aa r X i v : . [ m a t h . AG ] N ov WEAK BRILL-NOETHER FOR RATIONAL SURFACES
IZZET COSKUN AND JACK HUIZENGA
To Lawrence Ein, with gratitude and admiration, on the occasion of his sixtieth birthday
Abstract.
A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the modulispace has no cohomology. G¨ottsche and Hirschowitz prove that on P every moduli space of Giesekersemistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether.In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. Wecompletely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds.We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the firstChern class is nef. Contents
1. Introduction 12. Preliminaries 43. Strong exceptional collections and resolutions 74. Blowups of P revisited 165. Del Pezzo surfaces 17References 201. Introduction
Let X be a smooth, complex projective surface and let H be an ample divisor on X . Let v ∈ K num ( X ) be the (numerical) Chern character of a Gieseker semistable sheaf on X such that theEuler characteristic satisfies χ ( v ) = 0. Let M X,H ( v ) denote the moduli space of Gieseker semistablesheaves on X with Chern character v . We will call v stable if there is a semistable sheaf of character v . Definition 1.1.
The moduli space M X,H ( v ) satisfies weak Brill-Noether if there exists a sheaf E ∈ M X,H ( v ) such that H i ( X, E ) = 0 for all i .By semicontinuity, if E is any sheaf with no cohomology, then the cohomology also vanishes forthe general sheaf in any component of M X,H ( v ) that contains E . Since the cohomology vanishesidentically, for weak Brill-Noether to hold, we must have χ ( v ) = 0. Date : November 9, 2016.2010
Mathematics Subject Classification.
Primary: 14J60, 14J26. Secondary: 14D20, 14F05.
Key words and phrases.
Moduli spaces of sheaves, Brill-Noether theory, rational surfaces, Hirzebruch and del Pezzosurfaces.During the preparation of this article the first author was partially supported by the NSF CAREER grant DMS-0950951535 and NSF grant DMS-1500031, and the second author was partially supported by a National ScienceFoundation Mathematical Sciences Postdoctoral Research Fellowship DMS-1204066 and an NSA Young InvestigatorGrant H98230-16-1-0306.
The weak Brill-Noether property is the key ingredient in constructing effective theta divisors onthe moduli spaces M X,H ( v ) and plays a central role in describing effective cones of moduli spacesand strange duality. Assume that M X,H ( v ) is irreducible. The locusΘ = { E ∈ M X,H ( v ) | h ( X, E ) = 0 } is called the theta locus and is an effective divisor when weak Brill-Noether holds for M X,H ( v ). Inthis paper, we study when Chern characters v satisfy weak Brill-Noether on rational surfaces X .We completely classify Chern characters v satisfying weak Brill-Noether on Hirzebruch surfaces.For general rational surfaces, we give sufficient conditions on Chern characters that guarantee thatweak Brill-Noether holds. In particular, we show that if X is a del Pezzo surface of degree at least4 and ch ( v ) is nef, then weak Brill-Noether holds. We now summarize our results in greater detailand give some examples. Weak Brill-Noether for rank one sheaves.
When the sheaves have rank one, the story isparticularly simple.
Proposition 1.2.
Let L be a line bundle on a smooth projective surface X such that n = χ ( L ) ≥ .Let Z ⊂ X be a general zero-dimensional scheme of length n , and set v = ch( L ⊗ I Z ) , so that χ ( v ) = 0 . Then H i ( X, L ⊗ I Z ) = 0 for all i if and only if H ( X, L ) = H ( X, L ) = 0 . Inparticular, weak Brill-Noether holds for M X,H ( v ) if and only if there exists a line bundle M with ch ( M ) = ch ( v ) and H ( X, M ) = H ( X, M ) = 0 .Proof.
Any torsion-free rank one sheaf on a surface is isomorphic to L ⊗ I Z for a line bundle L and an ideal sheaf I Z of a zero-dimensional scheme. First, suppose H ( X, L ) = H ( X, L ) = 0 and h ( X, L ) = n . Let X [ n ] denote the Hilbert scheme of n points on X and let Z ∈ X [ n ] be a generalsubscheme of length n . Consider the restriction sequence0 → L ⊗ I Z → L → L | Z → . Since Z is general, the map H ( X, L ) → H ( X, L | Z ) is an isomorphism, and we find L ⊗ I Z has nocohomology.Conversely, suppose H ( X, L ) or H ( X, L ) is nonzero. If h ( X, L ) > n , then h ( X, L ⊗ I Z ) > Z ∈ X [ n ] . Thus we may assume h ( X, L ) ≤ n . Then since at least one of H ( X, L ) or H ( X, L ) is nonzero and χ ( L ) = n , we find H ( X, L ) = 0. But H ( X, L ⊗ I Z ) ∼ = H ( X, L ) forany Z ∈ X [ n ] . In particular, weak Brill-Noether holds for M X,H ( v ) if and only if there exists a linebundle M with ch ( M ) = ch ( v ) and H ( X, M ) = H ( X, M ) = 0. (cid:3)
Example 1.3. On P , weak Brill-Noether fails for moduli spaces of rank one sheaves when theslope is − v has rk( v ) ≥ χ ( v )is arbitrary, then the general sheaf E ∈ M P , O (1) ( v ) has at most one nonzero cohomology group.The signs of the Euler characteristic and the slope determine which cohomology group is nonzero.In particular, if χ ( v ) = 0, then for the general stable sheaf, all cohomology groups vanish andweak Brill-Noether holds. The purpose of this paper is to generalize this theorem to other rationalsurfaces.Our first main result classifies Chern characters on Hirzebruch surfaces that satisfy weak Brill-Noether. Let H be any ample class on the Hirzebruch surface F e = P ( O P ⊕ O P ( e )), e ≥
0. Let v be a stable Chern character of rank at least 2 on F e such that χ ( v ) = 0. Let ν ( v ) = c ( v ) r ( v ) = kr E + lr F EAK BRILL-NOETHER FOR RATIONAL SURFACES 3 denote the total slope of v , where E is the section of self-intersection − e and F is a fiber. ByWalter’s Theorem [Wal98, Theorem 1], the moduli spaces M F e ,H ( v ) are irreducible and the generalsheaf is locally free. By Serre duality, we may assume that kr ≥ − kr = − , then lr ≥ − − e . Theorem 1.4.
Let v be a stable Chern character that satisfies these inequalities. Then M F e ,H ( v ) satisfies weak Brill-Noether if and only if l − ker = ν ( v ) · E ≥ − . As Theorem 1.4 demonstrates, the existence of an effective curve C on X such that w = ch( O X ( C )) , χ ( w , v ) > , and ν ( v ) · H > ( K X + C ) · H provides an obstruction to weak Brill-Noether for M X,H ( v ). By Serre duality and stability,Ext ( O X ( C ) , E ) ∼ = Hom( E , O X ( K X + C )) ∗ = 0 . Since χ ( O ( C ) , E ) >
0, we have Hom( O X ( C ) , E ) = 0. Composing with the natural map O X −→ O X ( C ) , we see that Hom( O X , E ) = H ( X, E ) = 0 for every E ∈ M X,H ( v ). We remark that if M X,H ( v ) isnonempty, C must also satisfy ν ( v ) · H ≥ C · H .In order to prove weak Brill-Noether theorems, we need to ensure that these obstructions vanish.Let P X,F ( v ) denote the stack of F -prioritary sheaves on X . For blowups of P , our sharpest resultis the following. Theorem 1.5.
Let X be a blowup of P at k distinct points p , . . . , p k . Let L be the pullback of thehyperplane class of P and let E i be the exceptional divisor over p i . Let v ∈ K ( X ) with r = r ( v ) > ,and write ν ( v ) := c ( v ) r ( v ) = δL − α E − · · · − α k E k , so that the coefficients δ, α i ∈ Q . Assume that δ ≥ and α i ≥ for all i . Suppose that the linebundle ⌊ δ ⌋ L − ⌈ α ⌉ E − · · · − ⌈ α k ⌉ E k has no higher cohomology. If χ ( v ) = 0 , then the stack P X,L − E ( v ) is nonempty and the general E ∈ P
X,L − E ( v ) has no cohomology. In particular, when H is an ample divisor on X that satisfies H · ( K X + L − E ) < v is an H -stable Chern character satisfying the assumptions of Theorem 1.5, then M X,H ( v ) satisfies weakBrill-Noether. On del Pezzo surfaces of large degree we obtain sharper results. Theorem 1.6.
