Devops Ieee Project Ideas

Devops Ieee Project Ideas

Devops Automation Of Software Deployment Pipelin · ML Project

Python · Data Preprocessing · Model Development · Evaluation

Project focus: anomaly detection / classification using IoT sensor streams and timestamped device measurements.

THE BRILL-NOETHER THEORY OF THE MODULI SPACES OF SHEAVES

ON SURFACES

IZZET COSKUN, JACK HUIZENGA, AND HOWARD NUER

In honor of Peter Newstead’s 80th birthday, with great admiration

Abstract. In this paper, we survey recent developments in the Brill-Noether Theory of higher

rank vector bundles on complex projective surfaces. We focus on weak Brill-Noether Theorems on rational and K-trivial surfaces and their applications.

1. Introduction

In this paper, we survey recent developments in the Brill-Noether Theory of higher rank vector bundles on complex projective surfaces. Throughout the paper we work over the field of complex numbers C.

Brill-Noether Theory forms a cornerstone of classical curve theory. It describes representations

of curves in Pn and is a main tool in the study of moduli spaces of curves. The cohomological behavior of a general line bundle L of degree d on a smooth curve C of genus g is determined by d and g (see Proposition 3.1). More generally, the cohomological behavior of a general stable vector bundle V of rank r and degree d on C is determined by r, d and g (see Proposition 3.8). Conse- quently, Brill-Noether Theory focuses on the geometry of loci of vector bundles with unexpected

cohomological behavior. In contrast, the cohomological behavior of a general stable bundle on a surface is not well- understood except for special surfaces or for special Chern characters. Crucially, the cohomology of a general stable bundle on a surface is not in general determined by the Euler characteristic and the slope. Consequently, before one can study the behavior of special vector bundles, one has to understand the behavior of the generic stable bundle. After some preliminaries §2 and reviewing

the properties of the general stable bundles on a curve §3, in §4 we will contrast the behavior of bundles on curves and surfaces and give examples of new phenomena that occur. For example, the stack of coherent sheaves of rank r and degree d on a curve is irreducible. On the other hand, the stack of coherent sheaves with fixed invariants on a surface is almost never irreducible. In many nice settings, focusing on stable sheaves results in an irreducible moduli space, although this is not always the case (see §4). Hence, stability plays a more central role in

the case of surfaces. On a curve of genus g ≥ 2, there exist stable bundles of every rank r > 0 and every degree. In contrast, there are restrictions on the possible Chern characters of stable sheaves on surfaces such as the Bogomolov inequality. Furthermore, we do not have a classification of the

Chern characters of stable bundles on surfaces except in special cases. Finally, the cohomology

20 Mathematics Subject Classification. Primary: 14L30, 14M15, 14M17. Secondary: 14L35, 51N30.

Key words and phrases. Brill-Noether Theory, Moduli spaces of sheaves, Bridgeland stability, Ulrich bundles. During the preparation of this article the I.C. was partially supported by the NSF FRG grant DMS 16642 and NSF grant DMS 2200684, J.H. was partially supported by NSF FRG grant DMS 1664303.

of a general line bundle with fixed invariants on a surface may not be concentrated in a single degree. We often do not know the cohomology of certain line bundles even on relatively simple surfaces such as the blowup of P at 1 or more general points. This presents many challenges for studying the cohomology of higher rank bundles. In §5, we will survey recent developments in computing the cohomology of the general stable sheaf on a surface. We will concentrate on two techniques. First, we will describe results on rational

surfaces using the stack of prioritary sheaves. Second, we will describe results on K-trivial surfaces using Bridgeland stability conditions. We will survey cases when weak Brill-Noether Theorems hold, i.e., when the general stable sheaf has at most one nonzero cohomology group.

In §6, we will describe applications of weak Brill-Noether Theorems. We will concentrate on

the following applications. (1) The classification of Chern characters of stable bundles. After recalling the Drézet-Le Potier classification of stable bundles for P , we will describe recent progress on Hirzebruch surfaces. (2) The tensor product problem and the birational geometry of moduli spaces of sheaves. We will discuss the problem of computing the cohomology of the tensor product of two general stable sheaves and touch on applications to the construction of Brill-Noether divisors and

the birational geometry of the moduli spaces. (3) The classification of Ulrich bundles. We will discuss the relation between weak Brill- Noether Theorems and Ulrich bundles. Finally, in §7, we will survey some recent work on the cohomology jumping loci on moduli spaces of sheaves on P following [GLL22]. Very little is known (or even conjectured) about

Brill-Noether loci on moduli spaces of sheaves on surfaces. In general, the Brill-Noether loci

are reducible with components of different dimensions even in rank 1. When c (v) is a minimal positive class, however, then the Brill-Noether loci behave better as we will demonstrate.

Acknowledgments. We would like to thank Arend Bayer, Aaron Bertram, Lawrence Ein, Ben

Gould, Joe Harris, John Kopper, Daniel Levine, Yeqin Liu, Emanuele Macrı̀, Sayanta Mandal,

Geoffrey Smith and Matthew Woolf for many stimulating discussions on the geometry of moduli

2. Preliminaries

In this section, we collect basic facts and definitions concerning stable bundles. We refer the reader to [CH15, HL10, LeP97] for more details.

2.1. Stability. We begin by recalling the definition of stability.

Definition 2.1. Let X be a projective variety of dimension n equipped with an ample divisor H.

The H-slope of a torsion free coherent sheaf F is defined by

c (F) · H n−1 µH (F) = . rk(F)H n The sheaf F is called µH -(semi)stable if for every proper subsheaf E ⊂ F we have µH (E) < µH (F). (−)

When H is fixed or irrelevant for the discussion, we will omit it from the notation.

For curves, the ample H does not play a role. Taking H to have degree 1, the slope becomes the degree over the rank of F. For higher dimensional varieties, stability depends on H. Every torsion free sheaf F has a unique Harder-Narasimhan filtration

F ⊂ · · · ⊂ Fn = F

such that Ei = Fi /Fi−1 are µH -semistable with µH (E ) > µH (E ) > · · · > µH (En ).

Furthermore, µH -semistable sheaves have a (not necessarily unique) Jordan-Hölder filtration into

µH -stable sheaves. The reflexive hull of the associated graded sheaf is unique up to isomorphism [HL10, Corollary 1.6.10]. Two semistable sheaves are called S-equivalent if their associated graded sheaves are isomorphic. Consequently, stable vector bundles are the building blocks of all vector bundles. If V and W are two µH -semistable sheaves and ϕ : V → W is a nonzero homomorphism, then µH (W) ≥ µH (V). Furthermore, if both are stable and equality holds, then V ∼ = W. When X is a curve, Mumford constructed an irreducible, projective coarse moduli space MX (r, d)

parameterizing S-equivalence classes of semistable sheaves of rank r and degree d. For higher di- mensional varieties, a slightly different notion of stability is necessary. Definition 2.2. Let F be a pure dimensional coherent sheaf of dimension d. Then the Hilbert polynomial PF (m) of F has the form md PF (m) = χ(F(mH)) = ad + l.o.t. d!

The reduced Hilbert polynomial is defined by pF (m) = PFa(m)

d . The sheaf F is H-Gieseker (semi)stable if for every proper subsheaf E ⊂ F, we have pE (m) < pF (m) for m ≫ 0. (−)

For torsion free sheaves, adim(X) = rk(F). Consequently, by the Riemann-Roch Theorem, we

conclude that µH -stability implies H-Gieseker stability and H-Gieseker semistability implies µH - semistability. When rH n and c (F) · H n−1 are relatively prime, then these four notions coincide.

In general, all the implications are strict. Every torsion free sheaf has a Harder-Narasimhan

filtration with respect to H-Gieseker semistability, and every H-Gieseker semistable sheaf has a

Jordan-Hölder filtration with H-Gieseker stable quotients. Two H-Gieseker semistable sheaves

are called S-equivalent if their associated graded objects are isomorphic.

Given a fixed Chern character v, Gieseker [Gie77] and Maruyama [Mar78] constructed a projec-

tive coarse moduli space MX,H (v) parameterizing S-equivalence classes of H-Gieseker semistable sheaves on X. Compared to the case of curves, the geometry of these moduli spaces are less well understood. The purpose of these notes is to explain aspects of the geometry of MX,H (v) when X is a surface and compare and contrast it with the case of curves. Proposition 2.3. Let (X, H) be a polarized surface. If V and W are µH -semistable bundles such that µH (W) > µH (V) + H · KX , then Ext (V, W) = 0. In particular, if H · KX < 0, then

Proof. Let V and W be µH -semistable vector bundles. Then by Serre duality ext (V, W) = hom(W, V(KX )). This is zero if µH (W) > µH (V) + H · KX by stability. □

2.2. The Riemann-Roch Theorem. Let X be a smooth, projective curve of genus g and let V

be a vector bundle of rank r and degree d. Then the Riemann-Roch Theorem computes the Euler characteristic χ(V) χ(V) = h (X, V) − h (X, V) = d + r(1 − g). If X is a smooth projective surface and F is a sheaf of rank r, ch (F) = c and ch (F) = d, then the Euler characteristic is given by

KX · c

χ(F) = rχ(OX ) − + d, where KX is the class of the canonical bundle ωX of X. In these notes, we will be interested in χ(V ⊗ W). It convenient to make a change of coordinates to express the Chern character with the logarithmic invariants. Given the Chern character v of a torsion free sheaf F on a smooth polarized surface (X, H), define the total slope ν and discriminant ∆ of F by the following formulae c (v) ν(v)2 ch (v) ν(v) = , ∆(v) = − .

rk(v) 2 rk(v) Observe that these notions depend only on the Chern character of F. The Chern character can be easily recovered from r, ν and ∆. The advantage of these invariants is that they are additive on tensor products ν(V ⊗ W) = ν(V) + ν(W) and ∆(V ⊗ W) = ∆(V) + ∆(W).

In terms of these invariants, the Riemann-Roch formula reads

χ(V) = r(V)(P (ν(V)) − ∆(V)), where

P (ν) = χ(OX ) + (ν 2 − ν · KX )

is the Hilbert polynomial of OX . More generally, if V and W are two torsion free sheaves, X χ(V, W) = (−1)i exti (V, W) = r(V)r(W)(P (ν(W) − ν(V)) − ∆(V) − ∆(W)), i=0

where exti (V, W) = dim(Exti (V, W)).

3. Brill-Noether for curves

In this section, a curve will always mean a smooth, irreducible, complex projective curve. We will review basic properties of the cohomology of stable vector bundles on curves. In the next section, we will contrast the cases of curves and surfaces.

3.1. Line bundles on curves. Let C be a curve of genus g. Line bundles of degree d on C

are parameterized by Picd (C) which is isomorphic to a g-dimensional abelian variety. The Euler characteristic χ(L) = d − g + 1 determines the cohomology of the general L ∈ Picd (C). Proposition 3.1 (Weak Brill-Noether for line bundles). Let C be a curve of genus g. Let L be a general line bundle of degree d on C. Then h (C, L) = max(0, d − g + 1) and h (C, L) = max(0, g − d − 1).

Proof. If d < 0, then h (C, L) = 0 and h (C, L) = g − d − 1. If 0 ≤ d ≤ g − 1, let C (d) denote the d-th symmetric product of C. Since dim(C (d) ) = d < dim(Picd (C)) = g, the natural map d X d d ϕd : Sym (C) → Pic (C) sending D = pi 7→ OC (D) i=1

cannot be surjective. The fiber of ϕd is the linear system |OC (D)|. Hence, the general line bundle of degree d has no global sections. Finally, if d ≥ g, h (C, L) = d − g + 1 + h (C, L) ≥ 1. Letting ωC denote the canonical bundle of C, by Serre duality, h (C, L) = h (C, ωC ⊗ L−1 ). By Lemma 3.2, general points impose independent conditions on sections of a line bundle. Hence, if L = OC (D) for a general effective divisor D, then h (C, ωC (−D)) = 0 and h (C, L) = d − g + 1. □

Lemma 3.2. Let X be an irreducible projective variety and let L be a line bundle on X. If Z is a general set of m distinct points on X, then h (X, L ⊗IZ ) = max(h (X, L) − m, 0). Proof. If W is a zero-dimensional scheme of length m, then h (X, L ⊗IW ) ≥ max(h (X, L)−m, 0). We need to show that for a general set of points equality holds. This is trivially true if m = 0. By induction, suppose it holds for m ≤ m − 1. Since the symmetric product X (m) is irreducible, by the semi-continuity of cohomology, it suffices to exhibit one Z of length m for which the lemma

holds. Choose Z ′ of length m − 1 for which h (X, L ⊗IZ ′ ) = max(h (X, L) − m + 1, 0). If L ⊗IZ ′ has no global sections, then h (X, L ⊗IZ ) = 0 and we are done. Hence, we may assume that L ⊗IZ ′ has a nonzero section s. Take a point p where s does not vanish. Let Z = Z ′ ∪ p. Then h (X, L ⊗IZ ) ≤ h (X, L ⊗IZ ′ ) − 1 and the proof is complete. □ Definition 3.3. A sheaf F on a projective variety X satisfies weak Brill-Noether if F has at most one nonzero cohomology group. The sheaf F is nonspecial if H i (X, F) = 0 for i > 0. Otherwise,

F is called special. On any curve, a general line bundle L satisfies weak Brill-Noether and, if χ(L) ≥ 0, L is nonspecial. Consequently, classical Brill-Noether theory focuses on loci of line bundles in Picd (C) with unexpected cohomology. Recall that a gdr is a linear system of projective dimension r and degree d on the curve C of genus g. A base-point-free gdr corresponds to a nondegenerate morphism C → Pr of degree d. The curve C admits a nondegenerate map to Pr of degree at most d if and only if C has a gdr . The

Brill-Noether number is defined by

ρ(g, r, d) = g − (r + 1)(g − d + r). Let Wdr (C) be the locus of line bundles L ∈ Picd (C) such that h (C, L) ≥ r + 1. The scheme

Wdr (C) has a natural determinantal structure. The celebrated Brill-Noether Theorem determines

the structure of Wdr (C) on a general curve. Theorem 3.4 (Brill-Noether). Let C be a curve of genus g which is general in moduli. (1) The curve has a grd if and only if ρ(g, r, d) ≥ 0 [GH80, Griffiths and Harris].

(2) If ρ ≥ 0, then Wdr (C) is normal, Cohen-Macaulay of dimension ρ and smooth away from Wdr+1 (C) [Gie82, Gieseker]. (3) Wdr (C) is irreducible if ρ > 0 [FL81, Fulton and Lazarsfeld]. (4) When ρ ≥ 0, the universal space Wdr of gdr ’s has a unique component dominating the moduli space of curves [EH87, Eisenbud and Harris].

