Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles
aa r X i v : . [ m a t h . AG ] O c t EFFECTIVE DIVISORS ON THE HILBERT SCHEME OF POINTS IN THE PLANEAND INTERPOLATION FOR STABLE BUNDLES
JACK HUIZENGA
Abstract.
We compute the cone of effective divisors on the Hilbert scheme of n points in the projectiveplane. We show the sections of many stable vector bundles satisfy a natural interpolation condition, andthat these bundles always give rise to the edge of the effective cone of the Hilbert scheme. To do this, wegive a generalization of Gaeta’s theorem on the resolution of the ideal sheaf of a general collection of n points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and weobserve that ideal sheaves of collections of points are destabilized by exceptional bundles. By studying theBridgeland stability of exceptional bundles, we also show that our computation of the effective cone of theHilbert scheme is consistent with a conjecture in [ABCH] which predicts a correspondence between Moriand Bridgeland walls for the Hilbert scheme. Contents
1. Introduction 12. Preliminaries 53. Number-theoretic properties of exceptional slopes 104. The associated exceptional bundle 135. Resolution of the ideal sheaf of n points 166. Orthogonality of Kronecker modules 267. The effective cone of the Hilbert scheme of points 288. Connections with Bridgeland stability 319. Bridgeland stability of exceptional bundles 36References 411. Introduction
For a projective variety X , the Hilbert scheme X [ n ] parameterizes length n zero-dimensional sub-schemes of X . When X is a smooth surface, the Hilbert scheme is a useful compactification of the opensymmetric product ( X n \ ∆) /S n parameterizing distinct collections of n points. By a result of Fogarty, inthis case X [ n ] is a smooth projective variety of dimension 2 n , and X [ n ] → X n /S n resolves the singularitiesof the ordinary symmetric product [F1]. Date : December 7, 2018.2010
Mathematics Subject Classification.
Primary: 14C05. Secondary: 14E30, 14J60, 13D02.This material is based upon work supported under a National Science Foundation Graduate Research Fellowship and aNational Science Foundation Mathematical Sciences Postdoctoral Research Fellowship. n interesting topic in birational geometry is to describe the various birational models of moduli orparameter spaces. It is often the case that these models themselves admit interesting modular interpreta-tions, and describing these alternate compactifications is of particular interest. In [ABCH], the problemof carrying out the minimal model program for the Hilbert scheme P n ] is discussed.A first step in carrying out the minimal model program for the Hilbert scheme P n ] is to describethe full cone of effective divisors. In this paper, we compute the cone for every n , building on resultsfrom [H2] where a partial answer was obtained. Along the way, we will be led to consider naturalinterpolation questions for stable vector bundles. We will also develop a generalization of Gaeta’s theoremon resolutions of ideal sheaves of general collections of points in P . This generalization will illuminatemany cohomological properties of such ideal sheaves.The Picard group of P n ] has rank 2, and is generated over Z by classes H and ∆ /
2, where H is thelocus of schemes meeting a fixed line and ∆ is the locus of nonreduced schemes [F2]. The boundarydivisor ∆ always spans one edge of the cone Eff P n ] of effective divisors ([ABCH, H2, H3]). For manyvalues of n it is easy describe the other edge of this cone. For instance, if n = (cid:0) r +22 (cid:1) is a triangularnumber, then there is a divisor given as the locus of schemes which lie on some curve of degree r , and thisdivisor spans the edge of the cone. Alternately, the divisor is described as the locus of schemes which failto impose independent conditions on sections of the line bundle O P ( r ). Generalizing this constructionto allow more arbitrary vector bundles instead of only line bundles allows us to construct the nontrivialedge of Eff P n ] for every n .1.1. Interpolation for vector bundles.
Let E be a vector bundle of rank r on a smooth curve orsurface X . We say E satisfies interpolation for n points if a general collection Z of n points imposes rn conditions on sections of E , i.e. if h ( E ⊗ I Z ) = h ( E ) − rn. This forces h ( E ) ≥ rn ; let W ⊂ H ( E ) be a general subspace of dimension rn . Then the locus of Z ∈ X [ n ] which fail to impose independent conditions on sections in W forms an effective divisor D E ( n )in X [ n ] .In the particular case where X = P , one sees by an elementary Grothendieck-Riemann-Roch calcula-tion (see [ABCH, H2, H3]) that if E is a vector bundle which satisfies interpolation for n points then thedivisor D E ( n ) has class [ D E ( n )] = c ( E ) − rk( E ) ∆2 . If we let µ ( E ) = c ( E ) / rk( E ) be the slope , then this class spans the ray µ ( E ) −
12 ∆ . We are thus led to determine the minimum possible slope µ of a vector bundle satisfying interpolationfor n points. We will see that the extremal edge of the effective cone can always be realized as a divisorassociated to a vector bundle in this way. Furthermore, the minimum slope µ and vector bundles of slope µ with interpolation can be explicitly described.1.2. Stable vector bundles and interpolation.
The key to determining the minimum slope µ of avector bundle with interpolation for n points lies in considering stable vector bundles. A vector bundle E is (slope)-stable if every coherent subsheaf F ⊂ E with 0 < rk F < rk E has µ ( F ) < µ ( E ). Our generalexpectation is that a stable bundle E of rank r typically satisfies interpolation for n points so long as ithas at least rn sections. The number of sections of a general stable bundle E of slope µ ( E ) ≥ χ = χ ( E ) by [GH], so it is natural to try and determine the minimum possible slope of a stable bundlewith the property χ ≥ rn . A priori it could happen that no such minimum exists, since the infimum ofthe slopes of such bundles could be irrational. This, however, is not the case. heorem 1.1. For a fixed nonnegative integer n , the set of nonnegative slopes of stable bundles on P satisfying χ/r ≥ n has a minimum µ . The number µ can be explicitly computed for any given n , as we will see in Section 4. The maindifficulty is understanding when a given moduli space M ( r, c, d ) of stable vector bundles with Cherncharacter ( r, c, d ) is nonempty. The answer to this question is well-known, but depends intimately on thegeometry of the exceptional bundles on P ; these are the rigid stable bundles, i.e. the stable bundles E with Ext ( E, E ) = 0.It is particularly important to understand the set E of slopes of exceptional bundles on P . By resultsof Drezet and Le Potier [Dr2, DLP, LP], for any rational number µ ∈ Q there is an associated exceptionalslope α ∈ E such that the existence of stable vector bundles with slope µ is controlled by the (unique)exceptional bundle E α of slope α . Theorem 1.1 follows from a study of the number theory of exceptionalslopes. These slopes have many interesting properties; we highlight one result in this direction here. Theorem 1.2.
Let α ∈ E be the slope of an exceptional bundle on P . Every term in the even-lengthcontinued fraction expansion of the fractional part of α is a one or a two. Furthermore, these terms forma palindrome. A generalization of Gaeta’s theorem.
Let us recall Gaeta’s theorem on the resolution of theideal sheaf of n general points in P (see [E]). Let Z be a general collection of n points in P , and write n = r ( r + 1)2 + s (0 ≤ s ≤ r );there is a unique such decomposition. Then the ideal sheaf I Z admits one of the following two resolutions,depending on whether 2 s ≤ r or 2 s ≥ r :0 → O P ( − r − r − s ⊕ O P ( − r − s → O P ( − r ) r − s +1 → I Z → → O P ( − r − s → O P ( − r ) r − s +1 ⊕ O P ( − r − s − r → I Z → . Let us focus on the case where 2 s ≥ r ; a similar picture applies in the other case. We have r − s + 1 =dim Hom( O P ( − r ) , I Z ), so define a sheaf W to be the kernel of the canonical map O P ( − r ) ⊗ Hom( O P ( − r ) , I Z ) → I Z . Assuming the canonical map is surjective (which it is if say s ≤ r − W will be a vector bundle withresolution 0 → W → O P ( − r − s → O P ( − r − s − r → , where the map is the same one as in the resolution of I Z (and hence is general).The resolution 0 → W → O P ( − r ) ⊗ Hom( O P ( − r ) , I Z ) → I Z → W is stable ; however, W will be stable if and only if either ϕ − < s/r ≤ s/r is a convergent in the continued fraction expansion of ϕ − , where ϕ = (1 + √ / W is not stable, this resolution is somewhat unsatisfactory. In this case the terms of theGaeta resolution do not optimally reflect cohomological properties of the ideal sheaf I Z ; in particular, itmay very well happen that V ⊗ I Z has no cohomology for some vector bundle V , but that V ⊗ W and V ( − r ) have cohomology. If one wishes to use the Gaeta resolution to prove V ⊗ I Z has no cohomology, itbecomes necessary to analyze the maps in the resolution, and things become unwieldy. Our generalizationof Gaeta’s theorem will resolve the ideal sheaf into a pair of semistable bundles which are much moresuitable for such computations. heorem 1.3. Let µ be the minimum slope of a stable bundle on P satisfying χ ( E ) / rk( E ) = n . Let α ∈ E be the exceptional slope associated to µ . If µ < α , the general ideal sheaf I Z of n points admits acanonical resolution → W → E − α ⊗ Hom( E − α , I Z ) → I Z → where W is a stable bundle and E − α is the exceptional bundle of slope − α . Similarly, if µ > α , thegeneral ideal sheaf admits a canonical resolution → E − α − ⊗ Ext ( I Z , E − α − ) ∗ → W → I Z → , where W is a stable bundle. In each case, a resolution of W by semi-exceptional bundles can be explicitlydescribed. (For some sporadic values of n , W is actually an object of a derived category; see Theorem5.9 for a precise statement.) We also refer the reader to Theorem 5.9 for a statement in case µ = α . We note that the Gaetaresolution is recovered as the special case where the exceptional slope α is an integer.1.4. The effective cone of P n ] . By combining Theorem 1.3 with the main result from [H2] we canconstruct the extremal edge of the effective cone of P n ] . Theorem 1.4.
Let µ be the minimum slope of a stable bundle on P with χ/r = n . A general suchbundle V with sufficiently large and divisible rank satisfies interpolation for n points. Thus µH − ∆ isthe class of an effective divisor on P n ] . Furthermore, the effective cone of P n ] is spanned by µH −
12 ∆ and ∆ . Given the resolution of the ideal sheaf I Z , showing that such bundles V satisfy interpolation amountsto a previously studied problem about orthogonality of representations of the Kronecker quiver with twovertices and N arrows. We use results of Schofield and van den Bergh to prove interpolation holds.The computational value of the generalized Gaeta resolution is demonstrated by the fact that if say µ < α (as in Theorem 1.3) and V is a bundle as in Theorem 1.4, then both V ⊗ W and V ⊗ E − α turnout to have no cohomology. Thus there is no need to understand the map W → E − α ⊗ Hom( E − α , I Z ) toshow V ⊗ I Z has no cohomology. Such complication was unavoidable with the original Gaeta resolution.To show that the divisor µH − ∆ is actually extremal, we will study the rational map P n ] M (ch( W )) sending the general scheme Z to the bundle W in the resolution of I Z (or, more precisely,we study the map to a related moduli space of quiver representations). This map typically has positive-dimensional fibers, and our extremal divisors on the Hilbert scheme are pullbacks under this map.Since it is a bit tedious to determine the exact value of µ in the theorem by hand, we give a tabledescribing the effective cone and associated exceptional slopes for small n at the end of Section 7.1.5. Bridgeland stability.
The generalized Gaeta resolution has further relevance when one discussesthe Bridgeland stability of ideal sheaves I Z . In Sections 8 and 9 we will show that our computation ofthe effective cone Eff P n ] is consistent with a conjecture in [ABCH] predicting a correspondence betweenthe Mori walls for P n ] and the Bridgeland walls in a suitable half-plane of stability conditions. We willsee that our resolution shows that general ideal sheaves are always destabilized by certain exceptionalbundles. The main step in the proof consists of determining when exceptional bundles are Bridgelandstable. We give a fairly complete answer to this question in Section 9.1.6. Further work.
It appears that many of the results in this paper can be generalized to study conesof divisors on moduli spaces of semistable sheaves on the plane. The Picard group of a moduli space M ( ξ ) of semistable sheaves is naturally identified with a plane of orthogonal Chern characters [LP]. Interms of this description, the effective cone should correspond to Chern characters of stable orthogonalbundles. We will study this problem in upcoming work with Izzet Coskun and Matthew Woolf. .7. Acknowledgements.
I would like to thank Joe Harris, Izzet Coskun, and Daniele Arcara for theirmany helpful discussions regarding this work. Also, I am indebted to the anonymous referee of [H2],whose suggestions pointed me towards the methods used in this paper. The referees of this article alsoprovided very valuable advice, helping to greatly simplify the material from Sections 6 and 7. I wouldalso like to thank So Okada and RIMS Kyoto for organizing a very useful conference on related topics.Finally, I would like to thank the community of MathOverflow for their help with finding references forbasic facts regarding continued fractions. 2.
Preliminaries
In this section we set notation for the paper and review parts of the classification of stable vectorbundles on P that will be necessary throughout the paper. We predominantly choose notations to agreewith the papers of Drezet and Le Potier [Dr1, Dr2, DLP, LP], and summarize results from those sources.2.1. Invariants of coherent sheaves.
We collect here several formulas which will be used constantlythroughout the paper. Let E be a coherent sheaf on P , with Chern character (ch , ch , ch ) = ( r, c , ch ).When r >
0, the slope and discriminant are defined by µ ( E ) = c r and ∆( E ) = 12 µ − ch r , respectively. The Riemann-Roch formula relates the Chern character to the Euler characteristic by χ ( E ) = r ( P ( µ ) − ∆) , where P ( x ) = 12 ( x + 3 x + 2)is the Hilbert polynomial of the trivial sheaf O P .If F is another coherent sheaf, we put χ ( E, F ) = X i =0 ( − i dim Ext i ( E, F ) . In case both E and F have positive rank, a variant of Riemann-Roch shows χ ( E, F ) = r ( E ) r ( F )( P ( µ ( F ) − µ ( E )) − ∆( E ) − ∆( F )) . Finally, Serre duality for Ext-groups givesExt i ( E, F ) ∼ = Ext − i ( F, E ( − ∗ for each i [DLP, Proposition 1.2].We say that a sheaf E is acyclic if H i ( E ) = 0 for all i >
0. In practice, we will only consider thenotion of acyclicity for sheaves E with χ ( E ) = 0, in which case H ( E ) = 0 as well.2.2. Exceptional bundles.