Let X be a del Pezzo surface of degree at least 4. Let v ∈ K ( X ) with χ ( v ) = 0 ,and suppose c ( v ) is nef. Then the stack P X,L − E ( v ) is nonempty and a general E ∈ P
X,L − E ( v ) has no cohomology. The following conjecture may be thought of as a higher rank analogue of the celebrated Segre-Harbourne-Gimigliano-Hirschowitz conjecture [Seg60, Harb84, Gim87, Hir89].
Conjecture 1.7.
Assume X is a general blowup of P and c ( v ) is nef. Let F = L − E . If H isan ample class such that H · ( K X + F ) < , then M X,H ( v ) satisfies weak Brill-Noether. I. COSKUN AND J. HUIZENGA
We use two techniques to prove weak Brill-Noether Theorems. First, we give a resolution of thegeneral sheaf on M X,H ( v ) in terms of a strong exceptional collection satisfying certain cohomologyvanishing properties. This method allows us to prove Theorem 1.4 and show weak Brill-Noether onarbitrary rational surfaces provided v satisfies certain inequalities. The advantage of this methodis that it gives a convenient resolution of the general sheaf in M X,H ( v ). As a consequence, it showsthat the moduli space is unirational. The disadvantage is that this method is only applicable whenthe surface X admits a strong exceptional collection of the desired form.Second, we construct an explicit prioritary sheaf with vanishing cohomology as a sum of linebundles. Walter [Wal98, Proposition 2] proves that on a birationally ruled surface the stack pa-rameterizing sheaves prioritary with respect to the fiber class is smooth and irreducible. Assumingthat the stable sheaves in M X,H ( v ) are prioritary, to prove weak Brill-Noether, it suffices to exhibitone prioritary sheaf with vanishing cohomology. Constructing prioritary sheaves is much easier thanconstructing stable sheaves. In particular, under suitable assumptions, one may construct prioritarysheaves as sums of line bundles. The problem then reduces to the combinatorial problem of findinga prioritary combination of line bundles with no higher cohomology that has the same rank and firstChern class as v . We solve this problem explicitly for del Pezzo surfaces of degree at least 4. Bothof these techniques are applicable much more generally. However, to minimize the combinatorialcomplexity, we make additional assumptions on the Chern character v when convenient. The organization of the paper. In §
2, we collect basic facts about moduli spaces of sheavesand the cohomology of line bundles on rational surfaces. In §
3, we introduce our first method forproving weak Brill-Noether and characterize Chern characters on Hirzebruch surfaces that satisfyweak Brill-Noether. In § §
5, we introduce our second method and show that Chern characterswith nef ch satisfy weak Brill-Noether on del Pezzo surfaces of degree at least 4. Acknowledgements.
We would like to express our gratitude to Lawrence Ein whose unfailingsupport has been invaluable in our careers. We would also like to thank Daniel Levine for makingcomments and corrections on an earlier version of this work.2.
Preliminaries
In this section, we recall standard facts concerning Hirzebruch and del Pezzo surfaces and coho-mology of line bundles on rational surfaces. We refer the reader to [Bea83], [Cos06a], [Cos06b] or[Hart77] for more detailed expositions.
Hirzebruch surfaces.
Let e ≥ F e denote the Hirzebruch surface P ( O P ⊕ O P ( e )). When e ≥
1, let E be the class of the unique section of self intersection E = − e and let F denote the class of a fiber of the projection to P . The surface F is isomorphic to P × P .In that case, let E and F denote the classes of the two rulings. ThenPic( F e ) ∼ = Z E ⊕ Z F with E = − e, E · F = 1 , F = 0 . By adjunction, K F e = − E − ( e + 2) F. Consequently, the Riemann-Roch Theorem implies that χ ( O F e ( aE + bF )) = ( a + 1)( b + 1) − e a ( a + 1)2 . The effective cone of F e is generated by E and F , consequently H ( F e , O F e ( aE + bF )) = 0 if and only if a, b ≥ . By Serre duality, H ( F e , O F e ( aE + bF )) = 0 if and only if a ≤ − b ≤ − − e. EAK BRILL-NOETHER FOR RATIONAL SURFACES 5
Hence, to compute the cohomology of all line bundles, it suffices to assume that a ≥ −
1. In thiscase, since H vanishes and we have computed the Euler characteristic, specifying h determinesthe dimension of all cohomology groups. The following theorem (see [Cos06a], [Hart77, § V.2])summarizes the answer.
Theorem 2.1.
Let O F e ( aE + bF ) be a line bundle on the Hirzebruch surface F e with a ≥ − . Then:(1) h i ( F e , O F e ( − E + bF )) = 0 for ≤ i ≤ and all b .(2) h ( F e , O F e ( bF )) = b + 1 if b ≥ − and otherwise. In particular, h i ( F e , O F e ( − F )) = 0 for ≤ i ≤ .(3) We may assume that a ≥ and b ≥ . If b < ae , then h ( F e , O F e ( aE + bF )) = h ( F e , O F e (( a − E + bF )) . If b ≥ ae , then h ( F e , O F e ( aE + bF )) = χ ( O F e ( aE + bF )) = ( a + 1)( b + 1) − e a ( a + 1)2 . Proof.
We have already observed that H i ( O F e ( − E + bF )) = 0 for i = 0 ,
2. The case i = 1 followsfrom the fact that the Euler characteristic vanishes. This proves (1).Next, since F is a pullback from the base, we have H i ( F e , O F e ( bF )) ∼ = H i ( P , O P ( b )). Part (2)of the theorem follows.We can therefore assume that a ≥
1. If b <
0, then h i ( F e , O F e ( aE + bF )) = 0 for i = 0 , h is determined by the Euler characteristic. Hence, we may assume that a ≥ b ≥
0. If E · ( aE + bF ) = b − ae <
0, then E is in the base locus of the linear system and the map given bymultiplication by a section s E of O F e ( E ) H ( F e , O F e (( a − E + bF )) s E −→ H ( F e , O F e ( aE + bF ))induces an isomorphism. Repeating this process inductively, we reduce to the case when ae ≤ b .Consider the exact sequence0 −→ O F e (( a − E + bF ) −→ O F e ( aE + bF ) −→ O P ( b − ae ) −→ . If b − ae ≥ −
1, we have a surjection H ( F e , O F e (( a − E + bF )) → H ( F e , O F e ( aE + bF )) → . By inductively reducing a to 0, we conclude that h ( F e , O F e ( aE + bF )) = 0. Consequently, h ( F e , O F e ( aE + bF )) = χ ( O F e ( aE + bF )) . (cid:3) Blowups of P . We next record several basic facts concerning the cohomology of line bundles onblowups of P . Let X be the blowup of P at k distinct points p , . . . , p k . Let L denote the pullbackof the hyperplane class on P and let E i denote the exceptional divisor lying over p i . ThenPic( X ) ∼ = Z L ⊕ k M i =1 Z E i with L = 1 , L · E i = 0 , E i · E j = − δ i,j , where δ i,j is the Kr¨onecker delta function. Let D = δL − P ki =1 α i E i be an integral class on X . Since K X = − L + P ki =1 E i , by Riemann-Roch χ ( O X ( D )) = ( δ + 2)( δ + 1)2 − k X i =1 α i ( α i + 1)2 . If D is effective, then δ ≥
0. Otherwise, a general line with class L would be a moving curve with L · D <
0. In particular, by Serre duality, H ( X, O X ( D )) = 0 if δ ≥ − I. COSKUN AND J. HUIZENGA
Example 2.2 (Del Pezzo surfaces) . Del Pezzo surfaces are smooth complex surfaces X with ampleanti-canonical bundle − K X . They consist of P × P and the blowup of P in fewer than 9 pointsin general position. Since P × P is also a Hirzebruch surface, we will concentrate on the surfaces D n , the blowup of P at 9 − n general points. The effective cone of curves on D n is spanned bythe ( − − − C = aL − P − ni =1 b i E i on D n can be obtained by solvingthe equations C = a − − n X i =1 b i , − K D n · C = 3 a − − n X i =1 b i = 1 . For our purposes, it suffices to know that on D n for n ≥
3, the ( − E i for 1 ≤ i ≤ − n , L − E i − E j for i = j and 2 L − E a − E b − E c − E d − E e , where a, b, c, d, e are 5 distinct indices(whenever the number of points is large enough for these classes to exist) (see [Cos06b], [Hart77, § V.4]).We will need the following cohomology computations.