Unlike Proposition 3.1, the Brill-Noether Theorem requires C to be general in moduli. For

special curves, Wdr (C) may be reducible with components of larger than the expected dimension ρ. The structure of Wdr (C) on special curves is not fully understood and is an active area of research. For example, Larson, Larson and Vogt recently described Wdr (C) for a general curve C with fixed gonality [LLV20].

3.2. Higher rank vector bundles on curves. In this subsection, we will recall facts concerning

stable bundles on curves and their Brill-Noether theory.

3.2.1.LRational curves. A vector bundle V on P is isomorphic to a direct sum of line bundles

V ∼ r

= i=1 OP (ai ) for a unique sequence of integers a ≤ a ≤ · · · ≤ ar . In particular, there are no stable bundles of rank greater than one on P . There exists a semistable vector bundle V of rank r and degree d on P if and only if r divides d, in which case V ∼ = OP ( dr )⊕r . Lr The cohomology of the vector bundle V = i=1 OP (ai ) is given by

X X

h (P , V) = (ai + 1) and h (P , V) = (−1 − ai ). ai ≥0 ai ≤−2

3.2.2. Elliptic curves. Let E be an elliptic curve. Vector bundles on E have been classified by

Atiyah [Ati57]. There exists a stable vector bundle of rank r and degree d on E if and only if r and d are relatively prime. In this case, the determinant map gives an isomorphism between ME (r, d) and E. In particular, by taking direct sums, there exist semistable bundles of every rank r and degree d on E. If V and W are two semistable bundles on E with µ(V) > µ(W), then

Ext (W, V) ∼

= Hom(V, W)∗ = 0 by semistability. By induction on the length of the Harder-Narasimhan filtration, we conclude that a vector bundle on E is a direct sum of semistable vector bundles. Hence, the cohomology of any bundle on E is determined by the cohomology of the bundles appearing in its Harder-Narasimhan filtration. Proposition 3.5. Let V be a semistable bundle on an elliptic curve E. Then V satisfies weak Brill-Noether if and only if OE is not a Jordan-Hölder factor of V.

Proof. Let V be a stable vector bundle on E. If µ(V) < 0, then H 0 (E, V) = 0 by stability. Similarly, if µ(V) > 0, then H 1 (E, V) = H 0 (E, V ∗ ) = 0. If µ(V) = 0 and V is stable, then V must be a line bundle of degree 0. The cohomology of V in that case vanishes except when V = OE . We conclude that on an elliptic curve all stable bundles except OE satisfy weak Brill-Noether. By considering the Jordan-Hölder filtration, we conclude that any semistable bundle V on E satisfies

weak Brill-Noether provided that OE is not a Jordan-Hölder factor of V. It is easy to see that if

OE is a Jordan-Hölder factor, then V must have nonvanishing cohomology. □

Example 3.6. The cohomology of a semistable bundle whose Jordan-Hölder factors contain OE

depends on the extension class. Let

0 → OE → V → OE → 0

be the unique nontrivial extension of OE by OE . Then h (E, V) = h (E, V) = 1, whereas h (E, OE ⊕ OE ) = h (E, OE ⊕ OE ) = 2.

3.2.3. Curves of higher genus. For the rest of this section, let C be a curve of genus g ≥ 2. The

following theorem describes the moduli spaces of semistable sheaves on C. Theorem 3.7. Let C be a smooth curve of genus g ≥ 2. (1) Then the stack of coherent sheaves Cohd,r of degree d and rank r on C is a smooth, irre- ducible Artin stack. (2) The moduli space MC (r, d) is an irreducible, projective variety of dimension r (g − 1) + 1, which contains the locus of stable bundles as a dense open set. It is smooth at points corresponding to stable bundles. In particular, if r and d are relatively prime, then MC (r, d)

Sketch of proof. We refer the reader to [Hof10] for a detailed proof of (1). Briefly, using the

Grothendieck Quot scheme, one can show that Cohd,r is algebraic. Obstructions to deformations

of a coherent sheaf F are contained in Ext (F, F). Since Ext (F, F) = 0 on C, Cohd,r is smooth of dimension −χ(F, F). Finally, by induction on rank and degree, one can show that Cohd,r is connected to conclude (1). The irreducibility of MC (r, d) (when nonempty) follows from openness of semistability. We now construct semistable vector bundles of rank r and degree d on curves of genus g ≥ 1. First, assume r and d are coprime. Let es < dr < ft be the three consecutive fractions in the Farey

sequence for r. This implies that the middle fraction is the mediant of its two neighbors: d e+f = . r s+t Then s and t are strictly less than r and the e, s (similarly f, t) are coprime. By induction on the rank, there exist stable bundles V 1 and V 2 of rank and degree (s, e) and (t, f ), respectively. Since χ(V 2 , V 1 ) = et − sf + r(1 − g) < 0, there exists a nonsplit extension of the form

0 → V 1 → V → V 2 → 0.

We claim that V is stable. Otherwise, let W be the maximal destabilizing subbundle. Since µ(W) > µ(V), r(W) < r(V) and ft is the next smallest fraction in the Farey series for r, we must have µ(W) ≥ µ(V 2 ). Hence, either µ(W) > µ(V 2 ) and the natural map from W to V 2 is zero or µ(W) = µ(V 2 ). In the first case, there is an induced map from W to V 1 , which is a contradiction since W is stable and µ(V 1 ) < µ(W). In the second case, we must have W = V 2 and we get a splitting of the sequence, contrary to our assumptions.

Hence, when r and d are coprime, the moduli space MC (r, d) is nonempty. Furthermore, if V is a stable bundle, then hom(V, V) = 1, and ext (V, V) = r (g − 1) + 1.

By basic deformation theory, T[V] MC (r, d) = Ext (V, V) and MC (r, d) is smooth at V of dimension r (g − 1) + 1. If r and d are not coprime, let (r, d) = (kr′ , kd′ ), where r′ and d′ are coprime. There exists a stable vector bundle V of rank and degree (r′ , d′ ). Taking the direct sum of k copies of V, we obtain a semistable bundle of rank r and d. Hence, if g ≥ 1, MC (r, d) is nonempty. Now let g ≥ 2. By induction on the gcd k, assume that there are stable bundles for all k ′ < k.

The dimension of the stack MC (r, d) is r (g − 1). Consider the first step of a Jordan-Hölder filtration

0 → V → V → V → 0

with the rank and degree of V i equal to (ki r′ , ki d′ ) and k + k = k. We have ext (V 2 , V 1 ) = k k (r′ )2 (g − 1) + hom(V 2 , V 1 ). Since V 1 is stable, either hom(V 2 , V 1 ) = 0 or V 1 is a factor of the Jordan-Hölder filtration of V 2 .

In the first case, the dimension of such extensions is bounded above by

(k r′ )2 (g − 1) + k k (r′ )2 (g − 1) + (k r′ )2 (g − 1) < r (g − 1). In the latter case, V 1 is determined up to finitely many choices and hom(V 2 , V 1 ) ≤ kk . Hence such loci is bounded by k + k k (r′ )2 (g − 1) + (k r′ )2 (g − 1) < r (g − 1). k We conclude that there must be stable bundles of rank r and degree d on C. □ Proposition 3.8 (Weak Brill-Noether for curves). Let V be a general semistable vector bundle of rank r and degree d on a curve C of genus g. Then V satisfies weak Brill-Noether:

h (V) = max(0, d − r(g − 1)), and h (V) = max(0, r(g − 1) − d). Proof. The proposition is easy for g = 0 and follows from Proposition 3.5 for g = 1. Hence, we may assume that g ≥ 2. By the semicontinuity of cohomology and Theorem 3.7, it suffices to exhibit one coherent sheaf of rank r and degree d with the expected cohomology. By Riemann-Roch, the Euler characteristic is given by χ(V) = d + r(1 − g). By the division algorithm, write d = qr + s with 0 ≤ s < r. Consider a direct sum of line bundles

M s

M V= Li ⊕ L′j , i=1 j=1

where Li are general line bundles of degree q and L′j are general line bundles of degree q + 1. By Proposition 3.1, h (Li ) = max(0, q − g + 1) and h (L′j ) = max(0, q + 2 − g). Since h (V) =

P 0 P 0 ′ 0

i h (Li ) + j h (Lj ), we conclude that h (V) = 0 if q + 2 ≤ g or if s = 0 and q + 1 ≤ g. Finally, we have h (V) = d + r(1 − g) if q + 1 ≥ g. This proves the statement for h . The statement for h follows by Serre duality. □ Caution 3.9. We warn the reader that points of MC (r, d) correspond to S-equivalence classes of bundles. Certain properties of bundles such as cohomology or global generation are not invariant

under S-equivalence. The two bundles in Example 3.6 are S-equivalent, but they have different cohomology groups. Similarly, OE ⊕ OE is globally generated, whereas the nontrivial extension is not.

Since the cohomological behavior of the general stable bundle is as expected, Brill-Noether

theory concentrates on the cohomology jumping loci. In higher rank, much less is known about cohomology jumping loci. We refer the reader to [GM08] for a survey. As in the case of line bundles, there are no interesting jumping loci as soon as the slope of the vector bundle is sufficiently large. Proposition 3.10. Let C be a curve of genus g. If µ > 2g − 2, then every bundle V in MC (r, d) is nonspecial. Proof. By Serre duality, h (C, V) = h (C, V ∗ ⊗ωC ) = hom(V, ωC ).

Since V and ωC are semistable and

µ(V) > 2g − 2 = µ(ωC ), we conclude that hom(V, ωC ) = 0. Hence, every semistable bundle of slope greater than 2g − 2 has no higher cohomology. □ Consequently, the Brill-Noether problem is only interesting when µ(V ) ≤ 2g − 2. Since h (C, ωC ) = 1, the bound in the proposition is sharp. Several other properties of bundles such as global generation and ampleness play a central role in geometry. Recall that a vector bundle V is called ample if OP V (1) is an ample line bundle on

P V, or equivalently, if for every coherent sheaf F there exists an integer n such that F ⊗ Symn (V) is globally generated for all n ≥ n . For curves, it is easy to determine when the general stable bundle has these properties. Proposition 3.11. Let C be a curve of genus g ≥ 2. (1) [La91, Su87] A general stable bundle of rank r and degree d on C is globally generated if d − rg ≥ 1. (2) If µ > 2g − 1, then every semistable vector bundle of rank r and degree d on C is globally generated.

(3) [Har71, Theorem 2.4] If µ > 0, then every semistable bundle of rank r and degree d on C is ample. Remark 3.12. Global generation is not in general an open property. For example, let L be a line bundle of degree 0 on a positive genus curve C. Then L is globally generated if and only if L ∼ = OC . Hence, the locus of globally generated degree 0 line bundles is a single point in a g-dimensional abelian variety. However, the locus of globally generated bundles is open in the locus of nonspecial bundles.

Proof. If d − rg ≥ 1, then by Proposition 3.8, the general stable bundle has no higher cohomology and has at least r + 1 sections. By the irreducibility of Cohd,r , it suffices to exhibit one globally generated nonspecial bundle of rank r and degree d. A general line bundle L of degree d ≥ g + 1 has the property that h (C, L(−p)) = 0 for every p ∈ C. Indeed, consider the locus Xd−1 := {D ∈ C (d−1) |h (C, D) ≥ d − g + 1} ⊂ C (d) . By Proposition 3.8, the codimension of Xd−1 is at least 1. The fibers of the natural map ϕd :

C (d−1) → Picd−1 (C) of Xd−1 have dimension at least d − g. Consequently, the image of Xd−1 has dimension at most g − 2. Hence, the codimension of the locus of line bundles of degree d − 1 with

unexpected cohomology is at least 2. Therefore, for the general L ∈ Picd (C), h (C, L(−p)) = h (C, L) − 1 for every p ∈ C. By Serre duality, h (C, L(−p)) = 0 for every p ∈ C. By the Euclidean algorithm, write d = rm + s. If m ≥ g + 1, then we can take V to be an extension of the form 0 → Lr−s → V → Ks → 0, where L and K are general line bundles of degree m and m + 1, respectively. We then have that V is nonspecial and H 1 (C, V(−p)) = 0 for every p ∈ C. Taking the long exact sequence associated

to 0 → V(−p) → V → V |p → 0, we conclude that H (C, V) surjects onto H 0 (C, V |p ) for every p ∈ C and V is globally generated.

If m = g, a more subtle argument is needed. We claim that the Brill-Noether locus in MC (r, d) parameterizing special vector bundles has codimension at least 2 if d − r(g − 1) ≥ 1 (see [La91, Su87]). Granting this claim, deg(V(−p)) − r(g − 1) ≥ 1. Consequently, V(−p) does not have h for the general V and every p ∈ C. Hence, V is a globally generated, nonspecial bundle. To prove the claim, recall that the fibers of the determinant map MC (r, d) → Picd (C) are Fano varieties of Picard rank 1 [DN89]. Hence, it suffices to exhibit a complete curve in each fiber which is

nonspecial. Furthermore, by taking direct sums, it suffices to do this when the rank and the degree are coprime. Now it is easy to construct such complete families by induction on the rank as in the proof of Theorem 3.7. Now assume V is semistable and µ(V) > 2g − 1. For any point p ∈ C, consider the exact sequence 0 → V(−p) → V → V p → 0.

We have that H (C, V(−p)) ∼

= H (C, V ∗ ⊗ωC (p)). Since µ(V ∗ ⊗ωC (p)) < 0 and is semistable, it

cannot have any sections. Hence the evaluation map H 0 (C, V) → H 0 (V |p ) is surjective for every point p and V is globally generated. Hartshorne [Har71] proves (3) by showing that a vector bundle V of positive degree on a curve all of whose symmetric powers Symn (V) are semistable is ample. In characteristic 0, by the

Narasimhan-Seshadri Theorem, the symmetric powers of a semistable bundle are semistable. More

generally, Hartshorne uses this fact to show that any vector bundle of positive degree on a smooth curve all of whose quotients have positive degree is ample. □ The locus of stable vector bundles that fail to be globally generated has been studied by Kopper and Mandal [KM]. Their main theorem is the following. Theorem 3.13. [KM, Theorem 1.2] Let NC (r, d) be the locus of stable vector bundles of rank r ≥ 2 and degree d on a smooth curve of genus g ≥ 2 that fail to be globally generated. Assume

rg + 1 ≤ d ≤ r(2g − 1) − 1. (1) NC (r, d) is nonempty and it has a component of expected codimension d − rg and no component of smaller codimension. (2) If rg + g − 1 ≤ d ≤ r(2g − 1) − 1, then NC (r, d) is irreducible of the expected codimension d − rg.