The sources [DLP, LP] are good references for the material in this subsec-tion. A coherent sheaf E is said to be stable (resp. semi-stable ) if it is torsion free and every coherentsubsheaf F ⊂ E with 0 < r ( F ) < r ( E ) has µ ( F ) ≤ µ ( E ) with ∆( E ) < ∆( F ) (resp. ≤ ) in case of equality.For fixed values of the Chern character ch = (ch , ch , ch ), we denote by M (ch) the moduli space ofsemistable sheaves with ch( E ) = ch.The invariant ∆ is useful due to its connection with stable bundles. Bogomolov’s theorem shows that∆( E ) ≥ E . Furthermore, when the moduli space M (ch) is nonempty, it is irreducibleof dimension r (2∆ −
1) + 1. In particular, if M (ch) consists of a single point, then ∆ < / exceptional bundle E is a stable coherent sheaf such that M (ch( E )) is reduced to a point (it followsfrom this that E is homogeneous, hence locally free). Equivalently, it is a stable bundle with ∆( E ) < / r a rigid stable bundle (i.e. a stable bundle with Ext ( E, E ) = 0). A semi-exceptional bundle is a bundleof the form E k , with E exceptional.For any rational number α ∈ Q , denote by r α the denominator of α , i.e. the smallest positive integer r > rα ∈ Z . If there exists an exceptional bundle E α of slope α , then it is unique and its invariantsare given byrk( E α ) = r α c ( E α ) = αr α ∆ α := ∆( E α ) = 12 (cid:18) − r α (cid:19) χ α := χ ( E α ) = r α ( P ( α ) − ∆ α ) . Note that E ∗ α = E − α , and also E α (1) = E α +1 .Exceptional bundles play an important role in the problem of determining when the moduli spaces M (ch) are nonempty. In particular, it is necessary to understand the set E of slopes of exceptionalbundles.Clearly E is invariant under translation α α +1 and inversion α
7→ − α . If α, β ∈ E and 3+ α − β = 0,we define a rational number α.β = α + β β − ∆ α α − β , which should be thought of as a modification of the mean of α and β (note that there is a typo in [Dr2],and that ∆ β and ∆ α are reversed there). Let D = Z [ ] be the set of dyadic rational numbers. There isa bijection ε : D → E described inductively by setting ε ( n ) = n for n ∈ Z and ε (cid:18) p + 12 q (cid:19) = ε (cid:18) p q − (cid:19) .ε (cid:18) p + 12 q − (cid:19) . It is useful to keep several values of ε where q is small in mind, so we record them here. p q
18 14 38 12 58 34 78 ε (cid:0) p q (cid:1)
513 25 1229 12 1729 35 813
Lemma 2.1.
Suppose α = ε (cid:18) p q (cid:19) β = ε (cid:18) p + 12 q (cid:19) . Then (1) α < α.β < β , (2) r α.β = r α r β (3 − α + β ) , and (3) P ( α − β ) = ∆ α + ∆ β .Furthermore, the relations α.β − α = 1 r α (3 + α − β ) and β − α.β = 1 r β (3 + α − β ) hold. The “furthermore” part of the lemma is an elementary consequence of the previous properties. Mostproperties of exceptional slopes are more efficiently proved by using the identities in the lemma insteadof invoking the explicit definition of α.β . .3. Existence of stable coherent sheaves on P . For any α ∈ E , define a number x α = 32 − s − r α , which is the smaller of the two solutions of the equation P ( − x ) − ∆ α = . The number x α is alwaysirrational. We denote by I α ⊂ R the interval I α = ( α − x α , α + x α ) . The intervals I α are all disjoint, and they cover the rationals: Q = Q ∩ [ α ∈ E I α . If µ ∈ Q , then the unique slope α ∈ E with µ ∈ I α is called the associated exceptional slope to µ . Theorem 2.2 (Drezet [Dr2]) . Suppose r ≥ is an integer, and µ, ∆ ∈ Q are numbers such that rµ and r ( P ( µ ) − ∆) are integers. Define a function δ : Q → Q by the formula δ ( µ ) = P ( −| µ − α | ) − ∆ α if µ ∈ I α The moduli space M ( r, µ, ∆) of semistable sheaves with invariants ( r, µ, ∆) is nonempty if and only ifeither δ ( µ ) ≤ ∆ or ( r, µ, ∆) are the invariants of some semi-exceptional bundle. Write C = R \ S α ∈ E I α . We can view C as a generalized Cantor set, obtained by iteratively removingfrom R at step q all intervals I α where α is of the form ε ( p/ q ). It is easy to see from what has been saidso far that C is the closure of all the endpoints of the intervals I α (just as is true for the ordinary Cantorset). Remark 2.3.
As with the standard Cantor set, C is uncountable and most of its points are not endpointsof the intervals I α . This fact is a source of much technical difficulty.While the next result is well-known, the argument is fundamental to our discussion, so we include it. Proposition 2.4.
The function δ : Q → Q admits a unique continuous extension to a function R → R ,and δ − (1 /
2) = C .Proof. We can define δ on each interval I α by the formula δ ( µ ) = P ( −| µ − α | ) − ∆ α , so it is clear that this extension of δ is continuous everywhere except the points in C , where it has notyet been defined. Noting that lim µ → ( α + x α ) − δ ( µ ) = lim µ → ( α − x α ) + δ ( µ ) = 12by the definition of x α , we see that any continuous extension of δ to R must satisfy δ ( ξ ) = 1 / ξ ∈ C . Thus we define δ ( ξ ) = 1 / ξ ∈ C . We also observe that δ ( µ ) > / µ ∈ R \ C since δ is increasing on each interval ( α − x α , α ] and decreasing on each interval [ α, α + x α ). We must showcontinuity holds at ξ ∈ C .If ξ ∈ C is of the form α + x α , then clearly δ is left-continuous at ξ . Similarly, if ξ is of the form β − x β ,it is right-continuous there. Without loss of generality, suppose ξ is not of the form α + x α ; we show δ isleft-continuous at ξ . Since ξ is in C but not of the form α + x α , it is an increasing limit of exceptional lopes. If α ∈ E is any exceptional slope with α < ξ , then the maximum value of δ on the interval I α occurs at α , and equals δ ( α ) = 12 + 12 r α . For any ǫ >
0, we can choose an exceptional slope α < ξ sufficiently close to ξ such that all rationals µ ∈ [ α, ξ ) satisfy (2 r µ ) − < ǫ ; then for all x ∈ ( α, ξ ) we will have | δ ( ξ ) − δ ( x ) | < ǫ . (cid:3) Triads; resolutions of height 0 stable sheaves on P . A triad is a triple ( E, G, F ) of exceptionalbundles such that the slopes ( µ ( E ) , µ ( G ) , µ ( F )) are of the form ( α, α.β, β ), ( β − , α, α.β ), or ( α.β, β, α +3),where α, β are exceptional slopes of the form α = ε (cid:18) p q (cid:19) β = ε (cid:18) p + 12 q (cid:19) for some p, q (possibly with q = −
1, so that e.g. ( O P , O P (1) , O P (2)) is a triad). Any exceptional slopecan be written in the form α.β , so any exceptional bundle can be viewed as the bundle of slope α.β inany of the three types of triads. The results from the first part of this subsection can be found in [Dr1].For any triad ( E, G, F ), the canonical mapev ∗ : G → F ⊗ Hom(
G, F ) ∗ is injective, and the cokernel is an exceptional bundle S (for a discussion of which exceptional bundle S is, see either [Dr1] or Theorem 9.3 in this paper). On the other hand, the mapev : E ⊗ Hom(
E, G ) → G is surjective, with kernel S ( − V on P , there is a canonical complex E ⊗ Ext ( V, E ) ∗ A V → G ⊗ Ext ( S, V ) B V → F ⊗ Ext ( F, V )coming from a generalized version of the Beilinson spectral sequence. If Hom(
F, V ) = Hom(
V, E ) = 0,then the map A V is injective, the map B V is surjective, and the middle cohomology is just V .Many numerical invariants of pairs of members of a triad are easily computed, in light of the followingvanishing theorem. Theorem 2.5 (Drezet [Dr1, Theorem 6]) . If E, F are any exceptional bundles with µ ( E ) ≤ µ ( F ) , then Ext i ( E, F ) = 0 for i > . One then easily concludes the following facts by computing Euler characteristics: F ∗ ⊗ E, F ∗ ⊗ G, and G ∗ ⊗ E are acyclicdim Hom( E, G ) = 3 rk( F ) dim Hom( G, F ) = 3 rk( E ) rk( S ) = 3 rk( E ) rk( F ) − rk( G )Let V be a stable sheaf with invariants ( r, µ, ∆), and let α ∈ E be the exceptional slope associated to µ . The height of V is defined to be the integer h ( V ) = rr α (∆ − δ ( µ )) . In case µ ≤ α , this is just the number − χ ( E α , V ); similarly, in case µ ≥ α it equals − χ ( V, E α ).In the case where the height is zero, the above complex degenerates considerably, as discussed in [Dr2].To see this, suppose V has height zero, and first assume α − x α < µ ≤ α . Choose a triad ( E, G, F ) with F = E α . We have inequalities of slopes µ ( E ) < µ ( G ) < µ ( V ) ≤ µ ( F ) . he height zero hypothesis gives χ ( F, V ) = 0. Stability and the fact that V is non-exceptional givesHom( F, V ) = 0 and Ext ( F, V ) = 0 (by Serre duality). Thus also Ext ( F, V ) = 0. Stability also givesHom(
V, E ) = 0, so we conclude that the complex gives an exact sequence0 → E ⊗ Ext ( V, E ) ∗ → G ⊗ Ext ( S, V ) → V → . If we write this resolution in the form 0 → E m → G m → V → , then the hypothesis that α − x α < µ ≤ α is equivalent to the inequalities(1) r α x α < m m ≤ r α rk S .
In case α ≤ µ < α − x α , we choose a triad ( E, G, F ) with E = E α , and an identical argument gives anexact sequence 0 → V → G ⊗ Ext ( S, V ) → F ⊗ Ext ( F, V ) → . This time, writing the resolution in the form0 → V → G m → F m → S has changed because we are using adifferent triad.2.5. A Bertini-type statement.
Throughout the paper, the following setup will occur several times.Suppose E , F are vector bundles of ranks m, n on a smooth variety X and the sheaf H om ( E, F ) isglobally generated. For a map φ : E → F , denote by D k ( φ ) the degeneracy locus { x ∈ X : rk φ x ≤ k } . Proposition 2.6.
With the preceding setup, if φ is general then D k ( φ ) is empty or has the expectedcodimension ( m − k )( n − k ) . Furthermore, in case the general D k ( φ ) is nonempty, the locus of φ ∈ Hom(
E, F ) where D k ( φ ) has greater than the expected dimension is at least of codimension .Proof. We quickly sketch the argument, which is just an analysis of the proof of [O, Theorem 2.8]. Byglobal generation, we have a surjection H ( E ∗ ⊗ F ) ⊗ O X → E ∗ ⊗ F → , which shows that the natural evaluation mapev : X × P H ( E ∗ ⊗ F ) → P ( E ∗ ⊗ F )is surjective and has fibers isomorphic to P h ( E ∗ ⊗ F ) − mn . There is a subvariety Σ k ⊂ P ( E ∗ ⊗ F ) consistingof those points φ x : E x → F x such that rk( φ x ) ≤ k , and it is irreducible of codimension ( m − k )( n − k ).Then Z = ev − (Σ k ) is an irreducible variety of dimension h ( E ∗ ⊗ F ) − ( m − k )( n − k ) − . If the projection q : Z → P H ( E ∗ ⊗ F ) is surjective, then the general fiber has dimension dim X − ( m − k )( n − k ), so D k ( φ ) has codimension ( m − k )( n − k ). Furthermore, in this case the dimension of the fibers of q cannotjump in codimension 1, as this would violate the irreducibility of Z . Alternately, if q is not surjectivethen D k ( φ ) is empty for general φ . (cid:3) . Number-theoretic properties of exceptional slopes
The exceptional slopes α ∈ E have many surprising number-theoretic properties. These will beof utmost importance in proving that the set of nonnegative slopes of stable bundles V such that χ ( V ) / rk( V ) ≥ q, for q ∈ Q ≥ a fixed nonnegative rational, has a minimum.The main goal of this section is to describe nice properties of the continued fraction expansion of any α ∈ E . To do this, we essentially give an algorithm which computes the continued fraction expansion of α in terms of the binary expansion of the dyadic number p/ q with ε ( p/ q ) = α .Since the set E of exceptional slopes is invariant under translation by 1, it will suffice to consider onlythe case where 0 ≤ α <
1. For any real numbers a , . . . , a k for which it makes sense, define the number[ a ; a , . . . , a k ] := a + 1 a + 1. . . + 1 a k Recall that any rational number 0 ≤ α < α = [0; a , . . . , a k ]where the a i are positive integers and k is even. Indeed, if k is odd with a k = 1 then we can write α = [0; a , . . . , a k − + 1]; on the other hand if k is odd and a k > α = [0; a , . . . , a k − , . Following standard notation, we let p n and q n be the numerator and denominator of the rationalnumber [0; a , . . . , a n ], called the n th convergent of α . With this notation, α = p k /q k . The fundamentalrelation between convergents is encapsulated by the equality of matrices q n q n − p n p n − ! = a
11 0 ! a
11 0 ! · · · a n
11 0 ! . It is immediate from computing determinants that q n p n − − q n − p n = ( − n .We say that the continued fraction expansion of 0 ≤ α < palindromic if the word a , a , . . . , a k is apalindrome, i.e. if a i = a k +1 − i for each i . Taking transposes of the above equality of matrices and usingthe uniqueness of continued fraction expansions of a given length, we recover the following well-knownfact. Lemma 3.1.
A continued fraction expansion [0; a , . . . , a k ] for the number α is palindromic if and onlyif p k = q k − . That is, the denominator of the penultimate convergent equals the numerator of α . With preliminaries out of the way, we are now ready to state and prove our main result on the continuedfraction expansion of an exceptional slope α ∈ E . Theorem 3.2.