Lemma 2.3.
Let I ⊂ { , . . . , k } be a possibly empty index set. Then:(1) We have H i ( X, O X ( D )) = 0 for all i if D is one the following − H + X i ∈ I E i , − H + X i ∈ I E i , − E j + X i ∈ I,i = j E i . (2) Assume H i ( X, O X ( D )) = 0 for i > . If D · E j ≥ (respectively, D · L ≥ − ), then H i ( X, O X ( D + E j )) = 0 (respectively, H i ( X, O X ( D + L )) = 0 ) for i > .Proof. If an effective class is represented by a smooth rational curve C on a smooth rational surface X , then we claim that O X ( − C ) has no cohomology. Since H i ( X, O X ) = H i ( C, O C ) = 0 for i ≥ → O X ( − C ) → O X → O C → H i ( X, O X ( − C )) = 0 for all i . Since E i , H and 2 H can be represented by the exceptionalcurve, a line and a conic, respectively, the proposition is true when I = ∅ . If D is a class such that H i ( X, O X ( D )) = 0 for all i and E j · D = 0, then H i ( X, O X ( D + E j )) = 0 for all i . To see this,consider the exact sequence0 → O X ( D ) → O X ( D + E j ) → O P ( − → . Since O X ( D ) and O P ( −
1) have no cohomology, O X ( D + E j ) has no cohomology. Similar sequencesimply the last statement. (cid:3) Blowups of Hirzebruch surfaces.
Since the blowup of F at one point is isomorphic to theblowup of P at 2 points and F is isomorphic to the blowup of P at one point, we may assumethat e ≥
2. Let X be the blowup of F e along k distinct points p , . . . , p k which are not containedin the exceptional curve E . Then the Picard group of X is the free abelian group generated by E, F, E , . . . , E k , where E and F are the pullbacks of the two generators from F e and E , . . . , E k are the exceptional divisors lying over p , . . . , p k . The same argument as in Lemma 2.3 proves thefollowing. Lemma 2.4.
Let I ⊂ { , . . . , k } be a possibly empty index set. Then H i ( X, O X ( D )) = 0 for all i if D is one the following − E + mF + X i ∈ I E i ( m ∈ Z ) , − F + X i ∈ I E i , − E j + X i ∈ I,i = j E i . Moreover, assume H i ( X, O X ( D )) = 0 for i > and C is a rational curve with C · D ≥ − C − .Then H i ( X, O X ( D + C )) = 0 for i > . EAK BRILL-NOETHER FOR RATIONAL SURFACES 7
Moduli spaces of vector bundles.
Next, we recall some basic facts concerning moduli spacesof Gieseker semistable sheaves and prioritary sheaves. We refer the reader to [CH15], [Hui16b],[HuL10] and [LeP97] for details.Let (
X, H ) be a polarized, smooth projective surface. All the sheaves we consider will be pure-dimensional and coherent. If E is a pure d -dimensional, coherent sheaf, then the Hilbert polynomialhas the form P E ( m ) = χ ( E ( mH )) = a d m d d ! + l.o.t.The reduced Hilbert polynomial of E is defined by p E = P E /a d . A sheaf E is Gieseker semistable iffor every proper subsheaf F ( E , we have p F ≤ p E , where polynomials are compared for sufficientlylarge m . The sheaf is called Gieseker stable if for every proper subsheaf the inequality is strict.By theorems of Gieseker, Maruyama and Simpson, there exist projective moduli spaces M X,H ( v )parameterizing S -equivalence classes of Gieseker semistable sheaves on X with Chern character v (see [HuL10] or [LeP97]).It is often hard to verify the stability of a sheaf. The following notion provides a more flexiblealternative. Definition 2.5.
Let F be a line bundle on X . A torsion-free coherent sheaf E is F -prioritary ifExt ( E , E ⊗ F − ) = 0.We denote the stack of F -prioritary sheaves on X with Chern character v by P X,F ( v ). Thestack P X,F ( v ) is an open substack of the stack of coherent sheaves. In this paper, we will consider F -prioritary sheaves on (blowups of) F e and blowups of P , where F is the fiber class on F e andthe class L − E on a blowup of P . The class F endows these surfaces with the structure of abirationally ruled surface. The following theorem of Walter will be crucial to our arguments. Theorem 2.6 ([Wal98, Proposition 2]) . Let X be a birationally ruled surface, let F be the fiber classon X and let v be a fixed Chern character of rank at least 2. Then the stack P X,F ( v ) of F -prioritarysheaves is smooth and irreducible. In particular, if H is an ample divisor on a birationally ruled surface X such that H · ( K X + F ) < v is a stable Chern character of rank at least 2, then the moduli space M X,H ( v ) is irreducible andnormal [Wal98, Theorem 1]. The inequality H · ( K X + F ) < F -prioritary. When Walter’s Theorem applies, we can construct a prioritary sheaf withno cohomology to deduce that general Gieseker semistable sheaves with the same invariants have nocohomology. The advantage is that prioritary sheaves are much easier to construct than semistablesheaves.In our computations, we will use the following consequence of Riemann-Roch repeatedly. Lemma 2.7.
Let E be a sheaf of rank r on a surface X such that χ ( E ) = 0 and let M be a linebundle. Then χ ( E ⊗ M ) = ch ( E ) · ch ( M ) + r ( χ ( M ) − χ ( O X )) . Proof.
By the Hirzebruch-Riemann-Roch Theorem χ ( E ⊗ M ) = Z X ch( E ⊗ M ) td( X ) = Z X ch( E ) ch( M ) td( X ) . The formula follows immediately by expanding this expression. (cid:3) Strong exceptional collections and resolutions
In this section, we introduce our first method for proving weak Brill-Noether theorems. Thismethod provides a resolution of the general sheaf of the moduli space in terms of a strong excep-tional collection that satisfies certain cohomological properties. The method gives a unirational
I. COSKUN AND J. HUIZENGA parameterization of (a component of) the moduli space. The disadvantage is that it is only ap-plicable when a suitable strong exceptional collection exists. We begin by recalling some standardterminology.
Definition 3.1.
A sheaf A is exceptional if Hom( A, A ) = C and Ext i ( A, A ) = 0 for i = 0. Anordered collection ( A , . . . , A m ) of exceptional sheaves on a projective variety X is an exceptionalcollection if Ext i ( A t , A s ) = 0 for 1 ≤ s < t ≤ m and all i. The exceptional collection is strong if in additionExt i ( A s , A t ) = 0 for 1 ≤ s < t ≤ m and all i > . Example 3.2.
On the Hirzebruch surface F e , the collection of line bundles O F e ( − E − ( e + 1) F ) , O F e ( − E − eF ) , O F e ( − F ) , O F e is a strong exceptional collection. If 1 ≤ s < t ≤ i ( A t , A s ) ∼ = H i ( F e , − F ) or H i ( F e , − E + bF ) for some b. Since these cohomology groups vanish by Theorem 2.1, we conclude that the collection is exceptional.Similarly, Ext i ( A s , A t ) ∼ = H i ( F e , F ) or H i ( F e , E + bF ) for some b ≥ e − . Since these cohomology groups vanish for i >
Example 3.3.
Let Γ = { p , . . . , p k } be a set of k distinct points on P and let X be the blowupof P along Γ. Let E i denote the exceptional divisor lying over p i and let L be the pullback of thehyperplane class from P . Then O X ( − L ) , O X ( − L ) , O X ( − E ) , O X ( − E ) , . . . O X ( − E k ) , O X is a strong exceptional collection on X . This can be checked as follows (see also Bondal’s Theorem[Bon89], [KO95]). Let I ⊂ { , . . . , k } be an index set. By Lemma 2.3, H j ( X, O X ( D )) = 0 for all j and D of the form − L + X i ∈ I E i , − L + X i ∈ I E i , − E i , or E l − E i , l = i. Since for 1 ≤ s < t ≤ m , each Ext i ( A t , A s ) is isomorphic to one of these cohomology groups, weconclude that the collection is exceptional. Similarly, H j ( X, O X ( D )) = 0 for j > D of theform 2 L, L − E i , L, L − E i , E l − E i or E l . Since the groups Ext i ( A s , A t ) for 1 ≤ s < t ≤ m are isomorphic to one of these groups, we concludethat the collection is a strong exceptional collection. Notation 1.