4. Pathologies for surfaces

In this section, we will contrast the behavior of stable bundles on surfaces with that of curves discussed in the last section. We will see that many of the nice properties that are valid for curves

fail for surfaces. This makes studying Brill-Noether theory on surfaces much more challenging. Here are some of the new difficulties that arise. (1) A general line bundle with fixed invariants on a surface can have more than one nonzero cohomology group. Hence, unlike on curves, the cohomology of a general line bundle on a surface is more delicate and can be challenging to compute. (2) Unlike in the case of curves of genus g ≥ 2, there are restrictions on the possible invariants

of stable sheaves on surfaces. Determining the Chern characters of stable sheaves on a surface is a challenging problem. (3) Unlike in the case of curves, the stack of coherent sheaves with fixed invariants on a surface is almost always reducible. Hence, stability plays a much greater role in determining the cohomology of a ‘general’ stable sheaf on a surface. Furthermore, on a surface X, the moduli space MX,H (v) itself may be reducible and even disconnected. Hence, it does not

always even make sense to talk about a general stable sheaf with fixed invariants. In this section, we will give some simple examples of these phenomena.

4.1. The cohomology of line bundles on surfaces. Proposition 3.1 showed that a general

line bundle of degree d on any smooth curve has at most one nonzero cohomology group. This fails for surfaces. The general line bundle with a given class on a surface may have two or three nonzero cohomology groups. Moreover, whether weak Brill-Noether holds for a line bundle may depend on the isomorphism class of the surface. Example 4.1. We give several easy examples where weak Brill-Noether fails for line bundles on surfaces. (1) Let X be a K surface. Then the Picard group of X is discrete and

h (X, OX ) = h (X, OX ) = 1. (2) Let X be a very general surface of degree d ≥ 4 in P . By the Noether-Lefschetz Theorem, the Picard group of X is generated by OX (1). Let 0 ≤ k ≤ d − 4. The long exact sequence associated to 0 → OP (k − d) → OP (k) → OX (k) → 0 implies that    

0 k+3 1 2 d−k−1

h (X, OX (k)) = , h (X, OX (k)) = 0, h (X, OX (k)) = . 3 3 Observe that both h and h are nonzero. (3) Let Y ⊂ P be a very general surface of degree d ≥ 5 and ℓ be a general line in P . Let X denote the blowup of Y along ℓ ∩ Y = {p , . . . , pd } and let Ei denote the exceptional divisor over pi . Set E = di=1 Ei . Let H be the pullback of the hyperplane class from P Y . Then the Picard group of X is discrete. For an integer 1 ≤ k ≤ d − 4, the sections of

OX (kH − E) are given by hypersurfaces of degree k that vanish along ℓ ∩ Y . By Bézout’s Theorem, these hypersurfaces of degree k must vanish along ℓ. By Serre duality, h (X, kH − E) = h (X, (d − 4 − k)H + 2E). Since Ei ·Ej = −δi,j , 2E must be in the base locus of OX ((d−4−k)H +2E). Consequently, h (X, kH − E) = h (X, (d − 4 − k)H). By an easy Euler characteristic computation, we

0 k+3

h (X, kH − E) = − k − 1, h (X, kH − E) = d − k − 1,  

2 d−k−1

h (X, kH − E) = . Hence, all three cohomology groups are nonzero. Observe that if X were the blowup of Y along d general points, the cohomology would instead be       

0 k+3 1 k+3

h (X, kH − E) = max 0, − d , h (X, kH − E) = max 0, d − , 3 3  

2 d−k−1

h (X, kH − E) = . Hence, the generic cohomology of a line bundle depends on the isomorphism class of the surface. (4) Let A be an abelian surface of Picard rank 1. Let X be the blowup of A at a point with exceptional divisor E. Let H denote the pullback of a very ample divisor on A. Let m be a nonnegative integer. Then any line bundle algebraically equivalent to OX (H + mE) is of the form L ⊗ OX (H + mE), where L is the pullback of a line bundle of degree 0 from

A. The line bundle L ⊗ OX (H) is the pullback of an ample line bundle from A, which has positive Euler characteristic and no higher cohomology by the Kodaira Vanishing Theorem. Consequently, h (X, L ⊗ OX (H + mE)) > 0. However, χ(L ⊗ OX (H + mE)) = χ(OX ) + (H 2 − m − m) < 0 if m ≫ 0. We conclude that these line bundles have nontrivial h and h . In general, computing the cohomology of a line bundle on a surface can be a hard problem. For example, already for the blow-up of P in m ≥ 1 very general points, we do not know the

dimensions of the spaces of global sections of all line bundles. In this case there is a precise conjecture due Segre, Harbourne, Gimigliano and Hirschowitz. Conjecture 4.2 (SHGH Conjecture). [Se60, Ha86, Gim87, Hi89] Let X be a very general blow up of P at m ≥ 1 points p , . . . , pm . Let H denote the Ppullback of the class of a line and let Ei denote the exceptional divisor over pi . Let D = dH − m i=1 ni Ei be a divisor with d > 0 and ni ≥ 0. Then   X m  

0 d+2 ni + 1

2 i=1

if and only if OX (D) does not have a multiple (−1)-curve in its base locus.

The SHGH Conjecture, if true, provides an efficient algorithm for computing the cohomology

of any line bundle on X. Despite steady progress, the conjecture remains open in general. Given that weak Brill-Noether does not hold in general for line bundles on surfaces, we do not expect it to hold for higher rank bundles either. Unlike the case of curves, for surfaces already

computing the cohomology of a general stable sheaf in a component of the moduli space is an interesting and challenging problem.

Problem 4.3 (Weak Brill-Noether). Given an irreducible component of MX,H (v), compute the

cohomology of the general sheaf F in that component.

4.2. The cohomology of rank 1 sheaves. Once one knows the cohomology of line bundles

on a surface X, understanding the general cohomology of rank 1 sheaves does not present new difficulties. If V is a rank 1 torsion-free coherent sheaf on a smooth projective surface, then V ∼= IZ ⊗ L, where L is a line bundle and IZ is the ideal sheaf of a zero-dimensional subscheme of X. The Hilbert scheme X [n] parameterizing length n zero-dimensional subschemes of X is a smooth, irreducible, projective variety of dimension 2n. The long exact sequence associated to the standard exact sequence

0 → IZ ⊗ L → L → OZ → 0

implies that H (X, IZ ⊗ L) ∼ 2 2 = H (X, L). If Z is a general set of n points on X, then the map H (X, L) → H (X, OZ ) has maximal rank. We conclude that H 0 (X, IZ ⊗ L) = 0 for a general 0 0

set of points if and only if n ≥ H 0 (X, L). Similarly, h (X, IZ ⊗ L) = h (X, L) + n − h (X, L) for a general set of points Z. Hence, we can compute the cohomology of IZ ⊗ L for a general set of points Z purely based on the cohomology of L. Remark 4.4. We caution the reader that rank 1 sheaves behave slightly differently from higher rank sheaves. When n > 0, the moduli space consists entirely of non-locally-free sheaves. Consequently, one cannot directly apply Serre duality. In fact, if L is a line bundle with h (X, L) ̸= 0, then

L ⊗IZ has nonvanishing h and h as soon as n > h (X, L). In particular, if L is an ample line bundle on X with h (X, L) ̸= 0, then IZ ⊗ ωX ⊗ L∗ has nonvanishing h and h as soon as n ≥ 1. In particular, one can only hope for weak Brill-Noether to hold for rank 1 sheaves when h (X, L) ̸= 0. When the rank r > 1, there may be components of MX,H (v) that consist entirely of non-locally- free sheaves. However, such components do not exist for certain surfaces such as P and Hirzebruch

surfaces. Moreover, when ∆(v) ≫ 0, then the general member of the moduli space parameterizes a vector bundle. Consequently, one may apply Serre duality to compute the cohomology of the general sheaf.

4.3. The moduli space may be empty. Theorem 3.7 shows that on a curve of genus g ≥ 2,

the moduli space MC (r, d) is nonempty, irreducible and of the expected dimension for every r ≥ 1 and every d. In contrast, there are restrictions on the possible Chern characters of stable sheaves on surfaces. For example, the Bogomolov inequality provides an important constraint. Theorem 4.5 (The Bogomolov inequality). Let (X, H) be a polarized smooth surface and let F a µH -semistable sheaf. Then ∆(F) ≥ 0. For abelian surfaces Bogomolov’s inequality completely characterizes stable Chern characters.

For some surfaces such as P or K surfaces one can easily strengthen the Bogomolov inequality. Example 4.6. Let X = P and let V be a torsion-free stable sheaf. Then ext (V, V) = hom(V, V(−3)) = 0 by Serre duality and stability. Therefore, χ(V, V) = r (1 − 2∆(V)) = hom(V, V) − ext (V, V) ≤ 1.

Consequently,  

∆(V) ≥ 1− 2 .

2 r

We will later see that there are stronger restrictions on the Chern characters of stable sheaves on P . Example 4.7. Let X be a K surface and let V be a torsion-free stable sheaf. Since hom(V, V) = ext (V, V) = 1, we have χ(V, V) = r (2 − 2∆(V)) ≤ 2. Hence, ∆(V) ≥ 1 − r . On K surfaces this inequality characterizes the Chern characters of stable sheaves.

We will say that a Chern character is µH -(semi)stable (respectively, H-Gieseker (semi)stable)

if there exists a µH -(semi)stable (respectively, H-Gieseker (semi)stable) sheaf with that Chern character. The following is a central open problem. Problem 4.8. Given a polarized surface (X, H) classify the Chern characters of µH or H-Gieseker semistable sheaves. Determine when there exist stable sheaves with the given Chern character.

The results concerning Problem 4.8 have two flavors. First, there are important asymptotic

results for all surfaces guaranteeing the existence of stable bundles with sufficiently large discrim-

inant. Theorem 4.9 (O’Grady). Let (X, H) be a smooth polarized surface and let v be a Chern character with r(v) > 0. If ∆H (v) ≫ 0 (where the necessary inequality depends on r, X and H), then the moduli space MH (v) is normal, generically smooth, irreducible and nonempty of the expected dimension. Furthermore, the slope stable sheaves are dense in MH (v). Second, Problem 4.8 has been studied and solved on certain surfaces including P [DLP85], K surfaces [Yos99], Abelian surfaces [Yos01], Hirzebruch surfaces [CH21], Enriques surfaces [N16a,

N16b, NY], elliptic and bielliptic surfaces. There has been some progress for certain other classes of surfaces such as quintic and sextic surfaces in P at least in rank 2 [MS11]. However, the general problem remains wide open. Not knowing the Chern characters of stable sheaves often presents difficulties in constructions. One is often forced to assume inequalities on the discriminant in constructions. There is however a general way to construct rank 2 bundles via the Serre construction, which we now recall.

4.4. The Serre Construction. Let X be a smooth projective surface. Let Z ⊂ X be a zero-

dimensional local complete intersection scheme of length n. Then Z satisfies the Cayley-Bacharach property with respect to a line bundle L on X if for any subscheme W ⊂ Z of length n − 1, any section of L vanishing on W vanishes on Z. The name is inspired by the classical Cayley-Bacharach Theorem which asserts that any cubic curve which contains 8 of the 9 intersection points of two cubic curves in P also contains the ninth. Theorem 4.1 (The Serre correspondence). Let X be a smooth projective surface and Z ⊂ X be

a local complete intersection subscheme of dimension zero and length n. Let L be a line bundle on X. Then there exists an extension

0 → OX → V → IZ ⊗ L → 0

with V locally free if and only if Z satisfies the Cayley-Bacharach property for ωX ⊗ L.

The Serre construction allows one to construct stable vector bundles of rank 2 on surfaces. Corollary 4.11. Let H be an ample divisor on X. Let L be a line bundle such that c (L) · H > 0. Let Z be a nonempty set of distinct points such that Z satisfies the Cayley-Bacharach property for ωX ⊗ L. Assume that H 0 (X, N ⊗ IZ ) = 0 for any line bundle N with c (N ) · H ≤ c (L) · H. Then the general extension V

0 → OX → V → IZ ⊗ L → 0

is a µH -stable bundle. Proof. By Theorem 4.10, the general extension is a vector bundle. Suppose N → V is a destabiliz- ing line subbundle. Then there must exist a nonzero map N → IZ ⊗L. Hence, L⊗N −1 is effective and nontrivial. In particular, H 0 (X, IZ ⊗ L ⊗ N −1 ) ̸= 0 and yet 0 < H · c (L ⊗ N −1 ) < H · c (L), contrary to our assumptions. □ In general, it is hard to characterize loci of points on X that satisfy the Cayley-Bacharach

property for L ⊗ ωX . However, there are easy conditions that ensure that it holds. For instance, we may choose Z general and |Z| large, or we may choose L such that H 0 (X, L ⊗ ωX ) = 0.

4.5. The moduli space is not necessarily irreducible. Unlike in the case of curves, the stack

of coherent sheaves with a fixed Chern character on a surface is in general reducible. Consequently, one cannot construct a coherent sheaf with the expected cohomology and deduce that the general stable sheaf will have the expected cohomology. Furthermore even the substack of semistable or stable sheaves can be reducible. Example 4.1 (Mestrano [Me97]). Let X be a very general surface of degree d = 6 in P . By the Noether-Lefschetz Theorem, Pic(X) = ZH, where H is the hyperplane class. Let Z be a

zero-dimensional scheme of length 1 of one of the following types: I) Z is contained in the intersection of X with a twisted cubic curve C, or II) Z is a general set of points in a hyperplane section of X.

Consider an extension of the form

0 → OX → V → IZ (H) → 0.

Since ext (IZ (H), OX ) = h (X, IZ (3H)) = 1, up to scaling there exist unique nonsplit extensions of this form. Since KX = 2H, to apply Theorem 4.1 we need to check that Z satisfies the

Cayley-Bacharach property with respect to OX (3H). By Bézout’s Theorem a hypersurface of

degree 3 that contains at least 1 points of C contains all of C. Hence, Z satisfies the Cayley- Bacharach property in Case I. Similarly, in Case II, any hypersurface of degree 3 that contains 1 of the points contains the plane spanned by the points. Hence, Z satisfies the Cayley-Bacharach property in Case II. By Theorem 4.10, there are locally free sheaves V in both cases. Observe that V is necessarily µH -stable since V cannot admit a map from OX (kH) for k ≥ 1. By a monodromy argument, one can check that the locus of bundles in Case I is irreducible of

dimension 12, corresponding to a generically finite cover of the space of twisted cubics in P . The tangent space to the moduli space at V is given by Ext (V, V) ∼ = H 1 (X, V ⊗ V ∗ ). Since V is rank ∗ ∼ 2, we have V = V(−H) and consequently

V ⊗ V∗ ∼

= (V ⊗ V)(−H) ∼ V)(−H) ∼

^ = (Sym V ⊕ = Sym (V)(−H) ⊕ OX .