Let ≤ α < be an exceptional slope. The unique continued fraction expansion α =[0; a , . . . , a k ] with k even is palindromic, and every a i is either or . Furthermore, (1) every block of ones in the word a , . . . , a k has even length, and (2) every block of twos in the word a , . . . , a k − has even length.Proof. The theorem is clearly true for α = 0. Any exceptional slope in the interval (0 ,
1) can be writtenuniquely in the form α.β , where α = ε (cid:18) p q (cid:19) β = ε (cid:18) p + 12 q (cid:19) See [Sa] for another argument. or integers p, q with 0 ≤ p ≤ q − q ≥
0. We wish to induct on q . There is a slight difficulty, inthat perhaps β = 1, where the integer part of the even length continued fraction expansion is not 0. Tocircumvent this, we simply note that for every k ≥ ε (cid:16) − − k (cid:17) = F k F k +1 = [0; 1 , . . . , | {z } k copies ] , where F = 0, F = 1, F n +2 = F n +1 + F n is the Fibonacci sequence. Thus the theorem is true for allslopes of the form ε (1 − − k ) . For any α.β not of the form ε (1 − − k ) we will have β <
1, and thus wemay assume by induction that the theorem holds for α and β .We now describe how to compute the continued fraction expansion of α.β when the theorem holds for α = [0; a , . . . , a k ] and β = [0; b , . . . , b l ] (where k, l are even so that the expansions are palindromic).First suppose that β is of the form β = [0; b , b , . . . , b l − , . Consider the continued fraction [0; b , b , . . . , b l − , , , , a , . . . , a k ] . Visibly every term is a 1 or a 2, and one easily checks the hypotheses on the lengths of blocks of onesand twos are satisfied by this new word. The length of the word is also even. Furthermore, the equality[1; 1 , x ] = [2; − (3 + x )] , valid for any real x = − , −
3, shows that this continued fraction equals[0; b , . . . , b l − , , − (3 + α )] = [0; b , . . . , b l , − (3 + α )] . We’ll show in a minute that this number is precisely α.β . Before that, we handle the other possible formof β . If instead β = [0; b , . . . , b l − , , b , . . . , b l − , , , a , . . . , a k ] , again easily verifying that the condition on the parity of lengths of blocks is satisfied. Here the equality[2; 2 + x ] = [1 , , − (3 + x )] , valid again for real x = − , −
3, shows that this fraction equals[0; b , . . . , b l − , , , − (3 + α )] = [0; b , . . . , b l , − (3 + α )] . Thus, in either case we must show α.β = [0; b , . . . , b l , − (3 + α )] . Proving this relies on the palindromic property of the continued fraction expansion β = [0; b , . . . , b l ].Writing p i /q i for the convergents of [0; b , . . . , b l , − (3 + α )] , we have the relation p l +1 q l +1 = − (3 + α ) p l + p l − − (3 + α ) q l + q l − , and we must show this number equals α.β . From the palindromic property of β , we get q l − = p l . Writingeverything in terms of β and r β , we thus have p l = βr β q l = r β q l − = βr β p l − = 1 r β + β r β , where p l − is determined by the relation p l − q l − p l q l − = 1, recalling that l is even. Making thesubstitutions, basic algebra (using no special properties of α, β ) shows β − p l +1 q l +1 = 1 r β (3 + α − β ) . omparing this with Lemma 2.1, we conclude p l +1 /q l +1 = α.β .To complete the proof, it remains to show that our discovered continued fraction expansion for α.β is palindromic. To do this, we take the expansion we found and verify that reversing the terms givesa fraction that also equals α.β ; by uniqueness of even length expansions we conclude the expansion ispalindromic.So first suppose we are in the case where b l = 2. The fraction obtained by reversing the terms of[0; b , . . . , b l − , , , , a , . . . , a k ]is the fraction [0; a , . . . , a k , , , , b , . . . , b l ]making use of the palindromic hypothesis on α and β . Now [1; b , . . . , b l ] = β − − − β ] , and forany x = 0 , , − , x ] = 3 − x. Thus this fraction equals[0; a , . . . , a k , − β ] . Similarly, in case b l = b l − = 1, the fraction obtained by reversing the terms of[0; b , . . . , b l − , , , a , . . . , a k ]is [0; a , . . . , a k , , , b , . . . , b l ] . One easily checks [2; 2 , b , . . . , b l ] = 3 − β , so in this case the fraction also equals [0; a , . . . , a k , − β ]. Tocomplete the proof we must verify that α.β = [0; a , . . . , a k , − β ]. Letting p i /q i be the convergents, weuse the palindromic property of α to easily compute p k +1 q k +1 − α = 1 r α (3 + α − β ) . Again comparing with Lemma 2.1, we conclude p k +1 /q k +1 = α.β . (cid:3) An immediate consequence of our description of the continued fraction expansion of α ∈ E is thefollowing elementary congruence, which we will need later. We do not know of a simple proof of this factthat does not make use of continued fraction methods. Corollary 3.3. If α ∈ E is an exceptional slope, then ( αr α ) ≡ − r α ) .Proof. Clearly the congruence only depends on the fractional part of α , so we may assume 0 ≤ α <
1. If[0; a , . . . , a k ] is the even length palindromic continued fraction expansion of α , then q k p k − − q k − p k = 1,which in light of the palindrome condition gives r α p k − − ( αr α ) = 1. (cid:3) Corollary 3.4.
Let C = R \ S α ∈ E I α . If ξ ∈ C , then the fractional part of the continued fractionexpansion of ξ has only ones and twos in it.Proof. Since S α I α covers the rationals, ξ is irrational. Inspection of the continued fraction algorithmreveals that for each k , there is an ǫ > λ ∈ ( ξ − ǫ, ξ + ǫ ) have the same first k terms intheir continued fraction expansions as ξ . But every element of C is a limit of exceptional slopes. (cid:3) Corollary 3.5.
Let
D > be a rational number. The number ξ = − √ D lies in I α for some α ∈ E . That is, it is not in the generalized Cantor set C . Note that the numbers α ± x α are all quadratic irrationals, and they all lie in C . Thus many quadraticirrationals do not lie in some I α . Furthermore, when D = 5 the result is false, as then ξ = 0 − x . Wethank Henry Cohn for showing us the following argument [C]. roof. If D is a square the result is obvious, so we may assume ξ is irrational. Also, if 5 < D < − x < ξ <
0, so ξ ∈ I . Thus we assume D > r be the positive integer such that(2 r + 1) < D < (2 r + 3) . We claim the (repeating) continued fraction expansion of ξ takes the form[ r − a , . . . , a k ]with a k = 2 r + 1. Since 2 r + 1 ≥
3, we will conclude from Corollary 3.4 that ξ / ∈ C and hence ξ ∈ I α forsome α .Consider the number ξ + r + 2. The integer part of this number is 2 r + 1, so it will suffice to show thatit is purely periodic, i.e. that ξ + r + 2 = [2 r + 1; a , . . . , a k − ] . It is well known [Da] that a quadratic irrational is purely periodic if and only if it is larger than 1 andits algebraic conjugate is between − − < r + 2 + − − √ D < , and these are equivalent to (2 r + 1) < D < (2 r + 3) . (cid:3) The associated exceptional bundle
Let Z ∈ P n ] be a general point. In the next section, we will determine a particularly nice resolutionof the ideal sheaf I Z by a semi-exceptional bundle and a stable bundle. One of the more challengingaspects of finding this resolution is simply determining which exceptional bundle is the correct one. Thegoal of this section is to determine the slope of the exceptional bundle which is naturally associated tothe ideal sheaf I Z .As a first goal, we aim to determine the minimum possible slope µ of a stable bundle V with theproperty that χ ( V ) ≥ rk( V ) n . To do this, we introduce an auxiliary function γ : Q ≥ → Q ≥ on thenonnegative rationals by the formula γ ( µ ) = P ( µ ) − δ ( µ ) , noting that γ (0) = 0. By Theorem 2.2, for any rational numbers µ, ∆, there exists a non-exceptionalstable bundle V with slope µ and discriminant ∆ if and only if ∆ ≥ δ ( µ ). Equivalently, there exists sucha bundle if and only if γ ( µ ) ≥ P ( µ ) − ∆ = χ ( V )rk( V ) . Thus for each µ ∈ Q ≥ the maximum value of the ratio χ ( V ) / rk( V ) over all non-exceptional stablebundles V of slope µ is precisely γ ( µ ).Since δ admits a unique continuous extension to R by Proposition 2.4, we see immediately that γ alsoadmits a continuous extension to a function γ : R ≥ → R ≥ . Let us establish several other elementaryproperties of the function γ . Proposition 4.1.
The function γ : R ≥ → R ≥ (1) is strictly increasing, (2) is piecewise linear with rational coefficients on each interval I α , where α ∈ E , and (3) is unbounded.In particular, γ has an inverse. roof. We first record a more explicit formula for γ . On any interval I α , γ takes the form γ ( µ ) = ( α ( µ + 3) + 1 + ∆ α − P ( α ) if µ ∈ ( α − x α , α ]( α + 3) µ + 1 + ∆ α − P ( α ) if µ ∈ [ α, α + x α ) . Properties (2) and (3) follow immediately. Clearly γ is increasing on each interval ( α − x α , α + x α ).Thus to see γ is strictly increasing, it will suffice to see that if 0 ≤ α < β are exceptional slopes then γ ( α + x α ) < γ ( β − x β ). But γ ( α + x α ) = P ( α + x α ) − δ ( α + x α ) = P ( α + x α ) − < P ( β − x β ) −
12 = P ( β − x β ) − δ ( β − x β ) = γ ( β − x β )since the function P ( x ) is increasing on [ − / , ∞ ) and α + x α < β − x β . (cid:3) Let q ∈ Q ≥ be fixed. Since γ is increasing, we conclude that there exists a non-exceptional stablebundle V of slope µ ≥ q ≤ χ ( V ) / rk( V ) ≤ γ ( µ )if and only if γ − ( q ) ≤ µ . Thus the set of slopes of non-exceptional stable bundles with µ ( V ) ≥ χ ( V ) / rk( V ) ≥ q has a minimum if and only if γ − ( q ) is rational. That this is always the case followsfrom our investigation into the number theory of exceptional slopes, as we shall now see. Theorem 4.2.
The function γ : Q ≥ → Q ≥ is a bijection. We warn the reader that the properties of Proposition 4.1 alone are not sufficient to imply the theorem.In fact, it is a priori possible that there exists some q ∈ Q ≥ such that γ − ( q ) lies in the generalizedCantor set C = R \ S α I α . See [H1] for a discussion of several counterexamples. Proof.
Let q ∈ Q ≥ , and put ξ = γ − ( q ). If ξ ∈ I α for some α ∈ E , then since γ is piecewise linear withrational coefficients on I α we conclude that ξ ∈ Q , and we are done. If ξ lies in no I α , then we have q = γ ( ξ ) = P ( ξ ) − δ ( ξ ) = P ( ξ ) −
12 = 12 (1 + 3 ξ + ξ ) , and thus ξ = − √ q . Since q ≥
0, this contradicts Corollary 3.5 unless q = 0; in case q = 0 we trivially have ξ = 0, so we aredone. (cid:3) Remark 4.3.
In fact, keeping notation from the proof of the theorem, it is easy to show that if α is theassociated exceptional slope to γ − ( q ) then ( − √ q ) ∈ I α . Once the associated exceptional slopeto γ − ( q ) is known, it is easy to determine γ − ( q ) precisely via the formulas in the proof of Proposition4.1. Thus a fast method for determining γ − ( q ) is to first find the interval I α in which ( − √ q )lies. Corollary 4.4.
For every q ∈ Q ≥ , the set of nonnegative slopes of non-exceptional stable bundles V satisfying χ ( V ) / rk( V ) ≥ q has a minimum, namely γ − ( q ) . Furthermore, any such V of minimumpossible slope has χ ( V ) / rk( V ) = q . We now turn to including the exceptional bundles into the discussion. These bundles have an unusuallylarge ratio χ α /r α for their slopes, so they require special attention. We first collect some more necessaryfacts about the exceptional slopes in the following easy lemma. Lemma 4.5.
Let α ∈ E be nonnegative, and let n be a nonnegative integer. We have γ ( α ) < χ α r α < γ ( α + x α ) . (2) If γ ( α ) < n then χ α /r α ≤ n. (3) It is never the case that γ ( α ) = n unless α is an integer; in this case n = ( α + 2)( α + 1)2 − . Proof. (1) Note that χ α r α − γ ( α ) = 1 r α . The derivative of the function γ ( µ ) on the interval [ α, α + x α ) is α + 3, so it suffices to show1 r α < x α ( α + 3) . In fact, even 1 r α < x α is true, and is easily verified by basic algebra.(2) Assume that χ α /r α > n . Since χ α is an integer, we must actually have χ α r α ≥ n + 1 r α . But then γ ( α ) = χ α r α − r α ≥ n since r α ≥ γ ( α ) = − r α + 3( αr α ) r α + α r α r α . The numerator of this expression is congruent to − α r α (mod r α ), so since α r α ≡ − r α ) byCorollary 3.3 we see that γ ( α ) is not an integer, except possibly when α is an integer or half-integer.However, when r α = 2 we compute that the numerator is congruent to 2 mod 4. The expression for γ ( α )when α is an integer is elementary. (cid:3) We now remove our “non-exceptional” hypothesis from Corollary 4.4.
Theorem 4.6.
Let q ∈ Q ≥ . The set of nonnegative slopes of stable bundles V with the property χ ( V ) / rk( V ) ≥ q has a minimum µ .In case q = n is a positive integer, there exist stable V of slope µ with χ ( V ) / rk( V ) = n . In fact, unless n is of the form ( r + 2)( r + 1) / − for a positive integer r , no stable V of slope µ has χ ( V ) / rk( V ) > n .Proof. We already know from Corollary 4.4 that the set of nonnegative slopes of non-exceptional stablebundles with the property χ ( V ) / rk( V ) ≥ q has a minimum λ = γ − ( q ). Let α ∈ E be the associatedexceptional slope to λ . If β ∈ E has 0 ≤ β < α , then by Lemma 4.5 (1) χ β r β < γ ( β + x β ) < γ ( λ ) = q, so the exceptional bundle E β does not have χ ( E β ) / rk( E β ) ≥ q . On the other hand, it is possiblethat χ α /r α ≥ q and α < λ . It follows that the minimum nonnegative slope µ of stable bundles with χ ( V ) / rk( V ) ≥ q is either α or λ ; at any rate, the minimum exists. ow suppose q = n is a positive integer. With the notation of the previous paragraph, if µ = λ thenit follows from Corollary 4.4 that every non-exceptional stable bundle of slope µ with χ ( V ) / rk( V ) ≥ n actually has χ ( V ) / rk( V ) = n . If in fact we have µ = λ = α , then we must have γ ( α ) = n , so n is of theform ( r + 2)( r + 1) / − µ = α = λ we see α < λ , so γ ( α ) < n . ByLemma 4.5 (2), we conclude χ α /r α ≤ n , and thus χ α /r α = n . (cid:3) At last, we can make precise our notion of the associated exceptional bundle to the ideal sheaf I Z of n general points. Definition 4.7.