Throughout this section, let X be a smooth projective surface and let( A , . . . , A m , O X ) be a strong exceptional collection on X . Suppose that a sheaf E has a resolutionof the form(1) 0 −→ j M i =1 A ⊕ a i i φ −→ m M i = j +1 A ⊕ a i i −→ E −→ . Notice that O X is the last member of the strong exceptional collection and does not occur in theresolution of E . EAK BRILL-NOETHER FOR RATIONAL SURFACES 9
Lemma 3.4.
Let E be a sheaf with a resolution given by Sequence (1). Then H i ( X, E ) = 0 for all i . The exponents a i are determined by the following relations: a s = − χ ( E , A s ) − s − X i =1 a i hom( A i , A s ) for ≤ s ≤ j and a t = χ ( A t , E ) − m X i = t +1 a i hom( A t , A i ) for j + 1 ≤ t ≤ m. Proof.
Since ( A , . . . , A m , O X ) is a strong exceptional collection, Ext i ( O X , A s ) = 0 for all 1 ≤ s ≤ m and all i . Applying Ext( O X , − ) to the Sequence (1), we conclude thatExt i ( O X , E ) = H i ( X, E ) = 0 . To compute the exponents a s with 1 ≤ s ≤ j , we apply Ext( − , A s ) to the same sequence. SinceExt k ( A i , A s ) = 0 for all k if i > s and Ext k ( A i , A s ) = 0 for k >
0, we obtain the relation χ ( E , A s ) + s X i =1 a i hom( A i , A s ) = 0 . The desired formula follows from the fact that hom( A s , A s ) = 1. Similarly, to compute the exponents a t with j < t < m , we apply Ext( A t , − ) to the sequence. We deduce that χ ( A t , E ) = a t + m X i = t +1 a i hom( A t , A i ) . This concludes the proof of the lemma. (cid:3)
Lemma 3.5.
Let E be a locally free sheaf with a resolution given by Sequence (1) and let F be aline bundle on X . Assume that(1) Ext ( A i , A s ⊗ F − ) = 0 for ≤ i ≤ j and j < s ≤ m , and(2) Ext ( A i , A s ⊗ F − ) = 0 for j < i, s ≤ m .Then E is F -prioritary.Proof. We need to check that Ext ( E , E ⊗ F − ) = 0. By applying Ext( E , − ) to the sequence0 −→ j M i =1 A ⊕ a i i ⊗ F − −→ m M i = j +1 A ⊕ a i i ⊗ F − −→ E ⊗ F − −→ , it suffices to check Ext ( E , A s ⊗ F − ) = 0 for s > j . We now apply Ext( − , A s ⊗ F − ) to Sequence(1) to obtain M ≤ i ≤ j Ext ( A i , A s ⊗ F − ) ⊕ a i −→ Ext ( E , A s ⊗ F − ) −→ M j
Let E be a locally free sheaf with a resolution given by Sequence (1). Assume that E is F -prioritary and has Chern character v . Set U = j M i =1 A ⊕ a i i and V = m M i = j +1 A ⊕ a i i . Then the open set S ⊂ Hom(
U, V ) parameterizing locally free F -prioritary sheaves is a completefamily of F -prioritary sheaves. If the stack P X,F ( v ) is irreducible, then the general sheaf in P X,F ( v ) has no cohomology.Proof. To show that the family is complete, we need to show that the Kodaira-Spencer map issurjective. The Kodaira-Spencer map κ : T φ S = Hom( U, V ) → Ext ( E , E )factors as the composition of the two maps [HuL10], [LeP97]Hom( U, V ) µ −→ Hom( U, E ) ν −→ Ext ( E , E ) , where µ and ν are the morphisms that appear in the natural exact sequences obtained by applyingExt( U, − ) and Ext( − , E ), respectively:Hom( U, V ) µ −→ Hom(
U, E ) −→ Ext ( U, U ) , and Hom( U, E ) ν −→ Ext ( E , E ) −→ Ext ( V, E ) . Since ( A , . . . , A m , O X ) is a strong exceptional collection, we have thatExt ( U, U ) = 0 , Ext ( V, V ) = 0 and Ext ( V, U ) = 0We conclude that µ is surjective. Applying Ext( V, − ) to Sequence (1), we also conclude thatExt ( V, E ) = 0and the map ν is surjective. Therefore, the Kodaira-Spenser map is surjective. Since being F -prioritary is an open condition, we conclude that we have a complete family of F -prioritary sheaves.If P X,F ( v ) is irreducible, then prioritary sheaves having a resolution of the form (1) give a Zariski-dense open subset of P X,F ( v ). By Lemma 3.4, these sheaves have no cohomology. This concludesthe proof of the lemma. (cid:3) In view of Proposition 3.6, it is useful to know when E given by a resolution of the form (1) islocally free. We recall a standard Bertini-type theorem for the reader’s convenience (see [Hui16b,Proposition 2.6]). Lemma 3.7.
Assume that H om ( A s , A t ) is globally generated for ≤ s ≤ j and j < t ≤ m and that rk( E ) ≥ . Then a general sheaf E given by a resolution of the form (1) is locally free. Weak Brill-Noether for Hirzebruch surfaces.
As an application of our discussion, wedetermine when weak Brill-Noether holds for Hirzebruch surfaces. Let v be a Chern character ofrank r ( v ) ≥ χ ( v ) = 0 on a Hirzebruch surface F e . Let H be an ampledivisor on F e . Since F e and H will be fixed in this subsection, we denote M F e ,H ( v ) by M ( v ).By Walter’s Theorem 2.6, if the moduli space M ( v ) is nonempty, then it is irreducible and thegeneral sheaf is locally free. In particular, Serre duality gives a birational map between M ( v ) and M ( v D ), where v D is the Serre dual Chern character. Weak Brill-Noether for M ( v ) and M ( v D ) areequivalent problems.We write ν ( v ) = c ( v ) r ( v ) = kr E + lr F for the “total slope” of v , so µ H ( v ) = ν ( v ) · H . Since K F e = − E − (2 + e ) F, we may replace v by v D if necessary to assume kr ≥ − . Furthermore, if kr = − lr ≥ − − e . The Bogomolov inequality givesa further restriction on ν ( v ). EAK BRILL-NOETHER FOR RATIONAL SURFACES 11
Lemma 3.8.
With v as above, if M ( v ) is nonempty, then lr ≥ − ke r .Proof. There is nothing to prove in the case kr = −
1, since then we are already assuming lr ≥ − − e .Assume kr > − lr < − ke r . We show M ( v ) is empty. Let P ( ν ) = χ ( O F e ) + 12 ( ν − ν · K F e ) , so that the Riemann-Roch formula takes the form χ ( v ) = r ( P ( ν ( v )) − ∆( v )) . Then since χ ( v ) = 0, we have∆( v ) = P ( ν ( v )) = (cid:18) kr + 1 (cid:19) (cid:18) lr + 1 (cid:19) − e (cid:18) kr + 1 (cid:19) (cid:18) kr (cid:19) = (cid:18) kr + 1 (cid:19) (cid:18) lr + 1 − ek r (cid:19) < . Thus, by the Bogomolov inequality, M ( v ) is empty. (cid:3) Our assumptions on v now give some simple H -vanishing results for semistable sheaves. Lemma 3.9.
With v as above, if E is an H -semistable sheaf of character v , then H ( F e , E ) = H ( F e , E ( − E )) = 0 . Proof.