Since H 1 (X, OX ) = 0, Ext (V, V) = H 1 X, Sym (V)(−H) . Using the standard exact sequence
for symmetric powers 0 → V(−H) → Sym (V)(−H) → I2Z⊂X (H) → 0, where I2Z⊂X is the symbolic square of the ideal of Z in X, one shows that ext (V, V) = 12. Hence, the bundles in Case I lie on a generically smooth component of the moduli space of dimension 12. On the other hand, the locus of bundles in Case II has dimension at least 13. The choice of Z depends on 1 parameters, 3 for the choice of a hyperplane Λ and 1 for the choice of points

in Λ ∩ X. To go in the reverse direction, note that the bundles in Case II have h (X, V) = 2, and from µH -stability it follows that a choice of a nonzero global section determines a quotient V /OX ∼ = IZ (H) such that Z is in Case II from h (X, V) = 2 and h (X, OX ) = 0. Therefore, the locus in the moduli space coming from the bundles in Case I has dimension at least 13. We conclude that the moduli space has at least 2 irreducible components.

By using a similar construction, one can produce examples of moduli spaces with arbitrarily

many components parameterizing sheaves on surfaces in P . Theorem 4.13. [CH18c] For any integer k, there exists an integer dk such that if d ≥ dk , then a very general surface X ⊂ P of degree d has a moduli space of rank 2 sheaves with c = H that has at least k irreducible components. The moduli space does not even have to be connected. Okonek and Van de Ven [OVdV86] and Kotschick [Ko89] found examples of disconnected moduli spaces on elliptic surfaces with high Picard rank. These examples use ample classes that are very close to the fiber class. One can

even find disconnected examples on complete intersection surfaces of Picard rank 1. Example 4.14. Let d ≥ 5 and d > d be two integers. Let D be a smooth hypersurface in P that contains lines. Let D be a very general hypersurface of degree d . Let X be the complete intersection of D and D , which by Noether-Lefschetz Theory is a surface of Picard rank 1. Take a plane Λ containing a line ℓ of D . Then Λ ∩ D = C ∪ ℓ, where C is the residual curve of degree d − 1. Let Z = C ∩ D be the zero dimensional subscheme of length (d − 1)d . Then Z satisfies

the Cayley-Bacharach property for ωX (H) and up to scalars there is a unique nonsplit extension of the form

0 → OX → V → IZ (H) → 0.

One can compute that h (X, V) = 3 and h (X, V) = 0. Given a bundle W in the same irreducible

component as V, we have h (X, W) ≥ 3. The cokernel of the evaluation map of a section is IZ ′ (H), where Z ′ has length (d − 1)d . Furthermore, since h (X, IZ ′ (H)) ≥ 2, Z ′ lies on a plane Λ′ . By the Cayley-Bacharach Theorem, the residual d points must be collinear and by Bézout’s Theorem the line must be contained in D . With a slightly more careful analysis, one can deduce the following. Theorem 4.15. [CHK22, Theorem 3.4] Let v be the Chern character of the sheaf V constructed

in this example. For every connected (respectively, irreducible) component of the Fano scheme of lines F (D ) on D , MX,H (v) has a connected (respectively, irreducible) component of the same dimension. By letting the degree of D tend to infinity, we can find threefolds D in P that contain arbitrarily many isolated lines. We thus deduce the following.

Corollary 4.16. [CHK22, Corollary 1.1] For any integer k, there exists a smooth complete inter- section surface X ⊂ P with Picard rank 1 and a Chern character v on X such that MX,H (v) has at least k connected components. It is also interesting to note that the bundles V do not deform to bundles on complete intersec- tions where D does not contain a line.

Even on rational surfaces there are disconnected moduli spaces. Assuming the SHGH Conjec-

ture 4.2, one can construct moduli spaces with arbitrarily many components for certain special polarizations (see [CH23]).

In this subsection, we have discussed the connectedness and the irreducibility of the moduli

spaces. In general, there are many interesting questions about the topology of MX,H (v). A general expectation due to Donaldson, Gieseker and Li is that the moduli spaces become better behaved as ∆ becomes large. O’Grady’s Theorem 4.9 proves that the moduli space becomes irreducible for large ∆. There are several conjectures concerning the stabilization of other Betti numbers of the moduli spaces. Conjecture 4.17. [CW22, Conjecture 1.1] Let X be a smooth projective surface and let H be

an ample line bundle. Fix a rank r > 0 and a first Chern class c. Then the ith Betti number of MX,H (r, c, ∆) stabilizes to a constant bi,Stab (X), which is independent of r, c and H, as ∆ tends to ∞. The conjecture is known when X is a K or abelian surface and r and c are relatively prime. The conjecture is also known when X is a rational surface, MX,H (r, c, ∆) do not contain strictly semistable sheaves and H is a polarization such that H · KX < 0 (see [CW22] for a discussion and

references). The conjecture is wide open in general, especially for surfaces of general type.

5. Weak Brill-Noether

In this section, we will survey some results on the weak Brill-Noether Problem. Recall that an irreducible component of a moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in that component has at most one nonzero cohomology group. In particular, if χ(v) = 0 and a component of the moduli space M (v) satisfies weak Brill-Noether, then the general member of that component has no cohomology. The ideal situation occurs for P . Let L be the class of a line on P .

Theorem 5.1 (Göttsche-Hirschowitz [GH94]). A general stable bundle on P has at most one

Consequently, the slope and the Euler characteristic determine the cohomology of the general

stable bundle V ∈ MP ,L (v). • If χ(V) < 0, then h (P , V) = −χ(V) and h (P , V) = h (P , V) = 0. • If χ(V) ≥ 0 and µL (V) ≥ 0, then h (P , V) = χ(V) and all other cohomology vanishes. • If χ(V) ≥ 0 and µL (V) < 0, then h (P , V) = χ(V) and all other cohomology vanishes. • In particular, if χ(V) = 0, then the cohomology of V vanishes. Observe that if µL (V) < 0, then h (P , V) = 0 by stability. By Serre duality and stability, if µL (V) > −3, then h (P , V) = 0. Remark 5.2. The sheaves IZ (−d) have nonvanishing h and h provided |Z| ≥ 1 and d ≥ 3. The

general element of the corresponding moduli spaces are not locally free and Serre duality fails.

The general stable sheaf of rank at least 2 on P is locally free, hence weak Brill-Noether holds for MP ,L (v) if the rank of v is at least 2. We will discuss two general techniques for proving weak Brill-Noether Theorems on surfaces.

The first uses prioritary sheaves and the second uses Bridgeland stability conditions. We begin

5.1. Prioritary Sheaves. Let D be an effective divisor on a smooth surface X. A torsion free

sheaf V is called D-prioritary if Ext (V, V(−D)) = 0. We denote the stack of D-prioritary sheaves on X with Chern character v by P X,D (v). The stack P X,D (v) is an open substack of the stack of coherent sheaves on X with Chern character v.

The condition Ext (V, V(−D)) = 0 implies that the restriction map

Ext1X (V, V) → Ext1D (V |D , V |D )

is surjective. Consequently, the general first order deformation of the sheaf V on X gives a general first order deformation of V |D on D. For example, when D is a smooth rational curve and V is locally free on D, the restriction of a general deformation of V to D is balanced (i.e., the splitting type V |D = ⊕ri=1 OP (ai ) satisfies |ai − aj | ≤ 1 for all i, j). Let H be an ample divisor on X. Suppose that (KX + D) · H < 0. Then a µH -semistable sheaf V is D-prioritary. By Serre duality,

ext (V, V(−D)) = hom(V, V(KX + D)). Since (KX + D) · H < 0, stability implies that the latter group vanishes. Hence, V is D-prioritary. Remark 5.3. The condition (KX + D) · H < 0 implies that KX · H < 0. Hence, mKX cannot have any sections for m > 0. Therefore, this condition can only be satisfied for surfaces of Kodaira dimension −∞. Furthermore, if KX + D is effective, there cannot be any D prioritary sheaves on X. A projective surface X is ruled if it admits a morphism X → C onto a smooth curve where

all the fibers are isomorphic to P , equivalently if X is the projectivization of a vector bundle of rank 2 on C. A surface is birationally ruled if it admits a morphism X → C onto a smooth curve where the general fiber is P . The following theorem of Walter makes prioritary sheaves a useful tool for studying moduli spaces of sheaves on birationally ruled surfaces. Theorem 5.4 (Walter [Wal98]). Let X be a birationally ruled surface and let F be the fiber class. Then the stack of F -prioritary sheaves PF (v) is irreducible whenever it is nonempty.

This theorem generalizes an earlier theorem of Hirschowitz and Laszlo [HL93] which assets that

PP ,L (v) is irreducible when nonempty. The main advantage of working with prioritary sheaves is that they are easier than stable sheaves to construct. For example, OP (a) ⊕ OP (a + 1) is L-prioritary on P but not stable. Remark 5.5. For rational surfaces, there are always polarizations H for which (KX + F ) · H < 0. For minimal rational surfaces KX + F is anti-effective. Hence, (KX + F ) · H < 0 for every ample class H. Every rational surface is obtained by blowing up a minimal rational surface at finitely

many (possibly infinitely near) points. Let Y be the blowup of X at a point. If (KX + F ) · H < 0, then KY = π ∗ KX + E and (KY + F ) · π ∗ H = (KX + F ) · H < 0.

The divisor π ∗ H is not ample on Y , but nef. Any small perturbation H ′ of π ∗ H by adding an ample divisor is ample. Hence, for an ample H ′ sufficiently close to π ∗ H, we still have (KY + F ) · H ′ < 0. The utility of the notion of F -prioritary sheaves toward the weak Brill-Noether problem follows from openness of (semi)stability and semi-continuity of cohomology. Indeed, if (KX + F ) · H < 0 and there exist (semi)stable sheaves with Chern character v, then to show that v satisfies weak

Brill-Noether it suffices to show that there exists an F -prioritary sheaf with at most one nonzero cohomology group. By semi-continuity of cohomology and irreducibility of the moduli stack, it follows that the general stable sheaf has at most one nonzero cohomology group. Hence, the moduli space satisfies weak Brill-Noether. This strategy can be used to show weak Brill-Noether on certain rational or birationally ruled surfaces.

5.2. Elementary modifications. Given a torsion free sheaf V, a point p ∈ X and a surjection

ϕ : V ↠ Op , the kernel V ′ defined by the exact sequence ϕ (1) 0 → V ′ → V → Op → 0 is called the elementary modification of V with respect to ϕ. An elementary modification satisfies r(V ′ ) = r(V), ν(V ′ ) = ν(V), ∆(V ′ ) = ∆(V) + . r ′ ′ If W is a proper subsheaf of V of smaller rank with µH (W) ≥ µH (V ), then W is also a proper subsheaf of V with µH (W) ≥ µH (V). Hence, if V is a µH -(semi)stable sheaf, then V ′ is also

µH -(semi)stable. If V is a D-prioritary sheaf, then a general elementary modification is also D-prioritary.

Caution 5.6. Elementary modifications do not necessarily preserve Gieseker (semi)stability. For

example, on P the sheaf OP ⊕ OP is Gieseker semistable, however any elementary modification is isomorphic to Ip ⊕ OP , which is not Gieseker semistable. Since hi (Op ) = 0 for i > 0, we have that H 2 (X, V ′ ) = H 2 (X, V). The morphism ϕ corresponds to a choice of hyperplane Λ in the fiber of V over p. If h (X, V) > 0, we can choose p and the hyperplane Λ so that one of the sections at p is not contained in Λ. In that case, the map H 0 (X, V) → H 0 (X, Op ) induced by ϕ is surjective. By the long exact sequence of cohomology

associated to (1), we conclude that h (X, V ′ ) = h (X, V) − 1, h (X, V ′ ) = h (X, V).

The integrality of the Euler characteristic and the Riemann-Roch Theorem

χ(V) = r(V)(P (ν(V)) − ∆(V)) imply that the difference of the discriminants of any two sheaves with the same rank and first Chern class is a multiple of 1r . Our discussion on elementary modifications yields the following theorem. Theorem 5.7. Let V be a µH -(semi)stable (respectively, D-prioritary) sheaf of rank r, total slope ν and discriminant ∆0 on a smooth projective surface X such that h (X, V) = 0 and V has at most one nonzero cohomology group. Then there exists a µH -(semi)stable (respectively, D-prioritary)

sheaf with rank r, total slope ν with at most one nonzero cohomology group for every ∆ ≥ ∆0 for which (r, ν, ∆) is an integral Chern character.

5.3. Weak Brill-Noether on P . We are now ready to sketch the proof of Theorem 5.1] We will

prove the following more general statement. Theorem 5.8. Let V be a general prioritary sheaf on P with ∆(V) ≥ 0 and rank at least 2. Then V has at most one nonzero cohomology group. Proof. Since stable sheaves are prioritary and form a dense open substack of the stack of prioritary sheaves when nonempty, Theorem 5.1 follows by the semicontinuity of cohomology. By the irre- ducibility of the stack of prioritary sheaves and semicontinuity of cohomology, it suffices to exhibit one prioritary sheaf with at most one nonzero cohomology group. When the rank is at least 2, the

general prioritary sheaf is locally free. Hence, by Serre duality, we may assume that µ(V) ≥ − 2 . Given a rank r and a slope rc ≥ − 3 , we can find an L-prioritary sheaf V = OP (a)⊕r−s ⊕OP (a+1)⊕s with a ≥ −2 of rank r and slope rc . An easy computation shows that ∆(V) ≤ 0. The sheaf V is nonspecial. Taking general elementary modifications, we obtain a prioritary sheaf that has at most one nonzero cohomology group for every integral Chern character. This concludes the proof of the theorem. □

5.3.1. Gaeta resolutions. On P one can write down a resolution for the general prioritary sheaf

V with ∆ ≥ 0. By Riemann-Roch, there exists a unique integer n such that χ(V(−n)) ≥ 0 but χ(V(−n − 1)) < 0. Set n = a + 2. Then V has a resolution of the form 0 → OP (a)α → OP (a + 1)β ⊕ OP (a + 2)γ → V → 0, or 0 → OP (a)α ⊕ OP (a + 1)β → OP (a + 2)γ → V → 0. Here γ = χ(V(−a − 2)), α = −χ(V(−a − 3)), β = |γ − r(V) − α|. The resolution is of the first type if γ − r(V) − α ≥ 0 and otherwise of the second type. This generalizes Gaeta’s resolution for the ideal sheaf of a general set of points on P [Ga51].