For an integer n , let µ be the minimum nonnegative slope of a stable bundle V with χ ( V ) / rk( V ) = n . Let α ∈ E be the associated exceptional slope to µ . The associated exceptional slope to P n ] is α . 5. Resolution of the ideal sheaf of n points In this section we exhibit a particularly nice resolution of the ideal sheaf I Z of n general points in P .The strategy will be to first describe the resolution for some Z ∈ P n ] , then argue that in fact the general Z has a resolution of this form as well.Let n ≥ assume that n is not of the form (cid:18) r + 22 (cid:19) − . We will handle this very easy case later; while it is possible to handle it uniformly with the other cases,incorporating it into the main dialog makes things much more confusing due to its exceptional naturewith respect to Theorem 4.6.We begin by letting µ be the minimum nonnegative slope of a stable bundle with the property χ/r ≥ n .Consider the exceptional slope associated to µ . We may write it in the form α.β for some α, β ∈ E with α = ε (cid:18) p q (cid:19) β = ε (cid:18) p + 12 q (cid:19) . The triples ( E β − , E α , E α.β ) and ( E α.β , E β , E α +3 ) are then both triads. Lemma 5.1.
Suppose µ = α.β . (Since n = (cid:0) r +22 (cid:1) − , this is equivalent to assuming γ ( µ ) = n ). (1) If µ ∈ ( α.β − x α.β , α.β ) , there exists a stable bundle V of slope µ with χ ( V ) / rk( V ) = n and rk( V ) = ( α.β ) r α.β . (2) If µ ∈ ( α.β, α.β + x α.β ) , there exists a stable bundle V of slope µ with χ ( V ) / rk( V ) = n and rk( V ) = ( α.β + 3) r α.β . Proof.
Suppose µ ∈ ( α.β − x α.β , α.β ). By Proposition 4.1, we compute γ ( µ ) = ( α.β )( µ + 3) + 1 + ∆ α.β − P ( α.β ) = n, so µ = n − P ( α.β ) − ∆ α.β α.β − n − r α.β + χ α.β ( α.β ) r α.β − , and it follows that µ ( α.β ) r α.β is an integer. We conclude by Theorem 2.2 that the necessary V exists.The argument when µ ∈ ( α.β, α.β + x α.β ) is analogous. (cid:3) Definition 5.2.
In case µ = α.β , any bundle provided by Lemma 5.1 is called an associated orthogonalbundle to the general ideal sheaf I Z of n points. When γ ( µ ) < n , we have χ α.β /r α.β = n , and theexceptional bundle E α.β is called the associated orthogonal bundle. e denote by V an associated orthogonal bundle to I Z . The terminology comes from the numericalexpectation that if I Z is general then H i ( V ⊗ I Z ) = 0 for all i . Suppose V is not exceptional. In thiscase, we have P ( µ ) − δ ( µ ) = γ ( µ ) = n = χ ( V )rk( V ) = P ( µ ) − ∆( V ) , so ∆( V ) = δ ( µ ) and V has height zero. Thus V admits a nice resolution by semi-exceptional bundles. Proposition 5.3. If V is an associated orthogonal bundle, one of the following three possibilities holds. (1) µ = α.β and V is exceptional. (2) µ < α.β , and rk( V ) = ( α.β ) r α.β . In this case, V admits a resolution → E m β − → E m α → V → , where m = r α ( µ − α )( α.β ) and m = r β ( µ − β +3)( α.β ) . In particular, the numbers r α ( µ − α )( α.β ) and r β ( µ − β + 3)( α.β ) are positive integers. Furthermore, the inequalities r α.β x α.β < m m ≤ r α.β r β r α.β − r α hold. (3) µ > α.β , and rk( V ) = ( α.β + 3) r α.β . In this case, V admits a resolution → V → E m β → E m α +3 → with m = r α (3 + α − µ )( α.β + 3) and m = r β ( β − µ )( α.β + 3) , and the inequalities r α.β x α.β < m m ≤ r α.β r α r α.β − r β hold.Proof. Suppose we are in case (2), so that µ < α.β . Since V has height zero, associated exceptional slope α.β , and ( E β − , E α , E α.β ) is a triad, there exists a resolution of the form0 → E m β − → E m α → V → . It follows that α = µ ( E m α ) = c ( E m β − ) + c ( V )rk( E m β − ) + rk( V ) = m ( β − r β + µ ( α.β ) r α.β m r β + ( α.β ) r α.β , and by basic algebra m = r α.β ( µ − α )( α.β ) r β (3 + α − β ) = r α ( µ − α )( α.β ) , where we made use of the identity r α.β = r α r β (3 + α − β ). Comparing ranks, we have m r α = m r β + ( α.β ) r α.β , and using r α.β = r α r β (3 + α − β ) again we conclude m = r β ( µ − β + 3)( α.β ). The inequalities on m /m follow immediately from Section 2.4.In case (3) we use the triad ( E α.β , E β , E α +3 ) and perform an identical calculation. (cid:3) The discussion of Section 2.4 shows that the prior resolution of V is canonical, and that the numbers m and m are suitable Euler characteristics. .1. The resolution of I Z for µ < α.β . At this point we “guess” a resolution of an ideal sheaf I Z . Firstlet us suppose µ < α.β and rk( V ) = ( α.β ) r α.β . Put m = r α ( µ − α )( α.β ) and m = r β ( µ − β + 3)( α.β ),so that we have a resolution 0 → E m β − → E m α → V → . From the inequality m m > r α.β x α.β we find 3 r α.β m > m (proving this reduces to the inequality 3 x α.β r α.β > W defined by the exact sequence0 → W → E m − α − φ → E r α.β m − m − β → φ between semi-exceptional bundles is general. To ensure that φ is surjective, we restrictour attention to the case where the “expected rank” m r α − (3 r α.β m − m ) r β of W is at least 2; we willsee later that this is not very restrictive at all, and will handle the other cases separately. Despite theform of its resolution, we warn that W is not typically a height zero bundle. By Proposition 2.6, if weshow H om ( E − α − , E − β ) is globally generated, the hypothesis that the expected rank of W is at least 2implies φ is surjective and W is locally free of the expected rank. Lemma 5.4.
Let α, β be exceptional slopes of the form α = ε (cid:18) p q (cid:19) β = ε (cid:18) p + 12 q (cid:19) . Then the sheaf H om ( E β − , E α ) is globally generated.Proof. We induct on q , starting with q = − α = k and β = k + 2 for someinteger k then H om ( E β − , E α ) ∼ = O P (1) is globally generated.So suppose q ≥
0, and first assume p is odd. Putting η = ε (( p − / q ), the induction hypothesis shows H om ( E β − , E η ) is globally generated. We have α = η.β , so ( E η , E α , E β ) is a triad. The canonical map E η ⊗ Hom( E η , E α ) → E α → H om ( E β − , E η ) ⊗ Hom( E η , E α ) → H om ( E β − , E α ) → H om ( E β − , E α ) is a quotient of a globally generated bundle, and it is globallygenerated.When p is even, one reduces to the odd case by replacing ( α, β ) with ( − β, − α ) and then noting H om ( E − α − , E − β ) ∼ = H om ( E β − , E α ). (cid:3) Next, put m = r α.β ( P ( α.β ) − ∆ α.β − n ) = χ α.β − r α.β n = χ ( E α.β ⊗ I Z ) (which is clearly a positiveinteger) and consider a general map W ψ → E m − ( α.β ) . We will see in a moment by a purely numerical calculation that the expected rank of W equalsrk( E m − ( α.β ) ) −
1, so our hypothesis on the expected rank of W is equivalent to requiring m r α.β ≥ H om ( W, E − ( α.β ) ) is globally generated. Indeed, from the defining sequence of W we have asurjection H om ( E m − α − , E − ( α.β ) ) → H om ( W, E − ( α.β ) ) → , and global generation of the former bundle follows from the lemma. Thus the rank of ψ is only less thanrk W in codimension 2, and it is an injection. emma 5.5. Assume m r α.β ≥ , and consider the cokernel Q in the exact sequence → W → E m − ( α.β ) → Q → , where W is given by an exact sequence → W → E m − α − → E r α.β m − m − β → . We have rk( Q ) = 1 , c ( Q ) = 0 , and ch ( Q ) = − n .Proof. The proof is an entirely numerical calculation, albeit a complicated one. The basic strategy is touse the identities of Lemma 2.1 repeatedly to remove instances of r α , r β , r α.β , and α.β from expressions.Eventually we arrive at an expression only involving α and β , which turns out to not require any specialproperties of α, β . We first recollect m = r α ( µ − α )( α.β ) m = r β ( µ − β + 3)( α.β ) m = r α.β ( P ( α.β ) − ∆ α.β − n )and also n = γ ( µ ) = ( α.β )( µ + 3) + 1 + ∆ α.β − P ( α.β ) . Let us show rk( E m − ( α.β ) ) = 1+rk( W ). In fact, we will show this equality holds when rk( W ) is interpretedas the expected rank of W , so that the condition m r α.β ≥ W is at least2. Expanding, we haverk( E m − ( α.β ) ) = m r α.β = r α.β ( P ( α.β ) − ∆ α.β − n )= r α.β (2 P ( α.β ) − α.β − ( µ + 3)( α.β ) − W ) = 1 + rk( E m − α − ) − rk( E r α.β m − m − β )= 1 + m r α − (3 r α.β m − m ) r β = 1 + r α ( µ − α )( α.β ) − (3 r α.β r α ( µ − α )( α.β ) − r β ( µ − β + 3)( α.β )) r β = 1 + ( r α − r α.β r α r β + r β )( µ − α )( α.β ) + r β (3 + α − β )( α.β )= 1 + ( r α + r β − r α r β (3 + α − β ))( µ − α )( α.β ) + r β (3 + α − β )( α.β )= 1 + r α r β r β + 1 r α − α − β ) ! ( µ − α )( α.β ) + r β (3 + α − β )( α.β )= 1 + r α r β (2 − β − α − α − β ))( µ − α )( α.β ) + r β (3 + α − β )( α.β )= 1 + r α r β (2 − P ( α − β ) − α − β ))( µ − α )( α.β ) + r β (3 + α − β )( α.β )= 1 − r α r β (3 + α − β ) ( µ − α )( α.β ) + r β (3 + α − β )( α.β )= 1 − r α.β ( µ − α )( α.β ) + r β (3 + α − β )( α.β )Notice that our final expressions for rk( E m − ( α.β ) ) and 1 + rk( W ) both have the same coefficient of µ . Thuswe are reduced to showing r α.β (2 P ( α.β ) − α.β − α.β ) −
1) = 1 + r α.β α ( α.β ) + r β (3 + α − β )( α.β ) , n identity only involving exceptional slopes. Equivalently, we must show r α.β (2 P ( α.β ) − α.β − (3 + α )( α.β ) − r β (3 + α − β ) = 1 r β (3 + α − β ) + α.β or, applying Lemma 2.1, r α (3 + α − β )(2 P ( α.β ) − α.β − (3 + α )( α.β ) −
1) = β. Since βr α (3 + α − β ) = − r α + 3 + αr α (3 + α − β ) = 2∆ α − αr α (3 + α − β ) = 2∆ α − α )( α.β − α )this is the same as showing2 P ( α.β ) − α.β − α − (3 + α )( α.β ) = (3 + α )( α.β − α ) . Now using P ( α − α.β ) = ∆ α + ∆ α.β , we reduce to verifying2 P ( α.β ) − P ( α − α.β ) − (3 + α )( α.β ) = (3 + α )( α.β − α ) . This final equality is true with any numbers x, y in place of α and α.β , so we conclude the requiredidentity of ranks.To compute c ( Q ), we perform a similar calculation. In case α.β = 0, observe c ( Q ) = 0 is obvious, sowe divide by α.β freely. On the one hand, c ( E m − ( α.β ) ) α.β = rk( E m − ( α.β ) )We also have c ( W ) α.β = 1 α.β ( m c ( E − α − ) − (3 r α.β m − m ) c ( E − β ))= 1 α.β ( − m r α ( α + 3) + (3 r α.β m − m ) r β β )= − r α ( µ − α )( α + 3) + (3 r α.β r α ( µ − α ) − r β ( µ − β + 3)) r β β = − r α ( µ − α )( α + 3) + 3 r α.β r α r β ( µ − α ) β − r β ( µ − α ) β − r β (3 + α − β ) β = ( µ − α )( − r α ( α + 3) + 3 r α r β (3 + α − β ) β − r β β ) − r β (3 + α − β ) β = ( µ − α )( − r α (3 + α − β ) + 3 r α r β (3 + α − β ) β − ( r α + r β ) β ) − r β (3 + α − β ) β Let us verify the identity − r α (3 + α − β ) + 3 r α r β (3 + α − β ) β − ( r α + r β ) β = r α.β ( α.β ) . Dividing both sides by r α.β and applying P ( α − β ) = ∆ α + ∆ β shows that it is equivalent to − β + 3 β α − β + β (2 P ( α − β ) − α − β ) = 0 , which is valid for any real numbers α, β with 3 + α − β = 0. Thus c ( W ) α.β = r α.β ( µ − α )( α.β ) − r β (3 + α − β ) β = r α.β ( µ − α )( α.β ) − r β (3 + α − β )( α.β ) + r β (3 + α − β )( α.β − β )= r α.β ( µ − α )( α.β ) − r β (3 + α − β )( α.β ) − − (1 + rk( W )) , omparing with our final expression for 1 + rk( W ) in the previous calculation. Since we already knowrk( E m − ( α.β ) ) = 1 + rk( W ), we conclude c ( Q ) = 0.It is possible at this point to prove ch ( Q ) = − n by the same methods. However, we will later showthat there is a bundle V with χ ( V ) = rk( V ) n such that V ⊗ Q is acyclic, and this implies ch ( Q ) = − n .That result will not make use of any of the further discussion in the rest of this section, so we may safelyuse this fact at will. (cid:3) Corollary 5.6.