Since kr ≥ −
1, Lemma 3.8 in particular implies lr ≥ − − e . In fact, we prove the strongerresult that if v ∈ K ( F e ) is any character such that kr ≥ − lr ≥ − − e , and at least one of theinequalities is strict, then any H -semistable sheaf E of character v has H ( F e , E ) = 0.We use Serre duality to write H ( F e , E ) = Ext ( O F e , E ) = Hom( E , K F e ) ∗ . Since E and K F e are both H -semistable, the vanishing will follow if µ H ( E ) > µ H ( K F e ), which isequivalent to H · ( ν ( v ) − K F e ) >
0. The nef cone of F e is spanned by F and E + eF . We compute F · ( ν ( v ) − K F e ) = kr + 2( E + eF ) · ( ν ( v ) − K F e ) = lr + 2 + e. By our assumption on v , both of these intersection numbers are nonnegative, and at least one ofthem is positive. Since H is ample, it is a positive combination of the extremal nef classes, and itfollows that H · ( ν ( v ) − K F e ) > (cid:3) Next we describe characters v such that weak Brill-Noether fails for M ( v ) if the moduli space isnonempty. It will turn out that these are the only characters where weak Brill-Noether fails. Proposition 3.10.
Let v be a character as above, and suppose E is an H -semistable sheaf ofcharacter v . If χ ( E ( − E )) > , then H ( F e , E ) = 0 . More explicitly, if l − ker = ν ( v ) · E < − , then H ( F e , E ) = 0 .Proof. The restriction sequence 0 → E ( − E ) → E → E| E → H ( F e , E ( − E )) = 0, then H ( F e , E ) = 0. By Lemma 3.9, we have H ( F e , E ( − E )) = 0.Therefore, H ( F e , E ( − E )) = 0 since χ ( E ( − E )) >
0. The second statement follows immediatelyfrom Riemann-Roch. (cid:3)
Conversely, we have the main result of this section.
Theorem 3.11.
Let v be a character as above, and suppose E is a general H -semistable sheaf ofcharacter v . If χ ( E ( − E )) ≤ , then E has no cohomology.Moreover, unless E is a direct sum of copies of O P × P ( − , − , E admits a resolution of the form → O F e ( − E − ( e + 1) F ) a → O F e ( − E − eF ) b ⊕ O F e ( − F ) c → E → for some nonnegative numbers a, b, c .Proof. We may assume E is not a direct sum of copies of O P × P ( − , − E is a direct sum of copies of O P × P if and only if e = 0 and k = l = − r , so we will assume these equalities do not all hold.By Example 3.2, the collection A = O F e ( − E − ( e + 1) F ) , A = O F e ( − E − eF ) , A = O F e ( − F ) , O F e is a strong exceptional collection. Let E be the cokernel of a general map0 → O F e ( − E − ( e + 1) F ) a φ −→ O F e ( − E − eF ) b ⊕ O X ( − F ) c → E → . Step 1: Nonnegativity of the exponents.
First, we determine the exponents a, b, c that give thecorrect Chern class for E and we verify that they are nonnegative.By Lemma 3.4, c = χ ( E ( F )) . By Lemma 2.7, we get c = χ ( E ( F )) = c ( E ) · F + r = k + r ≥ kr ≥ −
1. Hence, c ≥ χ ( O X ( E ) , − ), we get b = − χ ( O X ( E ) , E ) = − χ ( E ( − E )) . Our assumption χ ( E ( − E )) ≤ b ≥ a is the most challenging. By Lemmas 2.7 and 3.4, a = − χ ( E ( − E − F )) = l − ke + k + r. By assumption, l − ke + r ≥ l − ke r ≥ . Hence, if either k ≥ , or k < e ≥
2, then a ≥ k < e = 0 or 1. Suppose a < k ≥ − r , we must have l <
0. We claim thatExt ( O ( E + F ) , E ) = Hom( E , O ( − E − ( e + 1) F )) ∗ = 0by stability. To see this compare the H = E + αF slopes (with α > e ) µ H ( E ) = kr α + lr − ker ≥ kr α + lr − ke r ≥ − α − µ H ( O ( − E − ( e + 1) F )) . The first inequality is strict unless e = 0. If e = 0, then the second inequality is strict unless k = l = − r . Since we are assuming E is not a direct sum of copies of O P × P ( − , − ( O ( E + F ) , E ) = 0 holds. Then since a is negative, we deduce Hom( O ( E + F ) , E ) = 0. This contradicts the H -semistability of E , since µ H ( O ( E + F )) > > µ H ( E ). Therefore a ≥ Step 2: A general sheaf with the specified resolution is locally free of class v . Since H om ( O F e ( − E − ( e + 1) F ) , O F e ( − E − eF )) ∼ = O F e ( F ) and H om ( O F e ( − E − ( e + 1) F ) , O F e ( − F )) ∼ = O F e ( E + eF ) EAK BRILL-NOETHER FOR RATIONAL SURFACES 13 are globally generated, by Lemma 3.7, the cokernel of a general map φ is locally free. Step 3: Any locally free sheaf E with the specified resolution is prioritary. By Theorem 2.1,Ext ( A , A ⊗ O F e ( − F )) = H ( F e , O F e ) = 0 , Ext ( A , A ⊗ O F e ( − F )) = H ( F e , O F e ( E + ( e − F )) = 0and Ext ( A , A ⊗ O F e ( − F )) = Ext ( A , A ⊗ O F e ( − F )) = H ( F e , O F e ( − F )) = 0 , Ext ( A , A ⊗ O F e ( − F )) = H ( F e , O F e ( E + ( e − F )) = 0 . By Lemma 3.5, we conclude that E is F -prioritary. Step 4: Conclusion of the proof.
By Proposition 3.6, the open subset S ⊂ Hom( A a , A b ⊕ A c )parameterizing locally free quotients parameterizes a complete family of prioritary sheaves. Anysheaf parameterized by S has no cohomology by Lemma 3.4. Since the stack P F e ,F ( v ) of prioritarysheaves of character v is irreducible and M ( v ) is a dense open substack of P F e ,F ( v ), we concludethat weak Brill-Noether holds for M ( v ). Furthermore, the general sheaf in M ( v ) has the requiredresolution. In particular, note that the moduli space is unirational since it is dominated by an openset in Hom( A a , A b ⊕ A c ). (cid:3) Applications to blowups of P . In this subsection, let Γ be a set of k distinct points p , . . . , p k on P and let X denote the blowup of P along Γ. Let L denote the pullback of the hyperplane classand let E i denote the exceptional divisor lying over p i . Our methods have the following consequence. Theorem 3.12.
Let v be a Chern character on X such that r ( v ) ≥ , χ ( v ) = 0 , ν ( v ) = δL − k X i =1 α i E i , where δ, α i ≥ and δ − k X i =1 α i ≥ − . Then the stack P X,L − E ( v ) is nonempty, and a general sheaf parameterized by P X,L − E ( v ) has nocohomology.If δ − P ki =1 α i + 2 ≥ , then the general sheaf E in P X,L − E ( v ) has a resolution of the form (2) 0 −→ O X ( − L ) a φ −→ O X ( − L ) b ⊕ k M i =1 O ( − E i ) c i −→ E −→ , Otherwise, E has a resolution of the form (3) 0 −→ O X ( − L ) a ⊕ O X ( − L ) b φ −→ k M i =1 O ( − E i ) c i −→ E −→ . The exponents are given by a = r ( v )( δ − k X i =1 α i + 1) , c i = r ( v ) α i , b = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( v ) + a − k X i =1 c i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Proof.