To see that such a resolution exists, given a Chern character v one can solve for a, α, β and γ and write a general Gaeta-type resolution. One then checks that V defined by such a resolution is prioritary and that the associated Kodaira-Spencer map is surjective (see [CH18b, §3] or [CH20a, §4] for more details). When r(V) ≥ 2, by a Bertini-type theorem, a sheaf defined by such a resolution is locally free. Consequently, we also deduce that the general prioritary sheaf of rank at least 2 is locally free. The existence of the Gaeta resolution also implies that the moduli spaces

MP ,L (v) are unirational. In general, we do not know a presentation for the general stable bundle on a surface. Having such presentations even for special families of surfaces would be very useful.

5.3.2. Global generation. The weak Brill-Noether Theorem allows one to classify Chern characters

of nonspecial stable bundles that are globally generated. Theorem 5.9. [CH20a, Corollary 5.3] (see also [BGJ16]) The general member of MP ,L (v) is globally generated if and only if one of the following holds: • v = r(1, 0, 0) and V = OP⊕r • µ(v) > 0 and χ(v(−1)) ≥ 0 • µ(v) > 0, χ(v(−1)) < 0 and χ(v) ≥ r + 2 • µ(v) > 0, χ(v(−1)) < 0, χ(v) ≥ r + 1, and v = (r + 1) ch(OP ) − ch(OP (−2)).

Proof. If V is globally generated, then its determinant is globally generated, hence µ(V) ≥ 0. If µ(V) = 0, then the Riemann-Roch Theorem implies that χ(V) ≤ rk(V) with equality if and only if ∆(V) = 0. If µ(V) = ∆(V) = 0, then V = OP⊕r . If χ(v(−1)) ≥ 0, then the general sheaf in P P ,L (v) has a Gaeta resolution with a ≥ 1. Then the general sheaf is clearly a quotient of a globally generated bundle. If χ(v(−1)) < 0 and χ(v) ≥ rk(v) + 2, then the general sheaf in P P ,L (v) has a Gaeta resolution of the form

0 → OP (−2)k ⊕ OP (−1)l → OPm → V → 0, or

0 → OP (−2)k → OP (−1)l ⊕ OPm → V → 0. In the first case, V is the quotient of a globally generated vector bundle, hence globally generated. The most interesting case is the second one. By the assumption that χ(V) ≥ rk(V) + 2, we have that m ≥ rk(V) + 2. Therefore, k ≥ l + 2. To show that V is globally generated, it suffices to show that H 1 (P , V ⊗Ip ) = 0 for every point p ∈ P . By the long exact sequence of cohomology, it suffices to show that the map ϕ : H 1 (P , Ip (−2))k → H 1 (P , Ip (−1))l

is surjective. Consider the sequence 0 → M → OP (−2)k → OP (−1)l → 0. Since the map is general, it is surjective and M is a vector bundle. Clearly M does not have any cohomology. Tensoring the standard exact sequence 0 → Ip → OP → Op → 0 with M , we see that H 2 (P , Ip ⊗ M ) = 0. Consequently, the map ϕ is surjective and V is globally generated. Finally, if χ(V) = rk(V) + 1 and V is globally generated, then there is a surjective map OPr+1

2 → V.

The kernel of this map is a line bundle OP (−d). If d = 1, then χ(V(−1)) = 0. If d ≥ 3, then χ(V) < r and it is not possible for the general prioritary sheaf with Chern character v to be globally generated. The only remaining possibility is for d = 2. In that case, χ(V) = r + 1 and this is the Gaeta resolution of the general sheaf. This concludes the classification of globally generated Chern characters on P . □

5.4. Weak Brill-Noether for Hirzebruch surfaces. Let e ≥ 0 be an integer. Let Fe :=

P(OP ⊕OP (e)) denote the Hirzebruch surface. We refer the reader to [Bea83, Cos06a] for detailed information on the geometry of Hirzebruch surfaces. The surface Fe has a section E with self- intersection −e. Let F denote the class of a fiber. Then Pic(Fe ) = ZF ⊕ ZE. The intersection numbers are E 2 = −e, E · F = 1, and F 2 = 0. The effective cone of Fe is spanned by E and F . The nef cone of Fe is spanned by E + eF and F . The minimal rational surfaces are P and Fe for e ≥ 0, e ̸= 1. The existence of curves

with negative self-intersection provides an obstruction for the weak Brill-Noether property on Hirzebruch surfaces Fe with e ≥ 1.

5.4.1. Negative curves on a surface and the cohomology of vector bundles. Suppose that a surface

X contains a smooth curve C with C 2 < 0. Then by the Riemann-Roch Theorem, m 2 m χ(OX (mC)) = χ(OX ) + C − K · C. 2 2

If m ≫ 0, χ(OX (mC)) < 0. We conclude that both H 0 (X, OX (mC)) and H 1 (X, OX (mC)) are nonzero. In fact, if X is a surface such that q = h (X, OX ) = 0 and pg = h (X, OX ) = 0, then OX (mC) is an effective line bundle with nonvanishing h provided that either (1) m ≥ 1 and either C 2 < −1 or the genus of C is at least 1, or (2) m ≥ 2 and C is an exceptional curve. A similar phenomenon persists for higher rank vector bundles. Proposition 5.10. Let X be a smooth projective surface that contains an irreducible curve C

with C 2 < 0. Then for every rank r there exist infinitely many Chern characters vi with rank r such that the moduli space MX,H (vi ) is nonempty and does not satisfy weak Brill-Noether.

Proof. Let MX,H (v) be an irreducible moduli space of rank r sheaves where the general sheaf is µH -stable and has sections. Such moduli spaces exist by O’Grady’s Theorem 4.9 and Serre vanishing. Let V be a general sheaf in MX,H (v). Let vi be the Chern character of V(iC). Since H 0 (X, V) ⊂ H 0 (X, V(iC)), the sheaves V(iC) have global sections. On the other hand, by Riemann-Roch if i ≫ 0, then χ(vi ) < 0, hence, they must also have h and cannot satisfy weak

Brill-Noether. □

Unlike P , by Proposition 5.10, we cannot expect weak Brill-Noether to always hold for moduli spaces on Fe with e ≥ 1. The next theorem shows that the existence of negative curves is the only obstruction for weak Brill-Noether on Fe . Theorem 5.11. [CH20a, Theorem 3.1] Let v be a Chern character with positive rank r(v) and ∆(v) ≥ 0. Then the stack of prioritary sheaves PF (v) is nonempty and irreducible. Let V be a general sheaf in PF (v). (1) If ν(v) · F ≥ −1, then h (Fe , V) = 0.

(2) If ν(v) · F ≤ −1, then h (Fe , V) = 0. (3) If ν(v) · F = −1, then h (Fe , V) = −χ(V) and all other cohomology vanishes. Assume that ν(v) · F > −1. (4) If ν(v) · E ≥ −1, then V has at most one nonzero cohomology group. If χ(V) ≥ 0, then h (Fe , V) = χ(V), and if χ(V) < 0, then h (Fe , V) = −χ(V). (5) If ν(v) · E < −1, then H 0 (Fe , V) ∼ = H 0 (Fe , V(−E)) and the Betti numbers of V are inductively determined by (2) and (4). Observe that if ν(v) · F < −1 and rk(v) ≥ 2, then the general prioritary sheaf is locally free

and Serre duality determines the cohomology.

Sketch of proof. The proof is similar to the proof of Theorem 5.1. A vector bundle V of the form OFe (−E − (e + 1)F )⊕a ⊕ OFe (−F )⊕b ⊕ OF⊕ce or OFe (−E − (e + 1)F )⊕a ⊕ OFe (−E − eF )⊕b ⊕ OF⊕ce is both E-prioritary and F -prioritary and has ∆(V) ≤ 0. Moreover, V has no higher cohomology. If v satisfies ν(v) · F ≥ −1 and ν(v) · E ≥ −1, we can find a prioritary bundle of Chern character v by tensoring v by a nef line bundle N on Fe since tensoring with a nef line bundle preserves

prioritariness and the discriminant. Furthermore, V ⊗N has no higher cohomology on Fe . We

conclude that weak Brill-Noether holds for slopes satisfying ν(v) · F ≥ −1 and ν(v) · E ≥ −1. For slopes in the range ν(v) · F > −1 and ν(v) · E < −1, one can use the exact sequence

0 → V(−E) → V → V |E → 0

and the fact that V |E is balanced on E to inductively compute the cohomology. □

As in the case of P , the weak Brill-Noether Theorem allows one to find a Gaeta-type resolution for the general prioritary sheaf on Hirzebruch surfaces.

Theorem 5.1 ([CH20a], Theorem 4.1). Let v be an integral Chern character on Fe of positive rank and assume that 1 1 ∆(v) ≥ if e = 0, ∆(v) ≥ if e = 1, ∆(v) ≥ 0 if e ≥ 2. 4 8

Then the general sheaf V ∈ P Fe ,F (v) admits a Gaeta-type resolution

(2) 0 → L(−E − (e + 1)F )a → L(−E − eF )b ⊕ L(−F )c ⊕ Ld → V → 0,

for some line bundle L and nonnegative integers a, b, c, d.

Theorems 5.1 and 5.1 allow one to classify moduli spaces where the general bundle is a nonspecial, globally generated bundle.

Theorem 5.13. [CH20a, Theorem 5.1] Suppose e ≥ 1. Let v be a Chern character on Fe such that r(v) ≥ 2, ∆(v) ≥ 0 and ν(v) is nef. Then the general member of PF (v) is globally generated if and only if one of the following holds. (1) We have ν(v) · F = 0 and

v = (r(v) − m) ch(OFe (aF )) + m ch(OFe ((a + 1)F ))

for integers a, m ≥ 0. (2) We have ν(v) · F > 0 and χ(v(−F )) ≥ 0. (3) We have ν(v) · F > 0 and χ(v(−F )) < 0 and χ(v) ≥ r(v) + 2. (4) We have e = 1, ν(v) · F > 0, χ(v(−F )) < 0, χ(v) ≥ r(v) + 1 and

v = (r(v + 1) ch (OF ) − ch (OF (−2E − 2F )) .

Remark 5.14. For F = P ×P , the statement needs to be slightly modified (see [CH20a, Theorem 5.2]). If F and F are the two rulings on P × P , then the general member of PF (v) is globally generated if and only if one of the following holds. (1) We have ν(v) · Fi = 0 for some 1 ≤ i ≤ 2 and

v = (r(v) − m) ch(OFe (aFi )) + m ch(OFe ((a + 1)Fi ))

for integers a, m ≥ 0. (2) We have ν(v) · Fi > 0 for 1 ≤ i ≤ 2 and χ(v(−Fj )) ≥ 0 for some 1 ≤ j ≤ 2. (3) We have ν(v) · Fi > 0, χ(v(−Fi )) < 0 for 1 ≤ i ≤ 2 and χ(v) ≥ r(v) + 2.

5.4.2. Applications to ample bundles. On curves stable vector bundles are ample if and only if their

slope is positive. On higher dimensional varieties, it is easy to see that there can be no numerical characterization of ample vector bundles of higher rank. Ampleness is an open condition and we can ask when a general member of an irreducible component of a moduli space is ample. Problem 5.15. Let X be a smooth, projective variety. Classify Chern characters v on X for which there exists an ample vector bundle of Chern character v. This problem is wide open even for surfaces. Recently, Huizenga and Kopper [HK22] have made

significant progress towards the solution of the problem for P and Hirzebruch surfaces. If V is an ample bundle on P or Fe , then the restrictions V |L , V |F and V |E are ample, hence split as a direct sum of line bundles on P of positive degree. Huizenga and Kopper prove that on minimal rational surfaces, asymptotically this is the only obstruction to ampleness of the general stable bundle. Theorem 5.16. [HK22, Theorem 4.1] Let X = P or Fe . Let v be a stable Chern character on

X such that

(1) if X = P , ν(v) · L > 1, (2) if X = P × P , ν(v) · F > 1 and ν(v) · E > 1, (3) if X = Fe with e ≥ 1, ν(v) · F > 1 and ν(v) · E ≥ 1. Then for n ≫ 0, the general bundle V ∈ MX,H (nv) is ample.

Huizenga and Kopper also classify globally generated ample bundles on minimal rational sur-

faces. Theorem 5.17. [HK22, Theorem 5.1] Let X = P or Fe . Let v be a Chern character such that the moduli space MX,H (v) is nonempty and the general sheaf is a globally generated vector bundle with no higher cohomology. (1) if X = P , ν(v) · L > 1 + 1r , (2) if X = P × P , ν(v) · F > 1 and ν(v) · E > 1, (3) if X = Fe with e ≥ 1, ν(v) · F > 1 and ν(v) · E ≥ 1. Then the general sheaf V ∈ MX,H (v) is ample.

5.5. Weak Brill-Noether for more general rational surfaces. The ideas and techniques

used to study the weak Brill-Noether Problem on P and Hirzebruch surfaces can be extended to non-minimal rational surfaces. Here we will discuss general blowups of P and refer the reader to [CH18b, CH18c] for analogous statements for general blowups of Hirzebruch surfaces. Let Xm denote the blowup of P at m general points p , . . . , pm . Let Ei denote the exceptional divisor lying over pi . Let H denote the pullback of the class of a line. Then Mm

Pic(Xm ) = ZH ⊕ Ei

and the intersection pairing satisfies H 2 = 1, H · Ei = 0, Ei · Ej = −δi,j , where δi,j is the Kronecker delta function. When m ≤ 8, then the corresponding surface is a del Pezzo surface. Del Pezzo surfaces and P × P are the Fano surfaces, that is the surfaces that have

an ample anti-canonical bundle. We refer the reader to [Bea83, Cos06b, Har77] for more details on the geometry of del Pezzo surfaces.

Proposition 5.1 shows that we cannot expect weak Brill-Noether to always hold. However, the

following higher rank generalization of the SHGH might hold. Conjecture 5.18. [CH18b, Conjecture 1.7] Let X be a blowup of P at m very general points. Let H be an ample class on X such that H · KX < 0. Let v be a Chern character such that ν(v) is nef. Then the general stable sheaf in MX,H (v) has at most one nonzero cohomology group. Remark 5.19. One can make a bolder conjecture by weakening the condition that ν(v) is nef to C · ν(v) ≥ −1 for every (−1)-curve on X. This would mimic the SHGH Conjecture more closely.

Even the weaker conjecture is open when m ≥ 5.