Assume m r α.β ≥ . The sheaf Q in the sequence → W ψ → E m − ( α.β ) → Q → is the ideal sheaf I Z of a zero-dimensional scheme Z ⊂ P of degree n .Proof. If we can show Q torsion free, then there is an embedding Q → Q ∗∗ which is an isomorphismoutside of codimension 2. Since rk( Q ) = 1 and c ( Q ) = 0, we have Q ∗∗ ∼ = O P , and it follows that Q ∼ = I Z for some zero-dimensional subscheme Z ⊂ P . Its degree must be n since ch ( Q ) = − n .To verify that Q is torsion free, first observe that by construction the rank of ψ only drops in codi-mension 2. Thus any torsion occurs in codimension 2. Let T ⊂ Q be the torsion subsheaf. If T = 0 then h ( T ) > T has zero-dimensional support) and therefore h ( Q ) > h ( Q ) = 0. For this it suffices to verify h ( E − ( α.β ) ) = 0 and h ( W ) = 0. For thefirst, we simply note that − ( α.β ) < E − ( α.β ) is stable. To see h ( W ) = 0 it is enough to check h ( E − β ) = 0 and h ( E − α − ) = 0. The first equality follows from stability. To see the second, we have H ( E − α − ) = H ( E α ) = Ext ( O P , E α ), which is zero by Theorem 2.5. (cid:3) The resolution of I Z for µ > α.β . At this point let us indicate how the previous arguments carryover to the case where µ > α.β and rk( V ) = ( α.β + 3) r α.β . Begin from the resolution0 → V → E m β → E m α +3 → m = r α (3 + α − µ )( α.β + 3) and m = r β ( β − µ )( α.β + 3). This time we consider a general sheaf0 → E r α.β m − m − α − φ → E m − β → W → , and assume the expected rank of W is at least 2 so that φ is injective and W is locally free. Let m = − r α.β ( P ( α.β ) − ∆ α.β − n ) = r α.β n − χ α.β = − χ ( E α.β ⊗ I Z ) >
0, and look at a general map E m − ( α.β ) − ψ → W, easily verifying the bundle of such maps is globally generated. Verify by a numerical calculation thatrk( W ) = 1 + rk( E m − ( α.β ) − ), so that ψ is injective and the cokernel Q has rank 1; furthermore we concludethat the expected rank of W is always at least 2, so there are no exceptional cases to consider here.Compute c ( Q ) = 0, and conclude as before that Q is an ideal sheaf of a zero-dimensional subscheme Z ⊂ P of degree n .5.3. The resolution of I Z for µ = α.β , with V exceptional. In case V is exceptional, things aresubstantially easier. We have γ ( µ ) < n , so let λ > µ be the rational number with γ ( λ ) = n (it hasassociated exceptional slope α.β by Lemma 4.5 (1)), and let U be a stable bundle of slope λ and rank( α.β +3) r α.β with χ ( U ) / rk( U ) = n . Apply the discussion of Section 5.2 to U instead of V (and λ instead of µ ); we see that m = 0, and it follows from the numerical calculations that W itself is already an ideal sheaf I Z . We can then further calculate m = χ ( E β ⊗ I Z ) and 3 r α.β m − m = − χ ( E α ⊗ I Z ) = − χ ( I Z , E − α − ) sing the techniques of the proof of Lemma 5.5 (where m and m are defined in terms of λ instead of µ ). Consider the resolution 0 → E r α.β m − m − α − → E m − β → I Z → . Applying H om ( − , E − α − ) to this sequence and taking cohomology shows that in fact − χ ( I Z , E − α − ) = dim Ext ( I Z , E − α − )since E β ⊗ E − α − is acyclic and E − α − is simple. Similarly, applying H om ( E − β , − ) and taking cohomologygives χ ( E − β , I Z ) = dim Hom( E − β , I Z ) . Thus we have a resolution0 → E − α − ⊗ Ext ( I Z , E − α − ) ∗ → E − β ⊗ Hom( E − β , I Z ) → I Z → µ = α.β .5.4. The remaining cases.
The only cases we have not yet covered are the cases m r α.β ≤ µ < α.β . We will discuss the case where r α.β = 2 and m = 1 in detail; the other cases can be handledin a similar but easier manner.If r α.β = 2 and m = 1, we must have ( α, α.β, β ) = ( k, k + , k + 1) for some positive integer k . Wehave m = r α.β ( P ( α.β ) − ∆ α.β − n ) = 1 , so n = 12 ( k + 4 k + 2) . In fact, k = 2 l must be even for n to be an integer. Write n = (2 l + 1)(2 l + 2)2 + l. Gaeta’s theorem asserts that the ideal sheaf I Z of n general points has a resolution0 → O P ( − l − ⊕ O P ( − l − l M → O P ( − l − l +2 → I Z → , where M is a general matrix of forms of the appropriate degrees. After applying automorphisms of thebundles, the matrix M can be brought into the form x q · · · q l y q · · · q l z q · · · q l q · · · q ... ... . . . ...0 q l +2 · · · q l +2 ,l where the q ij are quadrics. In other words, the resolution can be rewritten as0 → O P ( − l − l → O P ( − l − l − ⊕ T P ( − l − → I Z → . n particular, there is a map T P ( − l − → I Z (which is neither surjective nor injective). Consider themap of complexes 0 / / T P ( − l − / / I Z / / / / O P ( − l − l O O / / O P ( − l − l − O O / / I Z is placed in the degree 0 position. One can checkthis diagram commutes and is a quasi-isomorphism. Denoting the second complex by W • [1], this impliesthat W • [1] is isomorphic to the mapping cone of the morphism T P ( − l − → I Z in the derived category D b (Coh( P )). Thus there is a distinguished triangle W • → T P ( − l − → I Z → . Note that we have proved this result for every general Z , and the other outstanding cases can also behandled for general Z in this fashion.By working in the heart of a suitable t -structure on D b (Coh( P )), it is possible recover exactness. Thisidea will play a prominent role in Section 8.We note that the case where n = (cid:0) r +22 (cid:1) − µ < α.β = r using a non-exceptional stable bundle V of rank r having χ/r = n . It is necessary tointerpret W as a complex, but we obtain a distinguished triangle W • → O P ( − r ) → I Z → . Let us recap what has been proved to this point.
Proposition 5.7.
In case µ < α.β or n = (cid:0) µ +22 (cid:1) − there is a distinguished triangle W • → E m − ( α.β ) → I Z → for some Z ∈ P n ] , where W • is a complex E m − α − → E r α.β m − m − β concentrated in degrees and . So long as χ ( E α.β ⊗ I Z ) r α.β ≥ , W • is a vector bundle (sitting in degree0) and the distinguished triangle becomes an exact sequence.When µ > α.β and n = (cid:0) µ +22 (cid:1) − , there is an exact sequence → E m − ( α.β ) − → W → I Z → for some I Z , where W fits into an exact sequence → E r α.β m − m − α − → E m − β → W → . In case µ = α.β , some I Z admits a resolution → E − α − ⊗ Ext ( I Z , E − α − ) ∗ → E − β ⊗ Hom( E − β , I Z ) → I Z → . Interpolation for exceptional bundles.
We now know enough about resolutions of ideal sheavesto show that the general I Z imposes the “expected” number of conditions on sections of a large familyof exceptional bundles. This result will allow us to clarify the nature of our resolution of I Z . Theorem 5.8.
Let I Z be a general ideal sheaf of n points, and let α.β be the exceptional slope associatedto the rational number µ with γ ( µ ) = n . Let η be an exceptional slope which satisfies (1) η ≤ α , (2) η = α.β , or η ≥ β .If H ( E η ⊗ I Z ) = 0 , then H ( E η ⊗ I Z ) = 0 . That is, vanishing at a general collection of n points imposes the expected number of conditions onsections of E η . We suspect the theorem is true for all η , but the cases where α < η < β and η = α.β aretypically more difficult. Proof.
This is an open property of Z , so it suffices to show the result holds for a specific Z . Suppose weare in the case µ < α.β and χ ( E α.β ⊗ I Z ) r α.β ≥
3, and consider the exact sequence0 → W → E m − ( α.β ) → I Z → . When η ≤ α , we claim H ( E η ⊗ I Z ) = 0. We have Hom( E − η , E − ( α.β ) ) = 0 by stability since − η > − ( α.β ),so it is enough to show Ext ( E − η , W ) = 0. From the exact sequence0 → W → E m − α − → E r α.β m − m − β → E − η , E − β ) = 0 and Ext ( E − η , E − α − ) = 0. The first group is zerosince − η > − β , while the second is isomorphic to Ext ( E − α , E − η ), which is zero by Theorem 2.5 because − α ≤ − η. Thus H ( E η ⊗ I Z ) = 0. The case where η ≥ β is similar. In case η = α.β , note that E α.β ⊗ W is acyclic since E α.β ⊗ E − α − and E α.β ⊗ E − β are both acyclic. Thus H ( E α.β ⊗ I Z ) ∼ = Ext ( E − ( α.β ) , E − ( α.β ) ) = 0by rigidity.The other possibilities for µ are handled in the same way. (cid:3) In particular, we see that for the general ideal sheaf I Z we havedim Hom( E − ( α.β ) , I Z ) = χ ( E α.β ⊗ I Z ) = m in case µ < α.β and m = − χ ( E α.β ⊗ I Z ) = dim Ext ( E − ( α.β ) , I Z ) = dim Ext ( I Z , E − ( α.β ) − )in case µ > α.β .We now conclude the section with our final result on the resolution of I Z . Theorem 5.9.
Let Z be a general collection of n points. (1) In case µ < α.β and χ ( E α.β ⊗ I Z ) r α.β ≥ , we have a resolution → W → E − ( α.β ) ⊗ Hom( E − ( α.β ) , I Z ) → I Z → where the map E − ( α.β ) ⊗ Hom( E − ( α.β ) , I Z ) → I Z is the canonical one. The isomorphism class of W depends only on Z , and W is a stable bundle with resolution → W → E m − α − → E r α.β m − m − β → , where m = r α ( µ − α )( α.β ) and m = r β ( µ − β + 3)( α.β ) . (2) In case µ > α.β , we have a resolution → E − ( α.β ) − ⊗ Ext ( I Z , E − ( α.β ) − ) ∗ → W → I Z → . The isomorphism class of W depends only on Z , and W is a stable bundle with resolution → E r α.βm − m − α − → E m − β → W → , where m = r β ( β − µ )( α.β + 3) and m = r α (3 + α − µ )( α.β + 3) . In case µ = α.β , we have a resolution → E − α − ⊗ Ext ( I Z , E − α − ) ∗ → E − β ⊗ Hom( E − β , I Z ) → I Z → . Recall that we have already discussed how to prove the natural analog of this theorem in the caseswhere µ < α.β and χ ( E α.β ⊗ I Z ) r α.β ≤ Proof.
Let us focus on cases (1) and (2); the third case is easier. The key fact is that in either case ageneral bundle W with resolution of the prescribed form is stable, and, conversely, a general stable bundle W ′ ∈ M (ch( W )) admits a resolution of the same form as W . This will follow from Brambilla [Brm1,Proposition 4.4 and Theorem 8.2] if we check Ext ( E − α − , E − β ( − m /m ). This vanishing guaranteesthat the bundle W is prioritary , i.e. that Ext ( W, W ( − H ( E α +3 ⊗ E − β − ) = 0, first observe that it is obvious in case α and β are both integers. Thus we may assume β − α <
1. Write F = E α +3 ⊗ E − β − and observe that F is acyclic and µ ( F ) = α − β > −
1; we must show H ( F (2)) = 0. So let C ⊂ P be a plane conic, andconsider the exact sequences0 → F → F (2) → F (2) | C → → F ( − → F → F | C → . Since F is acyclic, H ( F (2)) ∼ = H ( F (2) | C ). We know H ( F | C ) = 0 since F is acyclic and H ( F ( − F ( −
2) is stable with slope greater than − H ( F | C ) surjects onto H ( F (2) | C ), so we conclude H ( F (2) | C ) = 0.Suppose we are in the case µ < α.β . By using Proposition 5.7 we see that there is some Z ∈ P n ] suchthat there is a resolution 0 → W → E m − ( α.β ) → I Z → W stable (since it is general by construction) and having the specified resolution. By Theo-rem 5.8, m = dim Hom( E − ( α.β ) , I Z ), so after performing an appropriate identification of C m withHom( E − ( α.β ) , I Z ) we see that either the map E − ( α.β ) ⊗ C m → I Z is the canonical one or there is somefactor E − ( α.β ) which maps to zero, and hence is a summand of W . But W is stable, so this is impossible,and the map is the canonical one.For a general Z with dim Hom( E − ( α.β ) , I Z ) = χ ( E − ( α.β ) , I Z ), we consider the canonical map E − ( α.β ) ⊗ Hom( E − ( α.β ) , I Z ) → I Z . The property that this map is surjective is open in Z , the property that the kernel is stable is open in Z ,and the property that the kernel has the expected resolution is open in Z . Thus defining W to be thekernel, we obtain the desired resolution.The case where µ > α.β is similar. Sheaves W fitting into the sequence0 → E − ( α.β ) − ⊗ C m → W → I Z → ( I Z , E − ( α.β ) − ) m . If the m components of such an element do notform a basis for Ext ( I Z , E − ( α.β ) − ), then W will have E − ( α.β ) − as a direct summand and will not bestable. When the components do form a basis, the isomorphism class of W is independent of the choice ofelements, as different choices merely amount to different identifications C m ∼ = Ext ( I Z , E − ( α.β ) − ) ∗ . (cid:3) . Orthogonality of Kronecker modules
Let N ≥ general Steiner bundle E on P N − = P V is a vector bundleadmitting a resolution of the form 0 → O b P N − M → O a P N − (1) → E → , where the a × b matrix M of linear forms is general. Consider the following fundamental problem. Givena second general Steiner bundle 0 → O b ′ P N − → O a ′ P N − (1) → F → , compute the dimension of the space Hom( F, E ).Since Ext i ( O P N − (1) , O P N − ) = 0 for 0 ≤ i ≤ N −
1, it is easy to see that any homomorphism F → E lifts to a commutative diagram0 / / O b P N − / / O a P N − (1) / / E / / / / O b ′ P N − / / O O O a ′ P N − (1) / / O O F / / O O , and in particular determines a diagram0 / / O b P N − / / O a P N − (1)0 / / O b ′ P N − / / O O O a ′ P N − (1) O O On the other hand, any diagram of the latter form induces a homomorphism F → E , and these construc-tions are inverse to one another.6.1. Kronecker modules.