The linear system | L − E | defines a map from X to P giving X the structure of a birationallyruled surface. By [Wal98, Proposition 2], the stack of prioritary sheaves P X,L − E ( v ) is smooth andirreducible. By Example 3.3, A = O ( − L ) , A = O ( − L ) , A = O ( − E ) , . . . , A k +2 = O ( − E k ) , O is a strong exceptional collection on X . Lemma 3.4 computes the exponents of a sheaf with Cherncharacter v with the given resolutions (2) or (3). Our assumptions on v imply that these exponentsare positive; the requirement on the sign of δ − P ki =1 α i + 2 which is used to determine the formof the resolution ensures that the exponent b is positive. Lemma 3.4 also implies that sheaves withthese resolutions have no cohomology. Since H om ( O X ( − L ) , O X ( − L )) ∼ = O X ( L ) , H om ( O X ( − L ) , O X ( − E l )) ∼ = O X (2 L − E i )and H om ( O X ( − L ) , O X ( − E i )) = O X ( L − E i )are globally generated and the rank of v is at least 2, by Lemma 3.7, the cokernel E of a generalmap φ is locally free in both cases.By Lemma 2.3, H ( X, O X ( D )) = 0 for D a divisor of the form E , L, L + E − E i , ∅ , E − E i . Hence, Ext ( A , A l ⊗ O X ( − L + E )) = 0 for 2 ≤ l ≤ k + 2and Ext ( A , A l ⊗ O X ( − L + E )) = 0 for 3 ≤ l ≤ k + 2 . Similarly, H ( X, O X ( D )) = 0 when D is an integral of the form dL − P a i E i with d ≥ −
2. Hence,Ext ( A i , A l ⊗ O X ( − L + E i )) = 0 for 2 ≤ i, l ≤ k + 2 . By Lemma 3.5, all locally-free sheaves with resolutions given by (2) or (3) are prioritary with respectto O ( L − E ). We conclude that the stack of prioritary sheaves P X,L − E ( v ) is nonempty. Finally, byLemma 3.6, this is a complete family of prioritary sheaves. Since the stack P X,L − E ( v ) is irreducible,the general prioritary sheaf with Chern character v has no cohomology and has a resolution of theform (2) or (3) depending on the sign of the exponent of O X ( − L ). (cid:3) Corollary 3.13.
Let H be an ample divisor on X such that H · ( − L + P ki =2 E i ) < . Let v bea stable Chern character on X satisfying the hypotheses of Theorem 3.12. Then the moduli space M X,H ( v ) is unirational and satisfies weak Brill-Noether.Proof. By Walter [Wal98], an H -semistable sheaf is prioritary with respect to L − E . Consequently, M X,H ( v ) is an open substack of P X,L − E ( v ). The corollary follows from Theorem 3.12. (cid:3) Remark 3.14.
Let X be the blowup of P along k collinear points. Let E be a coherent sheaf withChern character v with µ ( v ) = δL − k X i =1 α i E i , δ − k X i =1 α i < − . Assume E is semistable with respect to an ample divisor H such that µ ( E ) · H > − L · H. Then we claim that the cohomology of E does not vanish and M X,H ( v ) does not satisfy weakBrill-Noether. By stability,Ext ( O ( L − k X i =1 E i ) , E )) = Hom( E , O ( − L )) ∗ = 0 . By Lemma 2.7, χ ( O ( L − k X i =1 E i ) , E ) = rk( E )( − δ + k X i =1 α i − > . We conclude that Hom( O ( L − k X i =1 E i ) , E ) = 0 . EAK BRILL-NOETHER FOR RATIONAL SURFACES 15
However, since the k points are collinear, composing the natural map Hom( O , O ( L − P ki =1 E i )) witha nonzero morphism to E , we see that H ( X, E ) = 0. Hence, without further assumptions on thepositions of the points, Theorem 3.12 is sharp. However, if we assume that the points are general,we can use different strong exceptional collections to extend the range where weak Brill-Noetherholds.3.3. Applications to blowups of Hirzebruch surfaces.
A very similar theorem holds forblowups of Hirzebruch surfaces. Let X be the blowup of a Hirzebruch surface F e , e ≥
1, along k distinct points not lying on the exceptional curve E . Denote by E and F the pullbacks of thecorresponding classes on F e and let E , . . . , E k denote the exceptional divisors lying over the points p , . . . , p k . Let E be a locally sheaf of rank at least 2 with total slope ν ( E ) := ch ( E )rk( E ) = αE + βF − k X i =1 α i E i . Theorem 3.15.
Let v be a Chern character on X such that ν ( v ) satisfies α i ≥ , α − k X i =1 α i ≥ − , and β − k X i =1 α i + 1 ≥ max(( e − α, eα ) . Then the stack P X,F ( v ) is nonempty and the general sheaf in P X,F ( v ) has no cohomology. Further-more, the general sheaf in P X,F ( v ) admits a resolution of the form → O X ( − E − ( e + 1) F ) a φ → O X ( − E − eF ) b ⊕ O X ( − F ) c ⊕ k M i =1 O X ( − E i ) d i → E → , where a = r ( v )( β − ( e − α − k X i =1 α i + 1) , b = r ( v )( β − eα − k X i =1 α i + 1) ,c = r ( v )( α − k X i =1 α i + 1) d i = r ( v ) α i . Proof.
Since the proof of this theorem is similar to the proof of Theorems 3.11 and 3.12, we leavethe routine verifications to the reader. By Lemma 2.4, the sequence O X ( − E − ( e + 1) F ) , O X ( − E − eF ) O X ( − F ) , O X ( − E ) , . . . , O X ( − E k ) , O X is a strong exceptional collection. By Lemma 3.4, the exponents are as claimed and are positiveby assumption. Lemma 3.7 applies to show that the cokernel of a general map φ is locally free.By Lemma 2.4, Lemma 3.5 applies. By Lemmas 3.5 and 3.6, one obtains a complete family of F -prioritary sheaves. Since the stack of F -prioritary sheaves is irreducible, the theorem follows. (cid:3) As usual, we obtain the following corollary.
Corollary 3.16.
Let H be an ample divisor on X such that H · ( − E − ( e + 1) F + P ki =1 E i ) < .Let v be a stable Chern character satisfying the assumptions of Theorem 3.15. Then M X,H ( v ) isunirational and satisfies weak Brill-Noether. Blowups of P revisited Let X = Bl p ,...,p k P be the blowup of P at k distinct points p , . . . , p k ∈ P . In this section westudy the weak Brill-Noether problem for X . In particular, we give some sufficient conditions ona character v of Euler characteristic 0 for weak Brill-Noether to hold for certain moduli spaces ofsheaves of character v . We will prove sharper results when X is a del Pezzo surface of degree atleast 4 in the next section, using the basic tools developed in this section as a starting point. Fix F = L − E , so that the complete series | F | induces a map X → P with fiber class F , exhibiting X as a birationally ruled surface.If H is an ample divisor such that H · ( K X + F ) <
0, then any H -semistable sheaf is automatically F -prioritary, so that P X,F ( v ) contains the stack M X,H ( v ) of H -semistable sheaves as a dense opensubstack whenever H -semistable sheaves of character v exist. Note that such polarizations H alwaysexist; since F · ( K X + F ) = −
2, any ample divisor H spanning a ray sufficiently close to the rayspanned by the nef divisor F will do the trick. In what follows we work primarily with prioritarysheaves instead of semistable sheaves. Prioritary sheaves have the advantage that they are mucheasier to construct than semistable sheaves.4.1. Prioritary direct sums of line bundles.
By Walter’s irreducibility theorem, in order toprove a general prioritary sheaf of some character v has no cohomology, it suffices to construct aparticular such sheaf. It is often possible to do this by considering elementary modifications of directsums of line bundles. First we give a criterion for determining when a direct sum of line bundles isprioritary. Lemma 4.1.
Let E = L ⊕ · · · ⊕ L r be a direct sum of line bundles. Suppose that N is a nef divisorsuch that N · ( L i − L j ) < − N · ( F + K X ) for all i, j . Then E is F -prioritary.Proof. The vanishing Ext ( E , E ( − F )) = 0 will follow if Ext ( L i , L j ( − F )) = 0 for all i, j . We haveExt ( L i , L j ( − F )) ∼ = H ( − L i + L j − F ) ∼ = H ( L i − L j + F + K X ) ∗ . Then N · ( L i − L j + F + K X ) < , and since N is nef we conclude L i − L j + F + K X is not effective. Therefore Ext ( L i , L j ( − F )) = 0. (cid:3) Further prioritary sheaves of lower Euler characteristic can be constructed by elementary modi-fications.
Lemma 4.2.