When ν(v) is not too close to the boundary of the nef cone, one can find prioritary direct sums of line bundles with no higher cohomology.

Pk More precisely, let v be a Chern character of rank r

and let the total slope be ν(v) = δH − i=1 αi Ei . Then we have that q qi δ = d + , αi = ai + r r for some integers d, q, ai and qi with 0 ≤ q < r and 0 ≤ qi < r. Set k  q qi  q X qi γ(v) = 2 − + − . 2r 2r i=1 2r 2r

Theorem 5.20. [CH18c, Theorem 4.12] Let X be the blowup of P at m distinct points. Let v be a positive rank Chern character on X with total slope ν(v) = δH − α E · · · − αk Ek with δ ≥ 0 and αi ≥ 0. Suppose that the line bundle ⌊δ⌋H − ⌈α ⌉E · · · − ⌈αk ⌉Ek does not have higher cohomology. Assume that ∆(v) ≥ γ(v). Then the stack of prioritary sheaves P X,H−E (v) is nonempty and the general sheaf in P X,H−E (v) has at most one nonzero cohomology group.

Levine and Zhang in [LZ19] have studied the case of del Pezzo surfaces in greater detail. Let Wm denote the Weyl group acting on Pic(Xm ). Let P H W (v) denote the stack parameterizing torsion free sheaves V such that Ext (V, V(−σ(H))) = 0 for every σ ∈ Wm . Theorem 5.21. [LZ19, Theorem 1.2] Let Xm be a del Pezzo surface with m ≤ 5. Let v be a Chern character such that H · ν(v) ≥ −2 and P H W (v) ̸= ∅. (1) If C · ν(v) ≥ −1 for all (−1)-curves C on Xm , then v is nonspecial. (2) If there exists σ ∈ Wm such that ν(v) · σ(H) ≤ −1 or ν(v) · σ(H − Ei ) ≤ −1 for some i,

then v is nonspecial. (3) Let ν(v) · σ(H) > −1 and ν(v) · σ(H − Ei ) > −1 for all i and all σ ∈ Wm . Suppose C is a (−1)-curve such that C · ν(v) < −1 and let π denote the map contracting C. Then v is nonspecial if and only if π∗ (V) is nonspecial for the general V and χ(π∗ (V)) ≤ 0.

As an application, when m ≤ 6, Levine and Zhang classify −KXm -stable Chern characters in terms of a Drézet-Le Potier type condition. More generally, by studying blowups of P along collinear points and taking deformations, Zhao has studied the weak Brill-Noether Problem on general blowups of P [Zha22, Theorems 4.7 and 7.4] and has shown the nonemptiness of moduli spaces for certain Chern characters and polarizations [Zha22, Theorems 6.1 and 7.3].

5.6. Weak Brill-Noether for K-trivial surfaces. There has been recent progress for comput-

ing the cohomology of the general sheaf on K-trivial surfaces using Bridgeland stability conditions.

In this subsection, we will introduce Bridgeland stability conditions and briefly explain the strategy

for using them to prove weak Brill-Noether Theorems, concentrating on the case of K surfaces.

5.6.1. Bridgeland stability. Let Db (X) denote the bounded derived category of coherent sheaves

on X. Let K(Db (X)) denote the K-group of Db (X) and let Z : K(Db (X)) → C be a group homomorphism called the central charge which factors through the Chern character. Definition 5.22. A Bridgeland stability condition on Db (X) is a pair σ = (A, Z), where A is an abelian category which is the heart of a bounded t-structure on Db (X), and Z is a central charge satisfying the following properties: (1) Positivity: If 0 ̸= E ∈ A, then Z(E) = reiθ with r > 0 and 0 < θ ≤ π. Using Z, we can

ℑ(Z(E))

. An object E ∈ A is σ-(semi)stable if for every subobject F in A, we have µσ (F ) < µσ (E). (−)

(2) Harder-Narasimhan property: Every object E ∈ A has a finite Harder-Narasimhan filtra- tion with σ-semistable quotients of strictly decreasing slope. ∗ (3) Support condition: For a fixed norm | · | on Halg (X, R), there exists a constant C > 0 such that for all σ-semistable E ∈ A we have |Z(E)| ≥ C| ch(E)|. The set Stab(X) of Bridgeland stability conditions on X has the structure of a complex manifold [Bri07, Corollary 1.3]. Here we will be interested in very special stability conditions on surfaces

constructed by Bridgeland [Bri08] and Arcara and Bertram [AB13]. Given an ample divisor H on a surface X, define two subcategories of the category of coherent sheaves Coh(X) by T s = {E ∈ Coh(X) : µH (G) > sH 2 for every quotient G of E} F s = {E ∈ Coh(X) : µH (F ) ≤ sH 2 for every subsheaf F of E}. The pair (T s , F s ) forms a torsion pair in Coh(X), that is Hom(T, F ) = 0 if T ∈ T s and F ∈ F s and every coherent sheaf V fits in a unique exact sequence

0 → T → V → F → 0,

where T ∈ T s and F ∈ F s . Given a torsion pair in the heart of a t-structure, one obtains the heart of a new t-structure by tilting. Explicitly, tilting Coh(X) with respect to (T s , F s ), we obtain the category As defined by As = {E • ∈ Db (X) : H −1 (E • ) ∈ F s , H 0 (E • ) ∈ T s , H i (E • ) = 0 for i ̸= −1, 0}.

Define Z

Zs,t (E) = − e−(s+it)H ch(E). X

Then the pair σs,t = (As , Zs,t ) is a Bridgeland stability condition for s, t ∈ R, t > 0 [AB13]. Given a Chern character v and a stability condition σ = (As , Zs,t ), let Mσ (v) be the moduli space of σ-semistable objects in As with Chern character v.

5.6.2. Walls. Given a Chern character v, there is a locally finite wall-and-chamber decomposition

of the upper (s, t)-half-plane such that within each chamber the σ-(semi)stable objects remain constant. If a semistable object E gets destabilized by a subobject F , then the σ-slopes of E and F must become equal for some stability conditions. Expressing the equality µσ (E) = µσ (F ) as a function of s and t, one obtains that the walls are either vertical lines or nested, disjoint semi-circles. Moreover, when t is sufficiently large (the large volume limit), the Bridgeland moduli

space is isomorphic to MX,H (v).

5.6.3. Weak Brill-Noether for K surfaces. Let X be a K surface. Stable sheaves on K surfaces

have been classified by Mukai [Muk87], O’Grady [O’G99] and Yoshioka [Yos99]. It is customary to phrase the classification in terms of the Mukai vector instead of the Chern character. Let p v(E) := ch(E) td(X) = (r(E), c (E), r(E) + ch (E)), where td(X) is the Todd class of X. There is a pairing between any two Mukai vectors given by ⟨v, v′ ⟩ = ⟨(r, c, a), (r′ , c′ , a′ )⟩ := c · c′ − ra′ − r′ a = −χ(v, v′ ), where c · c′ is the intersection product on H 2 (X, Z). A class is called spherical if v = −2 and

∗ isotropic if v = 0. A Mukai vector v is primitive if it is not divisible in Halg (X, Z). We say a primitive Mukai vector v = (r, c, a) is positive if v ≥ −2 and either (1) r > 0; or (2) r = 0, c is effective, and a ̸= 0; or (3) r = c = 0 and a > 0. Theorem 5.23. Let X be a K surface and let v = mv be a Mukai vector, where v is a primitive positive Mukai vector and m > 0. Then MX,H (v) is non-empty for any ample divisor H. If H is generic with respect to v, then: (1) The moduli space MX,H (v) is non-empty if and only if v ≥ −2.

(2) If m = 1 or v > 0, then dim MX,H (v) = v + 2. (3) When v = −2, then MX,H (v) is a single point parameterizing the direct sum of m copies of a spherical bundle. When v = 0, then dim MX,H (v) = 2m. (4) When v > 0, MX,H (v) is a normal irreducible projective variety with Q-factorial singu- larities. Now suppose that Pic(X) = ZH with H 2 = 2n. If V is a stable sheaf with ch (V) = dH with d > 0, then H 2 (X, V) = 0 by stability. Hence, we need to compute H 0 (X, V) and H 1 (X, V). The

fact that H 2 (X, OX ) = C allows one to construct many counterexamples to weak Brill-Noether on K surfaces. Example 5.24. [CNY23, Example 1.3] The linear system |H| defines a morphism f : X → Pn+1 . The sheaf V = f ∗ TPn+1 is the unique stable sheaf in its moduli space. The pullback of the Euler sequence by f 0 → OX → OX (H)n+2 → V → 0 implies that h (X, V) = n + 4n + 3 and h (X, V) = 1.

Example 5.25. [CNY23, Example 6.3] Let X be a double cover of P branched along a very general sextic curve. Then the pullback of an exceptional bundle on P is a spherical stable bundle on X, which is the unique member of its moduli space. For example, let fk be the kth Fibonacci number with f = f = 1. For k ≥ 2, the pullback of the corresponding exceptional bundle from P has resolution ⊕f 0 → OX 2k−2 → OX (H)⊕f2k → V k → 0.

The bundle V k has rank f2k−1 and

h (X, V k ) = 2f2k + f2k−1 and h (X, V k ) = f2k−2 . When k = 2, one recovers the n = 1 case of the previous example. Example 5.26. [CNY23, Theorem 10.1] Let C be a smooth member in |H|. Consider extensions of the form 0 → OX (H)r → V → OC (L) → 0, where L is a general line bundle on C with Euler characteristic 2n − r. Then v(V) = (r, (r + 1)H, n(r + 2)), and from the exact sequence we conclude that h (X, V) = r(n + 2) + max(0, 2n − r) and h (X, V) = max(0, r − 2n). As the general element in MX,H (v) is given by such an extension, we get counterexamples to the

weak Brill-Noether property when r > 2n. These examples should convince the reader that the structure of the set of counterexamples to the weak Brill-Noether property on K surfaces is quite complicated and depends on arithmetic properties of the Mukai lattice. We now explain the strategy to use Bridgeland stability conditions to prove weak Brill-Noether.

5.6.4. The strategy. Let ∆ ⊂ X × X be the diagonal and let π and π denote the two projections

from X × X to X. Let ΦIX→X ∆ : Db (X) → Db (X) be the Fourier-Mukai transform acting by

I∆ ∗

ΦX→X (E) = π2∗ (π (E) ⊗ I∆ ). For each E ∈ Coh(X), tensoring the exact sequence

0 → I∆ → OX×X → O∆ → 0

by π1∗ E and pushing forward by π gives the exact triangle

ΦIX→X

(E) → RΓ(X, E) ⊗ OX → E

which induces the long exact sequence f

0 → H (ΦIX→X

(E)) → H 0 (X, E) ⊗ OX → E → H (ΦIX→X

(E)) →

H 1 (X, E) ⊗ OX → 0 → H (ΦIX→X

(E)) → H 2 (X, E) ⊗ OX → 0.

Here f is the evaluation morphism. Let F := ΦIX→X

∆ (E))∨ be the derived dual. Then

Hi (ΦIX→X

∆ (E))) = Hi (F ∨ ) = Exti (F, OX ). Lemma 5.27. [CNY23, Lemma 3.1] Let E be a coherent sheaf with no zero-dimensional torsion and set F := ΦIX→X ∆ (E)∨ . Then (1) F is a coherent sheaf if and only if E is nonspecial and generically globally generated. (2) F is a torsion-free sheaf if and only if E is nonspecial and fails to be globally generated in at most finitely many points. (3) F is a locally free sheaf if and only if E is nonspecial and globally generated.

Lemma 5.2 transforms the weak Brill-Noether problem into showing that ΦIX→X ∆

(E)∨ is a sheaf, and this is how Bridgeland stability becomes useful. Given the Mukai vector (r, dH, a) of E, there is a special chamber C in the (s, t)-plane right above the wall defined by µσ (Ix∨ ) = µσ (E), where Ix is the ideal sheaf of a point x ∈ X. Minamide, Yanagida and Yoshioka prove the following result. Theorem 5.28. [MYY18, Theorem 4.9] If σ ∈ C, then the moduli space of Bridgeland stable ob- jects Mσ (r, dH, a) is isomorphic to the Gieseker moduli space MX,H (a, dH, r) via E 7→ ΦIX→X

(E)∨ .

Recall that for t ≫ 0, the moduli space Mσ (r, dH, a) is isomorphic to the Gieseker moduli space MX,H (r, dH, a). As we decrease t in order to reach the chamber C, we may cross a number of Bridgeland walls. If not all of the sheaves in MX,H (r, dH, a) get destabilized along the way, then for σ ∈ C, MX,H (r, dH, a) and Mσ (r, dH, a) have a common open subset, so MX,H (r, dH, a) satisfies weak Brill-Noether by Lemma 5.27. Furthermore, if the general sheaf in MX,H (a, dH, r) is locally free, then the general sheaf in MX,H (r, dH, a) is globally generated.

However, along the way we may encounter totally semistable walls, where every sheaf gets

destabilized. Bayer and Macrı̀ [BM14a, BM14b] have classified the totally semistable walls. Using this classification, one can show that if v = (r, dH, a) is a Mukai vector with r ≥ 0 ,d > 0, and v ≥ −2 such that MX,H (v) does not satisfy weak Brill-Noether, then there must exists a spherical

Mukai vector v = (r , d H, a ) satisfying

(1) 0 < d ≤ d (2) 0 > ⟨v, v ⟩ = 2ndd − ar − ra (3) 0 < dr − d r ≤ da − d a.

Determining whether there are totally semistable walls between the Gieseker chamber and C is

a purely numerical problem, albeit a complicated one. Moreover, we obtain a dichotomy. Either there are no totally semistable walls above the chamber C and weak Brill-Noether holds, or the largest totally semistable wall provides a resolution of the general sheaf in MX,H (r, dH, a) which can then be used to compute the cohomology. This strategy yields the following sharp qualitative theorem. Theorem 5.29. [CNY23] Let X be a K surface such that Pic(X) ∼ = ZH with H 2 = 2n. Let

v = (r, dH, a) be a Mukai vector with v ≥ −2, r ≥ 2 and d > 0. (1) For each r ≥ 2, there exists a finite set of tuples (n, r, d, a) for which v fails to satisfy weak Brill-Noether. (2) If n ≥ r, then v satisfies weak Brill-Noether. (3) If a ≤ 1, then  v satisfies weak Brill-Noether. (4) If d ≥ r nr + 2, then v satisfies weak Brill-Noether. (5) Assume a ≥ 2 and n > 1. If n ≥ 2r or d ≥ 2r   n + 2, then the general sheaf in MH (v) is globally generated. More importantly, the technique allows one to compute the cohomology of the general sheaf.