The matrix M defining the Steiner bundle E can be thought of as a linearmap e : B ⊗ V → A , where B, A are b - and a -dimensional vector spaces, respectively. The precedingdiscussion shows that the space Hom( F, E ) is naturally isomorphic the space of commutative diagramsof the form B ⊗ V / / AB ′ ⊗ V β ⊗ id O O / / A ′ α O O Let Q be the quiver with two vertices and N arrows from the first vertex to the second, i.e. the N -arrowed Kronecker quiver. A representation of Q (or a Kronecker V -module ) assigns to each vertex avector space and to each arrow a linear map from the first vector space to the second. A representation e of Q is therefore the same thing as a linear map e : B ⊗ V → A , where B, A are vector spaces. If f : B ′ ⊗ V → A ′ is a second representation, then the homomorphisms f → e are precisely the commutativediagrams as above. In particular, if E, F are Steiner bundles with corresponding Kronecker V -modules e, f , then Hom Q ( f, e ) ∼ = Hom( F, E ).The dimension vector of e : B ⊗ V → A is the element dim e = (dim B, dim A ) of N . The Eulercharacteristic of a pair f, e of representations is defined by χ ( f, e ) = dim Hom Q ( f, e ) − dim Ext Q ( f, e ); ll higher Ext terms vanish. The Euler characteristic can be computed numerically in terms of dimensionvectors; precisely, if dim e = ( b, a ) and dim f = ( b ′ , a ′ ) then χ ( f, e ) = b ′ b + a ′ a − N b ′ a. Fix a dimension vector ( b, a ) and vector spaces
B, A of dimensions b and a . There is a natural actionof SL( B ) × SL( A ) on the space P Hom( B ⊗ V, A ). Denote by Kr ( V, B, A ) = Kr ( N, b, a ) the semi-stableobjects in the GIT quotient of this action. If e : B ⊗ V → A is a Kronecker module, we will also denoteby Kr (dim e ) = Kr ( V, B, A ) the space corresponding to the dimension invariants of e . For a nonzeromodule e , we define the slope µ ( e ) ∈ [0 , ∞ ] to be the number b/a , interpreted as ∞ if a = 0 and b = 0.It is observed in [Dr2] that the general Kronecker V -module with slope µ will be GIT-stable whenever µ ∈ ( ψ − N , ψ N ) , where ψ N = N + √ N − . By work of Schofield and van den Bergh [Sc, SvdB], stability of quiver representations can be detectedby the existence of orthogonal representations. We state their result in the special case of the Kroneckerquiver.
Theorem 6.1 ([SvdB, Corollary 1.1]) . A Kronecker V -module e is GIT-semistable if and only if thereis a nontrivial Kronecker V -module f with Hom( f, e ) = Ext ( f, e ) = 0 . Several other authors have discussed similar results, such as Derksen-Weyman and ´Alvarez-C´onsul-King [DW, ACK]. The following restatement of the theorem is immediate by the computation of theEuler form.
Corollary 6.2.
Consider Kronecker modules C b ⊗ V e → C a C ka ⊗ V f → C k ( Na − b ) where e is semistable and f is general. If k is sufficiently large, then Hom( f, e ) = 0 .In particular, the conclusion holds if e is general and µ ( e ) ∈ ( ψ − N , ψ N ) . Remark 6.3.
Keep the notation from the corollary. In [Dr2] it is shown that Kr (dim e ) has Picardgroup Z . The corollary implies that for a general f the locus D f = { e ′ : Hom( f, e ′ ) = 0 } ⊂ Kr (dim e )forms a divisor, which must be a multiple of the generator of the Picard group. Furthermore, for any e ′ ∈ Kr (dim e ), the general divisor D f does not contain e ′ .6.2. Orthogonality of quotients of semi-exceptional bundles.
As a simple application of the or-thogonality result for Kronecker modules, consider a triad (
E, G, F ) of exceptional bundles on P , andput N = dim Hom( E, G ) = rk F . Let V, W be general quotients of the form0 → E b → G a → W → → E ka → G k ( Na − b ) → V → k sufficiently large. Since H om ( G, E ) is acyclic, homomorphisms V → W correspond to diagrams0 / / E b / / G a / / E ka / / O O G k ( Na − b ) . O O lternately, W corresponds to a general Kronecker Hom( E, G ) ∗ -module e : C b ⊗ Hom(
E, G ) ∗ → C a , and V corresponds to a general f : C ka ⊗ Hom(
E, G ) ∗ → C k ( Na − b ) . Since E and G are simple, Hom( V, W ) ∼ =Hom Q ( f, e ). Corollary 6.4. If b/a ∈ ( ψ − N , ψ N ) and k is sufficiently large, then H om ( V, W ) has no cohomology.Proof. The inequalities on b/a ensure that V and W are stable (as in the proof of Theorem 5.9). Then H om ( V, W ) is stable of slope µ ( W ) − µ ( V ) = ( N ab − a − b ) rk( E ) rk( G )( µ ( G ) − µ ( E ))rk( W ) rk( V ) , which is nonnegative by the hypothesis on b/a , so Ext ( V, W ) = 0. By Corollary 6.2 we see Hom(
V, W ) =0. One easily calculates χ ( V, W ) = 0 using the additivity of the Euler characteristic, so Ext ( V, W ) = 0follows. (cid:3) The effective cone of the Hilbert scheme of points
We now combine our results on the resolution of ideal sheaves I Z with the orthogonality of Kroneckermodules to construct extremal effective divisors on the Hilbert scheme of points P n ] .Consider a general ideal sheaf I Z of n points. Let µ be the minimum slope of a stable bundle with χ/r = n , and assume µ is not exceptional. In most cases, Theorem 5.9 associates to I Z a stable bundle W . Either W or its dual admits a resolution by a pair of semi-exceptional bundles. Such a resolutioninduces a stable Kronecker module e as in Subsection 6.2, and the isomorphism class of this Kroneckermodule depends only on W . In cases where the exact sequence of Theorem 5.9 must be interpreted inthe derived category instead, it is still the case that the complex W • corresponds to a stable Kroneckermodule.Thus, so long as µ is non-exceptional, we obtain a dominant rational map π : P n ] Kr (dim e ) , where e is a Kronecker module corresponding to W . In case µ is exceptional, the general I Z is alreadythe cokernel of a map of semi-exceptional bundles, and this map can be regarded as a Kronecker module.The Hilbert scheme therefore always admits a rational map to a suitable moduli space of semistableKronecker modules.It is clear that when µ is exceptional the map π is birational. In the general case, the map haspositive-dimensional fibers. Lemma 7.1. If µ is non-exceptional, then dim Kr (dim e ) < n . Thus the general fiber of the rationalmap P n ] Kr (dim e ) is positive-dimensional.Proof. Let α be the exceptional slope associated to µ , and suppose µ < α . Let V be the associatedorthogonal bundle of Proposition 5.3. Then V has height zero, so there is a Kronecker module f givingrise to V , and M (ch V ) ∼ = Kr (dim f ). In Drezet [Dr2] it is shown that there is a natural isomorphism Kr (dim f ) ∼ = Kr (dim e ). Thus it will be enough to show dim( M (ch( V ))) < n . The same reductionworks in case µ > α .Recall that M (ch( V )) has dimension r ( V ) (2∆( V ) −
1) + 1. We have γ ( µ ) = n and ∆( V ) = δ ( µ ), sowe must show r ( V ) (2 δ ( µ ) −
1) + 1 < γ ( µ ) . We check this inequality holds for µ ∈ ( α − x α , α ]. We have r ( V ) = αr α . The left-hand side is a convexfunction of µ , while the right-hand side is linear in µ . Thus it suffices to check the inequality holds at he endpoints. We have δ ( α − x α ) = 1 /
2, while2 γ ( α − x α ) = 1 + 3( α − x α ) + ( α − x α ) , so the inequality holds at α − x α . At α , we have r ( V ) (2 δ ( α ) −
1) + 1 = ( αr α ) (2( P (0) − ∆ α ) −
1) + 1 = ( αr α ) · r α + 1 = α + 1while 2 γ ( α ) = α + 3 α + 1 − r α . As n ≥ α ≥
1, so the required inequality follows.We also must check the inequality holds when µ ∈ ( α, α + x α ); here things are slightly trickier. Wehave r ( V ) = ( α + 3) r α , and we must verify(2) (( α + 3) r α ) (2 δ ( µ ) −
1) + 1 < γ ( µ ) . The issue is that this inequality does not hold when substituting µ = α (although it does still hold for µ = α + x α , as δ ( α + x α ) = 1 / µ − α = c ( V )( α + 3) r α − α = c ( V ) r α − ( α + 3) αr α ( α + 3) r α . The numerator and denominator of this last fraction are integers, so since µ = α we may assume µ ≥ µ := α + (( α + 3) r α ) − . But in fact, plugging in µ = µ to inequality (2) yields an equality , so theconvexity argument shows the inequality holds when µ ∈ ( µ , α + x α ); we must rule out the possibilitythat µ = µ can actually occur. So suppose µ = µ . We have2 n = 2 γ ( µ ) = α + 3 α + 1 + 1 r α = 2 · χ α r α , which means that in fact we must have µ = α , as χ α /r α = n . Thus this case never actually arises, andthe required inequality holds. (cid:3) Theorem 7.2.
Let µ be the minimum slope of a stable vector bundle on P having the property χ/r = n .Let V be a general stable bundle of slope µ with χ/r = n such that r is sufficiently large and divisible.Then V has interpolation for n points, and the effective divisor D V ( n ) is extremal. The effective cone of P n ] is spanned by µH −
12 ∆ and ∆ . Proof.
Let α.β be the exceptional slope associated to µ , as in Section 5, and let Z ∈ P n ] be general.First assume µ < α.β . We assume χ ( E α.β ⊗ I Z ) r α.β ≥
3; the details in the other “derived” cases areessentially the same. We have a resolution0 → W → E − ( α.β ) ⊗ Hom( E − ( α.β ) , I Z ) → I Z → m , m as in the theorem, let V be a general bundle with resolution0 → E km β − → E km α → V → , where k is a sufficiently large integer. Since E β − ⊗ E − ( α.β ) and E α ⊗ E − ( α.β ) are acyclic, V ⊗ E − ( α.β ) isacyclic. Thus it suffices to show V ⊗ W is acyclic. Now W has a resolution0 → W → E m − α − → E r α.β m − m − β → . To show V ⊗ W is acyclic it suffices to show H om ( V, W ∗ ( − W ∗ ( −
3) has a resolution0 → E r α.β m − m β − → E m α → W ∗ ( − → . riting N = 3 r α.β = dim Hom( E β − , E α ), a = m , and b = N m − m , we have N a − b = m , so we seeCorollary 6.4 implies the acyclicity of H om ( V, W ∗ ( − b/a ∈ ( ψ − N , ψ N ). But simplealgebra shows the inequality m m > r α.β x α.β = N − r − N ! from Proposition 5.3 is equivalent to the inequality ba = N − m m > ψ − N ;the other needed inequality is trivial to establish.To see that D V ( n ) is extremal, observe that it is the pullback of a divisor D f on Kr (dim e ) under therational map π , where f is the Kronecker module corresponding to the resolution of V (see Remark 6.3).If µ > α.β an identical argument works. In case µ = α.β , interpolation follows from Theorem 5.8. Tosee the divisor is extremal, recall that the general I Z has a resolution0 → E − α − ⊗ Ext ( I Z , E − α − ) ∗ → E − β ⊗ Hom( E − β , I Z ) → I Z → . Any I Z with a resolution of this form has V ⊗ I Z acyclic. By Proposition 2.6 and the methods of Section5, we can vary the map in the resolution to produce complete curves in the Hilbert scheme consistingentirely of schemes admitting resolutions as above. This gives a moving curve class dual to D V ( n ), sothe divisor is extremal. (cid:3) Remark 7.3.
Even in the general case, one can produce moving curves on the Hilbert scheme dual tothe extremal effective divisor as in the final part of the proof of the theorem. For instance, starting froma resolution of the form 0 → W ψ → E m − ( α.β ) → I Z , one can vary the map ψ to produce complete curves in fibers of the rational map π .In case the exceptional slope α.β is an integer, it is easy to construct a moving curve classically. Write n = r ( r + 1) / s , with 0 ≤ s ≤ r . The assumption that α.β is an integer amounts to requiring either s/r > ϕ − or s +1 r +2 < − ϕ − , where ϕ is the golden ratio; these two inequalities correspond to thepossibilities µ < α.β and µ ≥ α.β , respectively. In the former case, there is a dual moving curve givenby letting n points move in a linear pencil on a smooth curve of degree r ; in the latter case, we get adual moving curve by letting n points move in a linear pencil on a smooth curve of degree r + 2. See[ABCH, H2, H3] for details.We have also described a dual moving curve in case α.β is a half-integer k/
2, with k odd, and µ < α.β .Writing n as in the previous paragraph, this corresponds to the case where √ − < sr − < . In this case, we showed in [H2, H3] that for a general collection Z of n points there is a curve C of degree2 r − r − ( r − − n nodes and no further singularities, such that Z moves in a linear pencilon C . Allowing Z to move in such a linear pencil describes a moving curve on P n ] , and it is dual to theextremal divisor D V ( n ). Remark 7.4.