Let E be an F -prioritary sheaf on X , and let p ∈ X be a point where E is locally free.Pick a surjection E → O p , and consider the elementary modification E ′ defined by the sequence → E ′ → E → O p → . Then E ′ is F -prioritary with χ ( E ′ ) = χ ( E ) − . If furthermore h ( E ) > and p and the map E → O p are general, then additionally h ( E ′ ) = h ( E ) − .Proof. Clearly E ′ is torsion-free since E is. Then Ext ( E ′ , E ′ ( − F )) is a quotient of Ext ( E , E ′ ( − F )),so it suffices to prove the latter group vanishes. Tensoring the exact sequence by O X ( − F ) andapplying Hom( E , − ), we get an exact sequenceExt ( E , O p ) → Ext ( E , E ′ ( − F )) → Ext ( E , E ( − F )) . Then Ext ( E , E ( − F )) = 0 since E is prioritary and Ext ( E , O p ) = 0 since E is locally free at p .If h ( E ) > p is general, then E has a section which does not vanish at p . Therefore a EAK BRILL-NOETHER FOR RATIONAL SURFACES 17 general surjection
E → O p induces a surjective map H ( E ) → H ( O p ), and we conclude h ( E ′ ) = h ( E ) − (cid:3) These two lemmas motivate the next definition.
Definition 4.3.
Fix a nef divisor N such that − N · ( F + K X ) ≥
2. We call a direct sum E = L ⊕ · · · ⊕ L r of line bundles ( N -) good if the following properties are satisfied.(1) E has no higher cohomology: h i ( E ) = 0 for i > i, j we have N · ( L i − L j ) ≤
1. In particular, E is F -prioritary by Lemma 4.1.Fix an integer r ≥
1. We letΛ r,N = { c ( E ) : E is a rank r good direct sum of line bundles } ⊂ N ( X ) Z . The next result follows immediately from Lemma 4.2 and the definitions.
Corollary 4.4.
Suppose v ∈ K ( X ) has χ ( v ) = 0 and r = r ( v ) > , and fix a nef divisor N as inDefinition 4.3. If c ( v ) ∈ Λ r,N , then P X,F ( v ) is nonempty and a general sheaf E ∈ P
X,F ( v ) has nocohomology. Our next result is our strongest result on the weak Brill-Noether problem for an arbitrary blowupof P . Theorem 4.5.
Let X = Bl p ,...,p k P be a blowup of P at k distinct points. Let v ∈ K ( X ) with r = r ( v ) > , and write ν ( v ) := c ( v ) r ( v ) = δL − α E − · · · − α k E k , so that the coefficients δ, α i ∈ Q . Assume that δ ≥ and α i ≥ for all i . Suppose that the linebundle ⌊ δ ⌋ L − ⌈ α ⌉ E − · · · − ⌈ α k ⌉ E k has no higher cohomology. Then choosing N = L , we have c ( v ) ∈ Λ r,L . In particular, if χ ( v ) = 0 ,then P X,F ( v ) is nonempty and the general E ∈ P
X,F ( v ) has no cohomology.Proof. In more detail, write ν ( v ) = (cid:16) d + pr (cid:17) L − (cid:16) a + p r (cid:17) E − · · · − (cid:16) a k + p k r (cid:17) E k where d = ⌊ δ ⌋ and a i = ⌊ α i ⌋ . Suppose M = eL − b E − · · · − b k E k is a line bundle such that e ∈ { d, d + 1 } and b i ∈ { a i , a i + 1 } for all i . Our assumptions imply that M has no highercohomology. Now we can construct a direct sum E = L ⊕ · · · ⊕ L r of line bundles of this form suchthat c ( E ) = c ( v ). Indeed, we only need to ensure that exactly p of the L i ’s have a coefficient of d + 1 on L , and similarly for the other coefficients. By construction, E is L -good. (cid:3) Del Pezzo surfaces
In this section we improve on Theorem 4.5 in the special case of a smooth del Pezzo surface X of degree 4 ≤ d ≤
7. Thus X = Bl p ,...,p k P is a blowup of P at 2 ≤ k = 9 − d ≤ − K X = 3 L − E − · · · − E k is ample. As in the previous section we fix afiber class F = L − E and study F -prioritary sheaves. Our main theorem is the following. Theorem 5.1.
Let X be a del Pezzo surface of degree ≤ d ≤ . Let v ∈ K ( X ) with χ ( v ) = 0 ,and suppose c ( v ) is nef. Then the stack P X,F ( v ) of prioritary sheaves is nonempty and a general E ∈ P
X,F ( v ) has no cohomology. The cone of curves NE( X ) is spanned by the classes of the ( − X ; the ( − d = 4 (reviewExample 2.2). Dually, the nef cone Nef( X ) ⊂ N ( X ) is the subcone of classes ν such that ν · C ≥ − C .The main additional ingredient that goes into the proof of Theorem 5.1 is the action of the Weylgroup on Pic( X ). The Weyl group preserves the intersection pairing, so it preserves the nef cone,the Euler characteristic, and the dimensions of cohomology groups of line bundles. The Weyl groupacts transitively on length ℓ ≤ k configurations ( C , . . . , C ℓ ) of disjoint ( − ℓ = k − K X . For this reason, we will take N = − K X in Definition 4.3 and study ( − K X )-good direct sums of line bundles—note that theneeded inequality K X . ( F + K X ) ≥ d ≥
4. Then the Weyl groupadditionally preserves the set Λ r, − K X of first Chern classes of rank r ( − K X )-good direct sums ofline bundles. Theorem 5.1 is a direct consequence of the next result and Corollary 4.4. Proposition 5.2. If X is a del Pezzo surface of degree ≤ d ≤ we have Nef( X ) Z ⊂ Λ r, − K X . Since the Weyl group preserves both the nef cone and Λ r, − K X , we only need to show that if D ∈ Nef( X ) Z , then some translate of D by a Weyl group element is in Λ r, − K X . Our next lemmafurther allows us to repeatedly replace D with a different divisor D ′ until D ′ is on the boundary ofthe nef cone. Lemma 5.3.
Let D ∈ Nef( X ) Z be a nef divisor, and suppose D · E i ≥ D · E j . Suppose D ′ = D − E i + E j is nef and that D ′ ∈ Λ r, − K X . Then D ∈ Λ r, − K X .Proof. Since D ′ ∈ Λ r, − K X , there is a ( − K X )-good direct sum of line bundles E ′ such that c ( E ′ ) = D ′ .Since D · E i ≥ D · E j , we have D ′ · E i > D ′ · E j , and at least one of the line bundles L ′ in E ′ musthave L ′ .E i > L ′ .E j ≥ −
1. If we put M = L ′ + E i − E j , then since L ′ has no higher cohomologyit follows that M has no higher cohomology. Additionally, M · ( − K X ) = L ′ · ( − K X ). Then if wedefine E by taking E ′ and replacing L ′ with M , it follows that E is ( − K X )-good and c ( E ) = D . (cid:3) Now let D ∈ Nef( X ) Z be arbitrary. If there are i, j such that Lemma 5.3 can be applied,then we replace D with D ′ . We iterate this process until we arrive at a divisor D such that thelemma cannot be applied for any pair of indices i, j . Note that if C is any ( − X , then C · ( − E i + E j ) ≥ −
1. This implies that D must be orthogonal to some ( − k ≥
3, thenapplying an element of the Weyl group we may assume D is orthogonal to E k . In this case, wereduce to studying a divisor class on the blowup at k − k = 2 then by inspection D is orthogonal to either E or E . That is, up to the Weylgroup action, D takes the form D = dL − aE for some 0 ≤ a ≤ d . In the end, we have reduced theproof of Proposition 5.2 to the following statement. Proposition 5.4.
Let D = dL − aE with ≤ a ≤ d be a nef divisor on X = Bl p ,p P , and let r ≥ . Then D ∈ Λ r, − K X . Remark 5.5.
It is crucial to regard D as a divisor on the blowup at 2 points instead of as a divisoron the blowup at 1 point. For example, consider D = 2 L and r = 3. Then D · ( − K X ) = 6, so weneed to express 2 L as a sum of 3 line bundles of ( − K X )-degree 2 with no higher cohomology. Theonly such line bundle on the blowup at 1 point is L − E , so there is no way to achieve this. On theother hand, on the blowup at 2 points, there is the additional line bundle E + E of ( − K X )-degree2, and 2 L = ( L − E ) + ( L − E ) + ( E + E ) . EAK BRILL-NOETHER FOR RATIONAL SURFACES 19
Proof of Proposition 5.4.