For example, in [CNY23] the authors classify all the Mukai vectors of rank at most 2 on K surfaces of Picard rank 1 for which weak Brill-Noether fails and compute the cohomology of the general sheaf in these cases. Recently, Liu has developed these ideas further and obtained an algorithm for computing the cohomology of spherical bundles on K surfaces [Liu23]. Similar strategies can be applied to other K-trivial surfaces such as abelian surfaces and Enriques surfaces. For example, in the case of abelian surfaces of Picard rank 1, one obtains the following.

Theorem 5.30. [CN23, Theorem 4.1] Let X be an abelian surface with Pic(X) = ZH and let v = (r, dH, a) be a Mukai vector such that r ≥ 0, d > 0 and v ≥ 6. Then weak Brill-Noether holds for MX,H (v).

5.7. Asymptotic weak Brill-Noether. In general, the weak Brill-Noether problem is wide

open. Given a rank r ≥ 2 and a slope ν, if ∆ ≫ 0, then χ(v) < 0. Moreover, by O’Grady’s Theorem 4.9, we can assume that the moduli space MX,H (v) is irreducible and the general member is a slope-stable vector bundle. Potentially increasing ∆, after a series of elementary modifications, we can assume that the general sheaf has no global sections. By applying the same argument to the Serre dual, we can conclude that the general sheaf also has no h . We thus obtain an asymptotic weak Brill-Noether result.

Proposition 5.31. [CH18c, Proposition 7] Let v be a Chern character of rank at least 2. If ∆ ≫ 0, depending on X, H, r and ν(v), then H i (X, V) = 0 unless i = 1. Similarly, by applying Serre Vanishing, one can show that the general sheaf in MX,H (v ⊗ mA) has no higher cohomology if A is ample and m ≫ 0 (see [CH18c, Theorem 3.7]). It would be very useful to have good explicit bounds on m, especially for surfaces of general type.

6. Applications and generalizations of weak Brill-Noether

In this section, we discuss some applications and generalizations of weak Brill-Noether Theo-

6.1. Classification of stable Chern characters. The weak Brill-Noether problem is closely

tied to Problem 4.8, the problem of classifying Chern characters of stable sheaves.

6.1.1. The classification of stable vector bundles on P . Drézet and Le Potier classified Chern

characters of stable bundles on P [DLP85, LeP97]. We now briefly recall this classification. Exceptional bundles. A coherent sheaf V is exceptional if Hom(V, V) = C and Exti (V, V) = 0 for i > 0. Exceptional sheaves on P have been classified by Drézet [D87]. Let V be an exceptional sheaf on P of rank r. First, since Ext (V, V) = 0, every small deformation of V has to be trivial.

Consequently, g ∗ (V) ∼

= V for every g ∈ PGL(3). We conclude that V cannot have any singularities and must be a vector bundle. By Riemann-Roch, χ(V, V) = 1 = r (P (0) − 2∆) = r − c + 2r ch . Hence, r and c are relatively prime and ∆(V) = 2 1 − r . Observe that the discriminant of

an exceptional bundle is less than 1 and that the slope determines the Chern character of an exceptional bundle. If E ⊂ P is an elliptic curve embedded as a cubic, then applying Hom(V, −) to the exact sequence

0 → V(−3) → V → V |E → 0

and using Hom(V, V(−3)) = Ext (V(−3), V(−3))∗ = 0 and Ext (V, V(−3)) = Ext (V(−3), V(−3))∗ = 0, we conclude that Hom(V |E , V |E ) = C. Hence, the restriction of V to E is simple. Since a bundle on an elliptic curve is a direct sum of its Harder-Narasimhan factors, we conclude that V |E is semistable. As E was arbitrary, it follows that V is semistable and thus stable since gcd(r, c ) = 1.

Consequently, an exceptional bundle on P is stable. Conversely, a stable bundle V on P with ∆(V) < 2 is exceptional by Riemann-Roch. Moreover, two exceptional bundles with the same slope are isomorphic since χ(V 1 , V 2 ) = 1 and hence there is a nontrivial homomorphism between V 1 and V 2 which must be an isomorphism by stability. The line bundles OP (n) are exceptional. An exceptional collection on P is a triple of excep- tional bundles (V 1 , V 2 , V 3 ) such that Exti (V k , V j ) = 0 for 1 ≤ j < k ≤ 3 and all i. On P ,

(OP , OP (1), OP (2)) is the standard exceptional collection. Given a pair of adjacent bundles in an exceptional collection V, W, we can form the left and right mutations of the pair 0 → V → W ⊗ Hom(V, W)∗ → RW (V) → 0

0 → LV (W) → V ⊗ Hom(V, W) → W → 0. For example, the Euler sequence exhibits TP as ROP (1) (OP ). Drézet proves that on P the right RW (V) and left LV (W) mutations are again exceptional bundles. In fact, given an exceptional col- lection, (V 1 , V 2 , V 3 ), the collections (V 2 , RV 2 (V 1 ), V 3 ) and (V 1 , LV 2 (V 3 ), V 2 ) are again exceptional collections. This justifies the terminology right/left mutation. Furthermore, every exceptional bundle on P is obtained from line bundles by a sequence of mutations [D87]. One can systemati-

cally generate the slopes of all exceptional bundles on P by formalizing the process of mutations to obtain an explicit one-to-one correspondence ε : Z[ 2 ] → E between dyadic integers and the exceptional slopes, defined inductively by ε(n) = n for an integer n and    p  p + 1 2p + 1 ε = ε q .ε , 2q+1 2 2q where α+β ∆β − ∆α α.β = + . 2 3+α−β Stable bundles in general. Given two stable bundles on P with 0 < µ(V) − µ(W) < 3, we

have that Hom(V, W) = 0 and Ext (V, W) = Hom(W, V(−3)) = 0 by stability. Consequently, χ(V, W) = r(V)r(W) (P (µ(W) − µ(V)) − ∆(V) − ∆(W)) ≤ 0. Setting V to be an exceptional bundle Eα , gives an inequality for the discriminant ∆(W) of a stable bundle W in terms of its slope µ(W) provided 0 < µ(Eα ) − µ(W) < 3. Similarly, setting W to be an exceptional bundle Eα , gives an inequality for the discriminant ∆(V) of a stable bundle V in terms of its slope µ(V) provided 0 < µ(V) − µ(Eα ) < 3. Graphing the case of equality in the

(µ, ∆)-plane for all exceptional bundles yields a fractal curve called the Drézet-Le Potier curve (see Figure 1). The Chern character of any stable bundle which is not exceptional must lie above the Drézet-Le Potier curve depicted by the shaded region. Conversely, using dimension estimates, one can show that if the Chern character lies above the Drézet-Le Potier curve, then the general bundle defined by the Gaeta resolution is stable. One thus obtains the main classification theorem

on P due to Drézet and Le Potier. Theorem 6.1 (Drézet-Le Potier). Let v be an integral Chern character of positive rank. There exists a Gieseker semistable sheaf with Chern character v if and only if ∆(v) ≥ δ(µ(v)) or v is a multiple of the Chern character of an exceptional bundle. When ∆(v) ≥ δ(µ(v)), the moduli space MP ,L (v) is an irreducible, normal projective variety of dimension r (2∆ − 1) + 1.

TP (−1)

OP OP (1) µ

0 1

Figure 1. The Drézet–Le Potier curve. Chern characters of stable bundles with

positive dimensional moduli spaces lie in the shaded region above the curve. The Chern characters of exceptional bundles are below the line ∆ = 1 .

6.1.2. Classification of stable Chern characters on Hirzebruch surfaces. Using Theorems 5.1 and

5.1 as main tools, one can classify the Chern characters of stable sheaves on Hirzebruch surfaces

Fe for generic polarizations (see [CH21]). Let Hm = E + (m + e)F on Fe with m ∈ Q>0 . Then Hm is ample and every ample divisor is proportional to some Hm . To classify Chern characters, one first shows that an Hm -semistable sheaf is H⌈m⌉+1 -prioritary and thus one needs to classify Hk -prioritary sheaves for positive integers k. Then one computes the Harder-Narasimhan filtration of the general Hk -prioritary sheaf. The Gaeta-type resolution resolves the first issue.

Let ν(v) = ϵE + φF . Set

1 ∆ ψ := φ + e(⌈ϵ⌉ − ϵ) − , 2 1 − (⌈ϵ⌉ − ϵ) and let L := L⌈ϵ⌉,⌈ψ⌉ = ⌈ϵ⌉E + ⌈ψ⌉F. Then the general sheaf in P Fe ,F (v) admits a Gaeta-type resolution as in (2) with L = L . Ex- pressing the condition that the sheaf be Hk prioritary, one sees that we need χ(v(−L − Hk )) ≤ 0.

Conversely, one can construct Hk -prioritary sheaves satisfying this inequality to conclude the

following theorem. Theorem 6.2. [CH21, Theorem 4.16] Let v be an integral Chern character of positive rank on Fe with ∆(v) ≥ 0 and let k be a positive integer. Then the stack P Hk (v) is nonempty if and only if χ(v(−L − Hk )) ≤ 0.

This theorem already provides stronger Bogomolov inequalities for the existence of semistable

Next, we compute the Harder-Narasimhan filtration of the general Hm -prioritary sheaf. There

exist Hm -semistable sheaves if and only if the filtration is trivial. Suppose the general Hm -Harder- Narasimhan filtration has length ℓ and the graded pieces have Chern characters vi = (ri , νi , ∆i ). Then

X ℓ

Furthermore, the moduli spaces MFe ,Hm (vi ) are nonempty since the graded pieces are semistable sheaves. The fact that the sheaves are H⌈m⌉+1 -prioritary and the Schatz-stratum corresponding to this Harder-Narasimhan filtration has codimension 0 lead to additional inequalities that place strong restrictions on vi . First, the prioritary condition implies that the restriction of the general sheaf to a general rational curve in the class H⌈m⌉ or H⌊m⌋ has balanced splitting. This translates to the inequality

Next, a dimension computation shows that for the Schatz-stratum to have codimension 0, the

following orthogonality relations hold χ(vi , vj ) = 0 for i < j. Hence, the slopes νi are restricted to a bounded region, and since the ranks ri are bounded there are only finitely many possibilities for the νi . Furthermore, the orthogonality relations imply the discriminant ∆i has to be the minimal possible discriminant of an Hm -semistable sheaf with rank ri and total slope νi . Hence, there are finitely many possible vi that can be the Chern characters of the graded pieces of the generic Hm -Harder-Narasimhan filtration.

Conversely, if one can find Chern characters vi , 1 ≤ i ≤ ℓ, that satisfy these constraints, then the general Harder-Narasimhan filtration has factors with these Chern characters. Theorem 6.3. [CH21, Theorem 5.3] Let v be a Chern character such that P H⌈m⌉ (v) is nonempty. Let v , . . . , vℓ ∈ K(Fe ) be positive rank Chern characters satisfying the following properties: (1) ℓi=1 vi = v, P (2) The reduced Hilbert polynomials qi of vi are strictly decreasing q > · · · > qℓ , (3) µHm (v ) − µHm (vℓ ) ≤ 1,

(4) χ(vi , vj ) = 0 for i < j, (5) MFe ,Hm (vi ) is nonempty, Then the Harder-Narasimhan filtration of the general sheaf in P H⌈m⌉ (v) has length ℓ and the factors have Chern characters vi .

Thus determining the Harder-Narasimhan filtration of a general H⌈m⌉ -prioritary sheaf becomes

a finite computational problem. In particular, one obtains an algorithm for classifying Chern characters of Hm -semistable sheaves. Using the fact that K(Fe ) ∼ = Z , one can show ℓ ≤ 4 to further simplify the problem. Remark 6.4. On P , either the subbundle or the quotient bundle in the general Harder-Narasimhan filtration of a general prioritary bundle is exceptional. Hence, one can give explicit inequalities for the discriminants of semistable bundles without computing the semistable bundles of lower rank.

On Fe , it may happen that neither the quotient nor the subbundle in a generic Harder-Narasimhan filtration of length 2 is an exceptional bundle (see [CH21] for explicit examples). In general, it is not enough to consider the constraints given by exceptional bundles.

6.2. The tensor product problem. A natural generalization of the Brill-Noether problem which

plays a fundamental role in the birational geometry of moduli spaces of sheaves is the tensor product problem (see [ABCH13, Hui16]). Problem 6.5 (The tensor product problem). Let V ∈ MX,H (v) and W ∈ MX,H (w) be two stable sheaves on X. Compute the cohomology of V ⊗ W.

More generally, one can ask for the cohomology of the tensor product of Bridgeland stable objects in the derived category of X. The problem is already interesting when V and W are general elements in their moduli spaces. When W = OX , the problem reduces to computing the cohomology of V. When V is general in its moduli, we recover the weak Brill-Noether problem. When W = IZ is an ideal sheaf of points and V is a line bundle, then the problem reduces to the classical interpolation problem asking for when points impose independent conditions on

sections of a line bundle. More generally, when V is a higher rank sheaf, the problem is the higher rank interpolation problem asking for when points impose independent conditions on sections of a sheaf. In general, these problems are wide open, though some important special cases have been solved. For example, the higher rank interpolation problem has an explicit combinatorial solution for zero-dimensional monomial schemes on P [CH14, Theorem 1.4]. On P there is an almost complete answer to the tensor product problem for general sheaves.

Let v be a stable Chern character on P that lies on or above the Drézet-Le Potier curve and let V be a general stable sheaf in MP ,L (v). Then the associated exceptional bundle E+ is the exceptional bundle on P with the smallest slope such that if E ′ is any exceptional bundle with µL (E ′ ) > µL (E+ ), then χ(V ⊗E ′ ) > 0. It may happen that χ(V ⊗E+ ) < 0, χ(V ⊗E+ ) = 0 or χ(V ⊗E+ ) > 0. By Theorem [CHW17, Theorem 4.1], the associated exceptional bundle exists. Briefly, by Riemann-Roch, the locus v⊥ := {w : χ(v ⊗ w) = 0} is an upward parabola in the

slope-discriminant plane. This parabola intersects the line ∆ = 2 in two points which lie under peaks of the Drézet-Le Potier curve. The Drézet-Le Potier curve has infinitely many peaks (see

Figure 1), each determined by an exceptional bundle. The associated exceptional bundle E+ is

the exceptional bundle lying under the rightmost peak determined in this way. (The points on the line ∆ = 2 which do not lie under a peak form a generalized Cantor set C. The fact that the parabola v⊥ meets ∆ = 1 at points that are not in C relies on number-theoretic properties of C. For instance, the points of C which are not endpoints of peaks have transcendental µ-coordinate.) The answer to the problem is easiest to state in the case where χ(v ⊗ E+ ) ≤ 0. In the other case where χ(V ⊗E+ ) > 0, we need to fix some additional notation. In this case we let

k = v − χ(V ⊗E+ ) ch(E+ ) be the Chern character of the mapping cone of the canonical evaluation map E+∗ ⊗ Hom(E+∗ , V) → V . Theorem 6.6. [CHK21, Theorem 1.2] Let v and w be Chern characters of stable bundles on P . Suppose MP ,L (v) is positive dimensional and w is sufficiently divisible (depending on v). Let V ∈ MP ,L (v) and W ∈ MP ,L (w) be general bundles. (1) If χ(v ⊗ E+ ) ≤ 0, then either H 0 (V ⊗ W) = 0 or H 1 (V ⊗ W) = 0. (2) If χ(v ⊗ E+ ) > 0 and rk(k) ≤ 0, then either H 0 (V ⊗ W) = 0 or H 1 (V ⊗ W) = 0.