The map i q : P n ] P n +1] given by taking the union of a scheme with a fixed point q ∈ P induces an isomorphism Pic( P n ] ) ∼ = Pic( P n +1] ) identifying the divisors H and ∆ in each space.Up to this identification, there is a natural inclusion Eff( P n +1] ) ⊂ Eff( P n ] ). Combining the theoremwith the results of Section 4 we see that this inclusion is strict unless n is of the form (cid:0) r +22 (cid:1) −
1, whenboth effective cones are spanned by rH − ∆ and ∆. n Table 1, we explicitly compute the nontrivial edge of the effective cone of the Hilbert scheme P n ] for small n . This data can be generated very quickly by computer using the results of Section 4. Remark4.3 is especially useful for this.8. Connections with Bridgeland stability
In [ABCH], it was conjectured that there is a correspondence between the walls in the Mori chamberdecomposition of the Hilbert scheme P n ] and the walls in a suitable half-plane of Bridgeland stabilityconditions. Our goal for the rest of the article is to show that our computation of the effective coneof P n ] is consistent with this conjecture. The key step is to determine when exceptional bundles areBridgeland semistable.8.1. Preliminaries on Bridgeland stability.
We briefly summarize the necessary material from[ABCH]; we refer the reader to sections 5-9 of that paper for a full account. Let D b ( P ) = D b (coh P ) bethe bounded derived category of coherent sheaves on P . For any s ∈ R , we define full subcategories F s and Q s of coh( P ) by the requirements • Q ∈ Q s if and only if Q is torsion, or every quotient in the Harder-Narasimhan filtration of Q has slope larger than s . • F ∈ F s if and only if F is torsion-free, and each quotient in the Harder-Narasimhan filtration of F has slope no larger than s .The subcategories ( F s , Q s ) define a torsion pair for each s . Associated to this torsion pair is a corre-sponding t -structure on D b ( P ). The heart of this t -structure is the full abelian subcategory A s of D b ( P )given by complexes whose H − -term is in F s and whose H -term is in Q s , with all other cohomologysheaves equal to zero: A s = { E • : H − ( E • ) ∈ F s , H ( E • ) ∈ Q s , and H i ( E • ) = 0 for other i } . Next we define on the category A s a family of slope functions; these will depend only on the Cherncharacter ( r, c, d ) = (ch , ch , ch ) of a complex E • . For each real number t >
0, put µ s,t ( r, c, d ) = − t r + ( d − sc + s r ) t ( c − sr ) . Then the pair A s,t = ( A s , µ s,t ) forms a Bridgeland stability condition [Bri, AB, BM]. One defines slopesemistability of objects of A s,t in the obvious way. For any Chern character and choice of ( s, t ), themoduli space M P ( r, c, d ) of semi-stable objects of A s,t with given Chern character can be constructedas an Artin stack [AP, AB, L, T]. These spaces can also be constructed as projective schemes usinggeometric invariant theory [ABCH, BM2].When E ∈ Q s is a coherent sheaf, we regard it as an object of A s by viewing it as a 0-complex. Thefollowing fact from [ABCH] is particularly relevant to the present discussion, so we single it out. Theanalogous fact for K Proposition 8.1.
Suppose E ∈ Q s is a Mumford-semistable sheaf. There is a number t > such that E is a µ s,t -semi-stable object of A s for all t > t . Furthermore, there exists a uniform choice of t dependingonly on the Chern character of E . Conversely, it can be seen that if E • ∈ A s has ch ( E • ) ≥ E • is ( s, t )-semi-stable for large t ,then in fact H ( E • ) ∈ Q s is a Mumford-semistable sheaf and H − ( E • ) = 0. Thus E • is isomorphic toa Mumford-semistable sheaf in Q s . We conclude that if s < c/r and t ≫ able 1. For each n ≥
2, the nontrivial edge of Eff P n ] is spanned by µH − ∆ . Theassociated exceptional slope to µ is α . n α µ n α µ n α µ n α µ emistable objects of A s,t with Chern character ( r, c, d ) is just the ordinary moduli space of Mumford-semistable coherent sheaves. In particular, if s < t ≫
0, the moduli space of ( s, t )-semistableobjects of A s with Chern character (1 , , − n ) is isomorphic to P n ] for large t .To understand the birational geometry of P n ] , we study the problem of understanding how themoduli space of ( s, t )-semistable objects of A s with Chern character (1 , , − n ) varies as ( s, t ) varies in thequadrant { s < , t > } . When the collection of semistable objects changes, it is due to ( s, t ) crossing a potential wall where some ideal sheaf I Z is destabilized. Precisely, for two Chern characters ( r, c, d ) and( r ′ , c ′ , d ′ ), the corresponding potential wall is the subset W ( r,c,d ) , ( r ′ ,c ′ ,d ′ ) = { ( s, t ) : µ s,t ( r, c, d ) = µ s,t ( r ′ , c ′ , d ′ ) } . When E is a bundle with Chern character ( r, c, d ), we frequently write W E, ( r ′ ,c ′ ,d ′ ) for the precedingwall, and similarly with the second argument. If E ′ → E is an inclusion of objects of A s , then E ispotentially destabilized as ( s, t ) crosses the wall W E,E ′ . In fact, elementary calculus shows that if ( r, c, d )and ( r ′ , c ′ , d ′ ) are not proportional then on one side of the wall we have µ s,t ( E ′ ) > µ s,t ( E ) and on theother we have µ s,t ( E ) > µ s,t ( E ′ ). In particular, E cannot be semistable on both sides of the wall, but itcould potentially be unstable on both sides.The geometry of the walls we must consider is particularly nice. Fix a Mumford-stable sheaf E withChern character ( r, c, d ), and consider the family of walls W E, ( r ′ ,c ′ ,d ′ ) as ( r ′ , c ′ , d ′ ) varies. One wall isthe vertical line s = c/r = µ ( E ), corresponding to ( r ′ , c ′ , d ′ ) with the same slope c ′ /r ′ = µ ( E ). Theother walls form two nested families of semicircles on either side of the vertical wall, with each semicirclecentered on the s -axis in the st -plane. The center is positioned at the point (cid:18) rd ′ − r ′ drc ′ − r ′ c , (cid:19) , and it has radius s(cid:18) rd ′ − r ′ drc ′ − r ′ c (cid:19) − (cid:18) cd ′ − c ′ drc ′ − r ′ c (cid:19) . The formula for the radius of a wall becomes more transparent when one looks at a Mumford-stablesheaf E of slope c/r = 0. In this case, the Bogomolov inequality gives d ≤
0. If we let x = rd ′ − r ′ drc ′ , then the wall W E, ( r ′ ,c ′ ,d ′ ) has center ( x,
0) and radius r x + 2 dr ≤ | x | . Noting that 2 d/r is a fixed nonpositive number, we observe the following basic fact.
Lemma 8.2.
The radius of a semicircular wall to the left of the vertical wall decreases as the centermoves to the right, toward the vertical wall. Similarly, the radius of a wall to the right of the vertical walldecreases as the center moves to the left.
The restriction that µ ( E ) = 0 in the preceding discussion is not essential; formally twisting by − µ ( E )shifts all the walls by µ ( E ), so the general case follows from this. Thus in order to show one wall W E, ( r ′ ,c ′ ,d ′ ) is nested in another W E, ( r ′′ ,c ′′ ,d ′′ ) it is enough to show both walls lie on the same side of thevertical wall and that the center of the first wall is closer to the vertical wall than the center of thesecond wall is. This fact can be useful, as the expression for the radius is far more complicated than theexpression for the center. f E is a Mumford-stable sheaf in Q s , then we say an injection F → E in the category A s destabilizes E at ( s , t ) if E is ( s , t )-semistable and ( s , t ) lies on the wall W E,F , so that
E, F have the same( s , t )-slope. In this case, for every ( s, t ) ∈ W E,F the map F → E destabilizes E at ( s, t ); in particular, E and F are in A s . Thus we say F destabilizes E along the wall W E,F . Recall that µ s,t ( F ) > µ s,t ( E )for all ( s, t ) on one side of the wall and µ s,t ( F ) < µ s,t ( E ) for all ( s, t ) on the other side of the wall; since E is Mumford-stable we see that in fact E is ( s, t )-stable for all ( s, t ) outside of the wall and E is not( s, t )-semistable for any ( s, t ) inside the wall.We may now describe the conjectural correspondence between Bridgeland and Mori walls for P n ] discussed in [ABCH]. Let I Z be the ideal sheaf of some Z ∈ P n ] , and consider the family of walls W I Z , ( r ′ ,c ′ ,d ′ ) . We call a wall in this family a Bridgeland wall for the Hilbert scheme if some ideal sheaf I Z is destabilized along the wall. With n fixed, Bridgeland walls only depend on their centers, so wedenote the Bridgeland wall with center ( x,
0) by W x . On the other hand, a Mori wall is a ray H + y ∆in Eff P n ] where the stable base locus of a divisor in the ray changes; such walls depend only on theparameter y < Conjecture 8.3.
There is a one-to-one correspondence between Bridgeland walls W x and Mori walls H + y ∆ for P n ] given by the transformation x = y − . The collapsing wall is the Bridgeland wall where the general ideal sheaf I Z is destabilized; it is theinnermost Bridgeland wall. The proof of the next theorem will occupy the rest of this section. Theorem 8.4.
The center ( x, of the collapsing wall for P n ] corresponds to the nontrivial edge µH − ∆ of Eff( P n ] ) by x = − (cid:18) µ + 32 (cid:19) . The destabilizing object for a general ideal sheaf.
To prove Theorem 8.4, we need to identifythe collapsing wall and verify the relation between its center and the edge of the effective cone. It isrelatively easy to specify what the collapsing wall is; the difficult part is to show that the general idealsheaf I Z is actually semistable along the collapsing wall. Here we describe the collapsing wall, and leavethe proof of semistability of the general ideal sheaf for the next section.Let µ be the minimum slope of a stable bundle V with χ/r = n , and let α.β be the associatedexceptional slope to µ , as in Section 5. Assume for now that µ < α.β . Let I Z be a general ideal sheaf of n points. By Theorem 5.9 there is a distinguished triangle W • → E − ( α.β ) ⊗ Hom( E − ( α.β ) , I Z ) → I Z → where W • is the complex E m − α − → E r α.β m − m − β . The shift E − ( α.β ) ⊗ Hom( E − ( α.β ) ,I Z ) → I Z → W • [1] → is then also a distinguished triangle. In case χ ( E α.β ⊗ I Z ) r α.β ≥
3, so that W = W • is actually astable vector bundle, we observe that µ ( W ) < µ ( E − ( α.β ) ) since c ( W ) = c ( E − ( α.β ) ) < W ) =rk( E − ( α.β ) ) − . Thus if µ ( W ) < s < µ ( E − ( α.β ) ) we have E − ( α.β ) ∈ Q s and W ∈ F s , so W [1] ∈ A s . Thusall the terms in the above triangle are in A s , and we have an exact sequence0 → E − ( α.β ) ⊗ Hom( E − ( α.β ) ,I Z ) → I Z → W [1] → n the category A s . The cases with χ ( E α.β ⊗ I Z ) r α.β ≤ W is interpreted as a complex. Treating the cases where µ > α.β and µ = α.β similarly, it is natural to expect the following theorem is true. Theorem 8.5.
Let Z ∈ P n ] be general. When µ < α.β or n is of the form (cid:0) r +22 (cid:1) − , the canonicalhomomorphism E − ( α.β ) ⊗ Hom( E − ( α.β ) , I Z ) → I Z is a destabilizing subobject of I Z .If µ > α.β , the canonical homomorphism I Z → E − ( α.β ) − [1] ⊗ Ext ( I Z , E − ( α.β ) − ) ∗ is a destabilizing quotient object of I Z . In other words, this map is surjective in appropriate categories A s , and its kernel W is a destabilizing subobject.If µ = α.β and n is not of the form (cid:0) r +22 (cid:1) − , the canonical homomorphisms E − β ⊗ Hom( E − β , I Z ) → I Z and I Z → E − α − [1] ⊗ Ext ( I Z , E − α − ) ∗ are destabilizing sub- and quotient objects of I Z , respectively.In each case, the center ( x, of the corresponding wall is given by x = − ( µ + ) . Proving even one of the three cases of the theorem takes a large amount of calculation, so we focusexclusively on the case µ < α.β , and further assume χ ( E α.β ⊗ I Z ) r α.β ≥
3. Verifying the other caseswould be a good exercise to become comfortable with the arithmetic of exceptional slopes and Bridgelandstability. Let us point out the following general fact before beginning the proof: given any exact sequenceof sheaves 0 → A → B → C → , there is an equality W A,B = W B,C = W C,A . These several descriptions of a given wall are frequentlyuseful.