The proof is by induction on r . The result is clear for r = 1. Let r ≥ r = 2 and D = L − E , then D = ( L − E ) + E , so we may ignore this case. We will prove thefollowing claim. Claim:
Suppose it is not the case that r = 2 and D = L − E . Write D · ( − K X ) r = 3 d − ar = m + pr (0 ≤ p < r ) . (We clearly have m ≥ M with M · ( − K X ) = m such that M has nohigher cohomology and D − M is nef.First observe that the claim implies the result. Indeed, by induction and Lemma 5.3 it followsthat D − M ∈ Λ r − , − K X . We have( D − M ) · ( − K X ) r − m + pr − , so D − M can be written as c ( E ) for a good sum E of p line bundles of ( − K X )-degree m + 1 and( r − − p line bundles of ( − K X )-degree m . Then D = c ( E ⊕ M ), and E ⊕ M is a good sum.Next we prove the claim by constructing the line bundle M . Note that the claim is trivial if m = 0, since then we can take M = 0. In what follows we assume m >
0. Write m = 3 s + t (0 ≤ t < . For 0 ≤ t < M t as follows: M = 0 M = E M = E + E noting that M t . ( − K X ) = t . Then we put B = sL + M t , and observe B. ( − K X ) = m . The line bundle B is the basic line bundle of ( − K X ) -degree m . Notethat B has no higher cohomology.We now modify B by adding some number α of copies of L − E and some number β of copiesof L − E − E to it; call such a line bundle B ′ = B + α ( L − E ) + β ( L − E − E ) = d ′ L − a ′ E − b ′ E . Then B ′ still has B ′ . ( − K X ) = m since L − E and L − E − E are both orthogonal to K X . For B ′ to be the desired line bundle M that proves the claim, we need to choose α and β so that B ′ hasno higher cohomology and D − B ′ is nef.For D − B ′ to be nef we must have ( D − B ′ ) · E ≥
0, so we have − ≤ b ′ ≤
0. Then either β = 0,or t = 2 and β = 1. Similarly, considering E shows − ≤ a ′ ≤ a . Now suppose we first make β as large as possible, then make α as large as possible, without violating the inequalities b ′ ≤ a ′ ≤ a . We claim that then D − B ′ is nef, i.e. that ( D − B ′ ) · ( L − E − E ) ≥
0. There are a couplecases to consider.
Case 1: t = 2 and a = 0 . In this case α = β = 0. We have D · ( L − E − E ) = d and B ′ = sL + E + E , so B ′ · ( L − E − E ) = s + 2. Then3( d − ( s + 2)) = 3 d − m + t − mr + p − m − m ( r −
1) + p − . Now m ≡ m >
0, so d ≥ s + 2 unless m = 2, s = 0, and d = 1. But then m = ⌊ /r ⌋ ,so this is impossible since r ≥
2. Therefore d ≥ s + 2 and D − B ′ is nef. Note that also B ′ has nohigher cohomology, so we may take M = B ′ in this case to complete the proof. Case 2: t = 2 or a > . If t = 2 but a > β = 1, and therefore b ′ = 0. We alsohave b ′ = 0 if t = 2, so b ′ = 0 in every case. Then B ′ takes one of the following three forms: d ′ L − ( a − E d ′ L − ( a − E d ′ L − aE . In order to have ( D − B ′ ) · ( L − E − E ) ≥
0, we will need to compare d ′ with d . To do this wefirst compute d ′ . Observe that the total number of line bundles added to the basic line bundle is α + β = (cid:22) a + t (cid:23) ;write a + t = 3( α + β ) + ℓ (0 ≤ ℓ < . Then d ′ = s + α + β, and 3( d − d ′ ) = 3 d − s − α + β ) = ( mr + a + p ) − ( m − t ) − ( a + t − ℓ ) = m ( r −
1) + p + ℓ. Therefore d ′ < d in any case, which implies D − B ′ is nef unless B ′ = d ′ L − ( a − E . In this casewe need the stronger inequality d ′ ≤ d −
2, or equivalently we must show m ( r −
1) + p + ℓ > . But if B ′ = d ′ L − ( a − E then ℓ = 2, and the only way the inequality can fail is if m = 1, r = 2,and p = 0. In this case 3 d − a = 2, and since a ≤ d we have d = a = 1. Thus this case is the specialcase D = L − E , r = 2, which we have already excluded. Therefore D − B ′ is nef.Finally we are ready to construct the line bundle M that proves the claim in case t = 2 or a >
0. Starting from the basic line bundle B , add a single copy of L − E − E if it won’t violatethe inequality b ′ ≤ a ′ ≤ a since a > t = 2). After that,repeatedly add copies of L − E . Sometime before the inequality a ′ ≤ a is violated, we will have B ′ · ( L − E − E ) ≤ D · ( L − E − E ). Let M be the first B ′ where this inequality holds. Each copyof L − E − E or M − E that is added decreases the intersection number with M − E − E by2. Thus we will additionally have M · ( L − E − E ) ≥ D · ( L − E − E ) − ≥ −
1. Since b ′ = 0,this implies M has no higher cohomology and D − M is nef. (cid:3) References [Bea83] A. Beauville. Complex algebraic surfaces, volume 68 of London Mathematical Society Lecture Note Series.Cambridge University Press, Cambridge, 1983.[Bon89] A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad Nauk SSS Ser. Mat.53 (1989), 25-44; English transl. in Math. USSR Izv. 34 (1990).[Cos06a] I. Coskun, Degenerations of surface scrolls and the Gromov-Witten invariants of Grassmannians, J. AlgebraicGeom., (2006), 223–284.[Cos06b] I. Coskun, Enumerative geometry of Del Pezzo surfaces via degenerations, Amer. J. Math., no. 3 (2006),751–786.[CH15] I. Coskun and J. Huizenga, The birational geometry of the moduli spaces of sheaves on P , Proceedings ofthe G¨okova Geometry-Topology Conference 2014, (2015), 114–155.[Gim87] A. Gimigliano. On linear systems of plane curves. PhD thesis, Queen’s University, Kingston, 1987.[GHi94] L. G¨ottsche and A. Hirschowitz, Weak Brill-Noether for vector bundles on the projective plane, in Algebraicgeometry (Catania, 1993/Barcelona, 1994), 63–74, Lecture Notes in Pure and Appl. Math., 200 Dekker, NewYork.[Harb84] B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, in Proceedingsof the 1984 Vancouver conference in algebraic geometry , 95–111, CMS Conf. Proc., 6, Amer. Math. Soc.,Providence, RI.[Hart77] R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No.52.[Hir89] A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles g´en´eriques, J.Reine Angew. Math. (1989), 208–213.
EAK BRILL-NOETHER FOR RATIONAL SURFACES 21 [Hui16a] J. Huizenga, Effective divisors on the Hilbert scheme of points in the plane and interpolation for stablebundles, J. Algebraic Geom. (2016), no. 1, 19–75.[Hui16b] J. Huizenga, Birational geometry of moduli spaces of sheaves and Bridgeland stability, preprint (2016).[HuL10] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves , second edition, Cambridge Mathe-matical Library, Cambridge Univ. Press, Cambridge, 2010.[KO95] S. A. Kuleshov and D. O. Orlov, Exceptional sheaves on del Pezzo surfaces,
Russian Acad. Sci. Izv. Math. , no. 3 (1995), 479–513.[LeP97] J. Le Potier, Lectures on vector bundles , translated by A. Maciocia, Cambridge Studies in Advanced Math-ematics, 54, Cambridge Univ. Press, Cambridge, 1997.[Man97] Yu. I. Manin,
Cubic forms: algebra, geometry, arithmetic , translated from the Russian by M. Hazewinkel,North-Holland, Amsterdam, 1974.[Rud88] A. N. Rudakov, Exceptional vector bundles on a quadric, Izv. Akad. Nauk SSSR Ser. Mat., no. 4 (1988),788–812; English translation Math. USSR Izv. (1989), 115–138.[Seg60] B. Segre, Alcune questioni su insiemi finiti di punti in geometria algebrica, Univ. e Politec. Torino Rend.Sem. Mat. (1960/1961), 67–85.[Wal98] C. Walter, Irreducibility of moduli spaces of vector bundles on birrationally ruled surfaces, Algebraic Geom-etry (Catania, 1993/Barcelona, 1994), Lecture Notes in Pure and Appl. Math., (1998), 201–211. Department of Mathematics, Statistics and CS, University of Illinois at Chicago, Chicago, IL 60607
E-mail address : [email protected] Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
E-mail address ::