, the Gieseker wall for v is no larger than thewall defined by the extremal character w . Remark 7.12.
Actually computing the extremal character w can be extremely challenging. Mini-mizing the discriminant of w subject to the condition that the moduli space M H,D ( w ) is nonemptyessentially requires knowing the sharpest possible Bogomolov inequalities for semistable sheaves on X . Conversely, if the nef cones of moduli spaces of sheaves on X are known, strong Bogomolov-typeinequalities can be deduced. On surfaces such as P and K § § v defined by an extremal character w is an actual wall. In the Hilbert schemecase, it is trivial to produce ideal sheaves I Z which are destabilized by a rank 1 object O X ( − C ):we simply put Z on C , and get an exact sequence0 → O X ( − C ) → I Z → I Z ⊂ C → H, D )-semistable sheaves E of character v fitting in sequences of the form 0 → F → E → G → F is ( H, D )-semistable of character w . This is somewhat technical. Let u = ch G ; then u has r ( u ) < r ( v ), and ∆ H,D ( u ) ≫
0. Therefore, by induction on the rank, we may assume theGieseker wall of u has been computed. We then show that the Gieseker walls for w and u are nestedinside W := W ( v , w ) if ∆ H,D ( v ) ≫
0. Therefore, any sheaves F ∈ M H,D ( w ) and G ∈ M H,D ( u )are actually σ -semistable for any stability condition σ on W . Then any extension E of G by F is σ -semistable, and it can further be shown that a general such extension is actually ( H, D )-stable.By varying the extension class, we can produce curves in M H,D ( v ) parameterizing non-isomorphic( H, D )-stable sheaves; these curves are orthogonal to the nef divisor given by the Gieseker wall.See [CH16b, § Remark 7.13.
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