Proof for µ < α.β and χ ( E α.β ⊗ I Z ) r α.β ≥ . Consider the exact sequence0 → E − ( α.β ) ⊗ Hom( E − ( α.β ) , I Z ) → I Z → W [1] → , valid in any category A s with µ ( W ) < s < µ ( E − ( α.β ) ). We must show the following facts:(1) The wall W I Z ,E − ( α.β ) = W I Z ,W = W E − ( α.β ) ,W is nonempty and lies between the vertical lines s = µ ( W ) and s = − ( α.β ).(2) The sheaf E − ( α.β ) is ( s, t )-semistable along the wall W I Z ,E − ( α.β ) .(3) The object W [1] is ( s, t )-semistable along the wall W I Z ,W .From (2) and (3) it follows that I Z is semistable along the wall, since it is an extension of semistableobjects of the same slope.Let us prove (1). Since walls W ( r,c,d ) ,E − ( α.β ) fall into two nested families of semicircles on either side ofthe vertical wall s = − ( α.β ), to show that the wall W I Z ,E − ( α.β ) lies to the left of s = − ( α.β ) it is enoughto show that its center lies to the left of this line (provided it is nonempty). For any exceptional slope α ∈ E we have ch( E α ) = (cid:18) r α , r α α, r α (cid:18) α − ∆ α (cid:19)(cid:19) , o the center of the wall W I Z ,E − ( α.β ) is positioned at the point ( x,
0) with x = ch ( E − ( α.β ) ) + ch ( E − ( α.β ) ) n ch ( E − ( α.β ) ) = − α.β α.β α.β − nα.β = − (cid:18) µ + 32 (cid:19) , using the fact that n = γ ( µ ) = ( α.β )( µ + 3) + 1 + ∆ α.β − P ( α.β ) . Thus the statement that x < − ( α.β ) is equivalent to the inequality µ > α.β − , which is obvious since α.β is the associated exceptional slope to µ . We conclude that if W I Z ,E − ( α.β ) is nonempty, it lies to theleft of the vertical line s = − ( α.β ). Furthermore, the distance between the center of W I Z ,E − ( α.β ) and thisline is − ( α.β ) − x = − α.β + µ < / W I Z ,W lies to the right of the wall s = µ ( W ), it suffices to see x > µ ( W ). Byour final observation in the previous paragraph, we can show the distance µ ( E − ( α.β ) ) − µ ( W ) betweenthe two vertical walls exceeds 3 /
2. We have µ ( E − ( α.β ) ) − µ ( W ) = − ( α.β ) + m r α.β ( α.β ) m r α.β − α.βm r α.β − . Since m = χ ( E α.β ⊗ I Z ) = χ α.β − nr α.β = χ α.β − γ ( µ ) r α.β we see this distance exceeds the number α.β ( χ α.β − γ ( α.β − x α.β ) r α.β ) r α.β − x α.β r α.β . This final quantity depends only on the integer r α.β , and is an increasing function of r α.β . Already for r α.β = 1 it equals ϕ + 1 ≈ . W I Z ,W must lie to the right of s = µ ( W ) if it is nonempty.Finally let us show this wall is actually nonempty. The radius ρ of this wall satisfies ρ = x − n = (cid:18) µ + 32 (cid:19) − γ ( µ ) = 2 P ( µ ) + 14 − γ ( µ ) = 2 δ ( µ ) + 14 > δ ( µ ) > / µ ∈ Q . Thus the radius is at least √ /
2; in particular the wall is nonempty.To complete the proof, it will be sufficient to show that E − ( α.β ) and W [1] are ( s, t )-semistable outsideof semicircular walls of radius at most √ /
2. We will prove this in the next section. (cid:3) Bridgeland stability of exceptional bundles
In this section we investigate the Bridgeland semistability of exceptional bundles in order to completethe proof of Theorem 8.5. Our main result ensures that the locus of ( s, t ) where an exceptional bundleis not ( s, t )-semistable is not “too large.”
Theorem 9.1.
Let α, β be exceptional slopes of the form α = ε (cid:18) p q (cid:19) β = ε (cid:18) p + 12 q (cid:19) , where p is even. The exceptional bundle E β is semistable along the semicircular wall W E α ,E β , and stableoutside this wall. We will see later that the wall W E α ,E β is nonempty and lies to the left of the vertical wall s = µ ( E β ) = β so long as q ≥
1. In case β is an integer, we have E β = O P ( β ) and E β is ( s, t )-stable for all s < β by[ABCH, Proposition 6.2]. We thus assume q ≥ o prove the theorem it will be necessary to simultaneously address the stability of shifted exceptionalbundles of the form E β [1]. The next lemma will allow us to treat these on an essentially equal footingwith ordinary exceptional bundles. Lemma 9.2.
Let E be a Mumford-stable sheaf, and suppose either E ∈ Q s or E ∈ F s ; write E ′ = E inthe first case and E ′ = E [1] in the second case, so that E ′ ∈ A s . Then E ′ is ( s, t ) -stable for sufficientlylarge t . If E ′ is ( s , t ) -semistable for some ( s , t ) , then E ′ is ( s, t ) -stable whenever the semicircular wallpassing through ( s , t ) is nested inside the semicircular wall passing through ( s, t ) . In [ABCH, Section 6] the case where E ∈ Q s was handled; when E ∈ F s similar methods can be used,so we omit the proof. Note in particular that the walls for E [1] are the same as the walls for E ; however,while the relevant family of semicircles for a sheaf E ∈ Q s is the family to the left of the vertical wall s = µ ( E ), the relevant family of semicircles for E [1] when E ∈ F s is the family to the right of the verticalwall s = µ ( E ), as this is the region where E [1] ∈ A s .It will be useful to introduce some additional exceptional slopes. We put ζ = ε (cid:18) p + 42 q − (cid:19) ζ = ε (cid:18) p − q (cid:19) α = ε (cid:18) p q (cid:19) β = ε (cid:18) p + 12 q (cid:19) η = ε (cid:18) p + 22 q (cid:19) ω = ε (cid:18) p + 42 q (cid:19) ω = ε (cid:18) p − q + 3 (cid:19) , where p is even and q ≥
1. Observe that β = α.η , if p ≡ α = ζ .η , and if p ≡ η = α.ω . The significance of the slopes ζ i and ω i is provided by the following result from [Dr1]. Theorem 9.3.
Let i ∈ { , } be such that i ≡ p (mod 4) . There are exact sequences of vector bundles → E ζ i → E α ⊗ Hom( E α , E β ) → E β → and → E β → E η ⊗ Hom( E β , E η ) ∗ → E ω i → . The theorem provides an inductive description for building up E β in terms of simpler exceptionalbundles. We record how these exact sequences interact with the categories A s . Proposition 9.4.
For each of the following four cases, there exists some ( s, t ) such that the displayedsequence is an exact sequence of objects of A s with the same µ s,t -slope. (1) For p ≡ or q = 1 , → E α ⊗ Hom( E α , E β ) → E β → E ζ [1] → . (2) For p ≡ and q ≥ , → E ζ → E α ⊗ Hom( E α , E β ) → E β → . (3) For p ≡ and q ≥ , → E β [1] → E η [1] ⊗ Hom( E β , E η ) ∗ → E ω [1] → . (4) For p ≡ or q = 1 , → E ω → E β [1] → E η [1] ⊗ Hom( E β , E η ) ∗ → . Before proving the proposition let us isolate some numerical facts that will be useful. emma 9.5. Let α, β be two exceptional slopes as in this section, except do not require that p be even(so that, in particular, this result also applies to the pair of slopes ( β, η ) from this section). The centerof the wall W E α ,E β is located at the point ( x, with x = α + β β − ∆ α α − β , and the radius ρ satisfies ρ = (cid:18) α − β (cid:19) − P ( α − β ) + (cid:18) ∆ β − ∆ α α − β (cid:19) . In fact, the formula for the center is valid for any two exceptional slopes (the formula for the radius isnot).
The proof is a straightforward computation with the formulas of the previous section. Quantities ofthe form (∆ β − ∆ α ) / ( α − β ) occur frequently, and we must estimate them. Lemma 9.6.
With the notation of this section, suppose q ≥ . Then ∆ β − ∆ α α − β < − and ∆ η − ∆ β β − η > . When q = 1 , the first quantity equals − / and the second is / .Proof. We verify the first inequality. It is equivalent to α − α > β − ∆ β − ∆ α = β − P ( α − β ) . This can be rearranged to give 1 r α >
12 ( β − α )(5 + α − β ) . Since α < β , it is enough to show 52 ( β − α ) ≤ r α . Since β = α.η , we find β − α = 1 r α (3 + α − η ) . As q ≥ η − α ≤ /
2, and the required inequality follows. (cid:3)
Corollary 9.7.
The walls W E α ,E β and W E β ,E η are nonempty.Proof. Lemmas 9.5 and 9.6 show the radius ρ of each wall satisfies ρ > (cid:3) Proof of Proposition 9.4.
In each case, the result amounts to showing that an appropriate (nonempty)potential wall lies in the region where all the objects in the exact sequence lie in the category A s .(1) Suppose p ≡ W E α ,E β = W E ζ ,E β = W E ζ ,E α lies in the strip { ( s, t ) : ζ < s < α } , where the exact sequence is valid in A s . For this we may show the center of thewall lies in the strip. Using the description of the wall as W E ζ ,E α , we must verify the inequalities ζ < α + ζ ζ − ∆ α α − ζ < α. But α − ζ ≥ q = 1), so (cid:12)(cid:12)(cid:12)(cid:12) ∆ ζ − ∆ α α − ζ (cid:12)(cid:12)(cid:12)(cid:12) < , nd both inequalities hold.(2) If p ≡ q ≥
2, then noting ζ < α < β we claim the wall W E α ,E β = W E α ,E ζ lies tothe left of the vertical wall s = ζ , i.e. that α + ζ ζ − ∆ α α − ζ < ζ . Recalling ζ = ε (( p − / q ), we see that Lemma 9.6 gives∆ ζ − ∆ α α − ζ ≤ − q ≥
2. But ( α + ζ ) / − ζ = ( α − ζ ) / ≤ /
4, so the required inequality holds.Cases (3) and (4) are mirror images of the previous two cases. (cid:3)
Lemma 9.8.
The wall W E α ,E β is nested inside the wall W E α ,E α.β , and the wall W E η ,E β is nested insidethe wall W E η ,E β.η .Proof. We check the first statement. The center of the wall W E α ,E β is positioned at the point ( x,
0) with x = α + β β − ∆ α α − β . We have ( α + β ) / − α = ( β − α ) / ≤ / q ≥
1, so Lemma 9.6 shows x < α and thus the wall W E α ,E β is centered to the left of the vertical wall s = α . To show it is nested inside W E α ,E α.β , we useLemma 8.2 and show the center of the latter wall lies to the left of the center of the former. In symbols,we must establish the inequality α + α.β α.β − ∆ α α − α.β < α + β β − ∆ α α − β . Using the by now standard identities of Lemma 2.1, one easily shows this inequality is equivalent to theinequality 2∆ β < . Note that the wall W E η ,E β is located to the right of the wall s = η ; otherwise the argument isidentical. (cid:3) Corollary 9.9.
The walls W E α ,E β and W E η ,E β have radius smaller than √ / .Proof. Consider the sequence of walls W E α ,E β , W E α ,E α.β , W E α ,E α. ( α.β ) , W E α ,E α. ( α. ( α.β )) , . . . with each wall nested inside the next. The sequence β, α.β, α. ( α.β ) , α. ( α. ( α.β )) , . . . is decreasing and converges to α + x α , and the discriminants of the corresponding exceptional bundlesconverge to 1 /
2. Thus the squares of the radii increase and converge to (cid:16) x α (cid:17) − P ( − x α ) + − ∆ α x α ! = 54 , so the radius of W E α ,E β is smaller than √ / (cid:3) When combined with Proposition 9.4, Lemma 9.8 provides the main technical tool we need to completethe proof of Theorem 9.1. roof of Theorem 9.1. We will prove by induction on q that E β is ( s, t )-semistable along the wall W E α ,E β and that E β [1] is ( s, t )-semistable along the wall W E β ,E η . The conclusion is true for line bundles, soby induction we may assume the result is true for the exceptional slopes ζ i , α, η, ω i . We must show thecorresponding walls where these exceptional bundles and their shifts are destabilized are nested inside W E α ,E β or W E β ,E η as necessary in each case. Case 1: p ≡ or q = 1. Here we have an exact sequence0 → E α ⊗ Hom( E α , E β ) → E β → E ζ [1] → A s with the same µ s,t -slope for each ( s, t ) ∈ W E α ,E β . Decompose α = σ.τ for someexceptional slopes σ , τ with σ < τ . By induction, E α is ( s, t )-semistable outside the wall W E α ,E σ . Wemay write σ = ε (cid:18) p ′ − q ′ (cid:19) α = ε (cid:18) p ′ q ′ (cid:19) α.τ = ε (cid:18) p ′ + 12 q ′ (cid:19) τ = ε (cid:18) p ′ + 22 q ′ (cid:19) with p ′ ≡ q ′ < q . By Theorem 9.3 there is an exact sequence0 → E σ → E α ⊗ Hom( E α , E α.τ ) → E α.τ → , so there is an equality of walls W E α ,E σ = W E α ,E α.τ . Now the sequence of walls W E α ,E α.τ , W E α ,E α. ( α.τ ) , W E α ,E α. ( α. ( α.τ )) , . . . has each wall nested in the next. But β is one of the products α.τ, α. ( α.τ ) , α. ( α. ( α.τ )) , . . . so we conclude W E α ,E σ is nested in W E α ,E β , and E α is ( s, t )-semistable along the wall W E α ,E β .We must also show E ζ [1] is ( s, t )-semistable along W E α ,E β . From Theorem 9.3 we see W E α ,E β = W E α ,E ζ . Noting ζ = ω −
3, the center of W E α ,E ζ is located at the point ( x,
0) with x = α + ω −
32 + ∆ ω − ∆ α α − ω = α.ω −
32 = η − . We may write σ ′ = ε (cid:18) p ′ − q ′ (cid:19) σ ′ .ζ = ε (cid:18) p ′ − q ′ (cid:19) ζ = ε (cid:18) p ′ q ′ (cid:19) τ ′ = ε (cid:18) p ′ + 22 q ′ (cid:19) with p ′ ≡ , and observe ζ = σ ′ .τ ′ . By induction E ζ [1] is ( s, t )-semistable outside the wall W E ζ ,E τ ′ . From the exact sequence0 → E σ ′ .ζ → E ζ ⊗ Hom( E σ ′ .ζ , E ζ ) ∗ → E τ ′ → W E ζ ,E τ ′ = W E ζ ,E σ ′ .ζ . By the same argument as in the previous paragraph (making use ofthe other half of Lemma 9.8), this wall is nested inside W E ζ ,E η − . This semicircle lies in the same familyof semicircles as W E ζ ,E α , (and both are to the right of the vertical wall s = ζ ) so we show that thecenter of W E ζ ,E η − lies to the left of the center of W E ζ ,E α . This amounts to the inequality ω + η − ω − ∆ η η − ω < η − , which can be rearranged to ω + η ω − ∆ η η − ω > η, i.e. η.ω > η , which is true. Case 2: p ≡ and q ≥
2. This case is considerably easier than the previous one. We have anexact sequence 0 → E ζ → E α ⊗ Hom( E α , E β ) → E β → long the wall W E α ,E β . By induction, E α is semistable along the wall W E α ,E ζ = W E α ,E β . If we decompose ζ = σ.τ then E ζ is semistable along W E σ ,E ζ . This wall equals W E ζ ,E ζ .τ , so W E σ ,E ζ is nested inside W E ζ ,E α = W E α .E β . Thus both E α and E ζ are semistable along W E α ,E β .Shifted objects can be handled in the same manner. (cid:3) End of the proof of Theorem 8.5.
By Theorem 9.1 and Corollary 9.9, any exceptional bundle is ( s, t )-semistable outside a wall of radius smaller than √ /
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