(3) Suppose χ(v ⊗ E+ ) > 0 and rk(k) > 0. (a) If χ(k ⊗ w) ≥ 0 or χ(w ⊗ E+∗ ) ≤ 0, then either H 0 (V ⊗ W) = 0 or H 1 (V ⊗ W) = 0. (b) Otherwise V ⊗ W is special and h (V ⊗ W) = χ(v ⊗ E+ )χ(w ⊗ E+∗ ) and h (V ⊗ W) = −χ(k ⊗ w)

By applying Serre duality, the theorem computes the tensor product of two general stable

bundles subject to the requirement that w is sufficiently divisible. One can give an explicit bound on how divisible w needs to be depending on v. We do not know whether the divisibility is

necessary on P or an artifact of the proof. The following example shows that on Pn for n > 2, some divisibility is necessary. Example 6.7. [CHS23, Example 1.3] Let V m be the kernel of a general map 0 → V m → OP (2)⊕4m → OP (4)⊕m → 0. When m = 1, h (P , V 1 ) = 6 and h (P , V 1 ) = 1. On the other hand, when m ≥ 2, then h (P , V m ) = 0.

6.3. Construction of Brill-Noether divisors and birational geometry. Let v and w be two

stable Chern characters on X such that χ(v ⊗ w) = 0. Then given a general sheaf W ∈ MX,H (w), we can define the virtual Brill-Noether divisor DW := {V ∈ MX,H (v)|h (V ⊗ W) ̸= 0}. In general, DW may not be a divisor. If hi (W ⊗ V ′ ) = 0 for all i and a general V ′ , then DW is an effective divisor. Hence, the tensor product problem in the special case of χ(v ⊗ w) = 0 is the key ingredient for constructing effective Brill-Noether divisors. There has been significant progress in computing the ample and effective cones of MX,H (v) and

running the minimal model program for these moduli spaces in recent years, especially when X is P , a Hirzebruch, K3, abelian or Enriques surface. Surveying the developments in this direction would take us too far afield, so we refer the reader to [ABCH13, BM14a, BM14b, BC13, CH16, CH15, CH18a, CHW17, Hui16, LZ19, MYY18, N16a, NY] for some of these developments.

6.4. Constructions of Ulrich bundles on surfaces. A solution of the weak Brill-Noether

problem allows one to construct Ulrich bundles on surfaces. Definition 6.8. Let X ⊂ Pn be a smooth, projective variety of dimension d. An Ulrich bundle V on X is a bundle that satisfies H i (X, V (−j)) = 0 for 1 ≤ j ≤ d and all i. Ulrich bundles play a central role in the study of Chow forms of a variety [ESW03], the min- imal resolution conjecture (see [AGO17]) and Boij-Söderberg Theory (see [ES11]). For example, Eisenbud and Schreyer show that the cone of cohomology tables of X is the same as that of

Pd if and only if X admits an Ulrich bundle [ES11]. Eisenbud and Schreyer raise the question whether every projective variety admits an Ulrich bundle. Existence is known in some cases in- cluding smooth curves [ESW03], complete intersections [BaHU91], del Pezzo surfaces [CKM13], certain rational surfaces [ESW03, Kim16], K surfaces [AGO17, Fae18], abelian surfaces [Bea16] and certain Enriques surfaces [BN18] among many others. Let V be an Ulrich bundle for a polarized surface (X, H) with rank r, total slope ν and dis-

criminant ∆. Since V(−H) and V(−2H) have no cohomology, the Euler characteristics of V(−H) and V(−2H) must vanish. By Riemann-Roch, we conclude that 1 2 1 ν − ν · KX + χ(OX ) − ∆ = 0 2 2 1 1 (ν − H)2 − (ν − H) · KX + χ(OX ) − ∆ = 0 2 2

Consequently, an Ulrich bundle on a surface satisfies

(3) 2ν · H = H 2 + H · KX and 2∆ = ν 2 − ν · KX + 2χ(OX ).

Conversely, if we take 2ν = H + KX , MX,H (r, ν, ∆) contains locally free sheaves, and it satisfies weak Brill-Noether, then the general locally free sheaf in MX,H (r, ν, ∆) is an Ulrich bundle.

Combining O’Grady’s Theorem with Serre vanishing yields an asymptotic existence result on

any smooth projective surface. Theorem 6.9. [CH20b, Theorem 4.3] Let (X, H) be a smooth, polarized surface. There exists a positive integer m such that for all m ≥ m , the polarized surface (X, mH) admits an Ulrich bundle of every positive even rank. Moreover, if KX (respectively, KX + H) is divisible by 2 in Pic(X) and 2m ≥ m (respectively, 2m + 1 ≥ m ), then (X, 2mH) (respectively, (X, (2m + 1)H)) admits an Ulrich bundle of every rank r ≥ 2. A polarized variety (X, H) is of Ulrich wild representation type if one can find arbitrarily large

dimensional families of Ulrich bundles on X with respect to H. An easy corollary Theorem 6.9 is the following: Corollary 6.10. [CH20b, Corollary 4.5] Let (X, H) be a smooth, polarized surface. Then there exists an integer m such that for all m ≥ m , (X, mH) is of Ulrich wild representation type.

On specific surfaces, weak Brill-Noether gives detailed information on Chern classes of Ulrich

bundles. For example, on Hirzebruch surfaces one obtains the following (see also V. Antonelli [Ant18]). Theorem 6.11. [CH20b, Theorem 4.6] Let H = aE + bF be an ample divisor on Fe . Let v = (r, ν, ∆) = (r, αE + βF, ∆) be an integral Chern character with r ≥ 2. There exists a locally free

F -prioritary sheaf V with Chern character v satisfying

H i (Fe , V(−H)) = H i (Fe , V(−2H)) = 0 ∀i

if and only if ea(a − 1) ea(a − 1) a−1+ ≤ α ≤ 2a − 1 − ,  2b 2b b e β = e− (α + 1) + 3b − 1 − (3a + 1) a 2 and   e b ∆(V) = − (α + (2 − 3a)α + 2a − 3a + 1).

2 a

In particular, there exists an Ulrich bundle of rank 2 for H = aE + bF with invariants       3 3 e a v = (r, ν, ∆) = 2, a−1 E+ b − − 1 F, (2b − ae) . 2 2 2 8 Similarly detailed theorems can be proven on certain blowups of P or Fe using the corresponding weak Brill-Noether statements. We refer the reader to [CH20b, Theorem 4.1 and 4.11] for precise statements. The weak Brill-Noether Theorems for K or abelian surfaces similarly yield

classification results for Ulrich bundles. We state the result for K surfaces and refer the reader to [CN23, Corollary 6.1] for abelian surfaces. Proposition 6.12. [CNY23, Proposition 4.4] Let X be a K surface with Pic(X) = ZH. There exists an Ulrich bundle of rank r with respect to mH if and only 3rm
if 2 | rm. Moreover,
when an Ulrich bundle of rank r exists, it has Mukai vector v = r, 2 H, r(2m n − 1) . In particular, there exists an Ulrich bundle of any rank r ≥ 2 with respect to 2H.

One can also prove results on variants of Ulrich bundles. For example, Eisenbud and Schreyer [ES11] conjectured that there exist bundles V on P × P with natural cohomology, meaning that V(aF + bF ) has at most one nonzero cohomology group for any a, b ∈ Z. The weak Brill-Noether theorem on P × P immediately yields a positive solution (see also [Sol18]). Corollary 6.13. [CH20b, Corollary 4.13] Let F and F denote the two fiber classes on P ×P . Let v be an integral Chern character such that rk(v) ≥ 2 and ∆ ≥ 0. Then P P ×P ,F (v) is nonempty,

and the general V ∈ P P ×P ,F (v) is locally free and has at most one nonzero cohomology group. In particular, the very general member of P P ×P ,F (v) is a bundle with natural cohomology.

7. Cohomology jumping loci

When the weak Brill-Noether Theorem holds for MX,H (v), it is natural to try to describe the loci of bundles with unexpected cohomology. In this section, we will review recent developments on the Brill-Noether loci on P due to Gould, Lee and Liu [GLL22] and describe some generalizations to other surfaces. Let MX,H (v) be an irreducible moduli space such that every sheaf V has H 2 (X, V) = 0. For example, this condition automatically holds on moduli spaces of sheaves on P , a K3, an abelian

or a Hirzebruch surface if ν(v) is effective. Assume that the locus of stable sheaves MX,H (v)s k is nonempty. Define the k-th Brill-Noether locus BX,H (v) as the closure of the locus of stable sheaves with at least k-independent sections: k BX,H (v) := {V ∈ MX,H (v)s |h (X, V) ≥ k} ⊂ MX,H (v). k The locus BX,H (v) ∩ MX,H (v)s has a natural determinantal structure (see [CHW17, Proposition 2.7] or [CoMR10, §2]), so every irreducible component of BX,H k (v) has codimension at most the

k+1 k expected value k(k − χ(v)). By definition, we have BX,H (v) ⊂ BX,H (v) for k ≥ 0. Problem 7.1. One can immediately raise the following natural problems. k (1) Determine when BX,H (v) is nonempty, equivalently determine maxV∈MX,H (v)s (h (X, V)). k (2) Classify the irreducible components of BX,H (v) and determine their dimensions. In par- k ticular, determine when BX,H (v) is irreducible. k (3) Describe the singularities of BX,H (v). k (4) When do Torelli type theorems hold for BX,H (v), i.e., when can X be recovered from

BX,H (v)?

k In general, very little is known about these problems. Even in rank 1, the loci BX,H (v) can have many components of different dimensions and are poorly understood. Example 7.2. [GLL22, Proposition 5.3] Let v = ch(IZ (3)) on P , where Z is a zero-dimensional scheme of length |Z| = n ≥ 10. The Brill-Noether locus BP ,L (v) parameterizing IZ (3) with h (P , IZ (3)) ≥ 2 has at least two irreducible components. (1) Let W ⊂ P2[n] be the closure of the locus with n − 4 points lying on a line and 4 general

points. Then dim(W ) = n + 6. (2) Let W ⊂ P2[n] be the closure of the locus with n − 1 points lying on a smooth conic and

1 general point. Then dim(W ) = n + 6.

We claim that if IZ (3) ∈ BP ,L (v) where Z has distinct points, then Z ∈ W or Z ∈ W . Let C and C be two distinct curves of degree 3 vanishing on Z. By Bézout’s Theorem, C and C must have a common curve D. Then D is either a conic (possibly reducible or nonreduced) or a line. If D is a conic, then the residual curves must be lines and they intersect in at most one point. Hence, Z ∈ W . If D is a line, then the residual curves must be conics. Since they do not have a common curve, their intersection must be a scheme of length 4. Hence, Z ∈ W . Since W and

W have the same dimension, they form distinct irreducible components of BP ,L (v). A slightly more subtle argument allows Gould, Lee and Liu to deduce that when v = ch(IZ (k)), where |Z| > k 2 , the Brill-Noether locus BP ,L (v) has at least k components, typically not all of the same dimension. Example 7.3. The previous example is not special to P . For simplicity, let X be a surface of Picard rank 1 generated by an effective ample divisor H and assume H 1 (X, OX ) = 0. Let m, n be two integers such that m > 2 is a sufficiently large odd integer and n > m H 2 . Let

v = ch(IZ (mH)) with |Z| = n. For 0 < s < m, let Ws ⊂ X [n] be the closure of the locus of points with h (X, OX ((m − s)H)) − 2 general points on X and n + 2 − h (X, OX ((m − s)H)) general points on a curve with class sH. Then dim(Ws ) = h (X, OX (sH)) + h (X, OX ((m − s)H)) + n − 2. Note that dim(Ws ) = dim(Wm−s ) and if s ≫ 0, then we can assume that the curve of degree s is irreducible. Hence, there are at least two components Ws and Wm−s with maximal dimension. The inclusion IZ (mH) ⊂ OX (mH) induces two independent curves C and C in |mH| that contain

Z. Let IZ (mH) be a general point of an irreducible component of BX,H (v). Since |Z| > m H 2 , by Bézout’s Theorem, C and C must have a common component of class kH. By the semicontinuity of the degree of the base curve, this component cannot contain Ws for s > k. Hence, if BX,H (v) is irreducible, we must have k = 1. There can be at most h (X, OX ((m − 1)H)) − 2 general points which do not lie on the common component. Hence, if this locus contains W , it must coincide with W . Then at least one of Ws or Wm−s is a component distinct from W and we conclude

that BX,H (v) is reducible. In fact, by the semicontinuity of the degree of the base locus, observe that Ws ̸⊂ Wt if s > t. Assuming that there are irreducible sections of H 0 (X, OX (kH)) for every k ≥ 1, then the loci Ws have no containments among each other and one can show that BX,H (v) has at least k components. Once the Brill-Noether loci in rank 1 are reducible, one can make examples of higher rank reducible Brill-Noether loci.

Example 7.4. [GLL22, Example 5.5] Consider extensions of the form

0 → OP → V → IZ (3) → 0,

where |Z| ≥ 10. Then χ(V) = 1 − |Z|. The Cayley-Bacharach condition is trivially satisfied, so there exists vector bundles of this form. Furthermore, if the points of Z are not collinear then there are no nonzero maps OP (2) → IZ (3), so V is stable. We have ext (IZ (3), OP ) = |Z| − 1. Set v = ch(V) and w = ch(IZ (3)). Now assume that Z ∈ Wi for i = 1 or 2 as in Example 7.2. Then V ∈ BP ,L (v). We thus obtain two loci in BP ,L (v) of dimension 2|Z| + 2. The expected codimension of BP ,L (v) is 3(|Z|−8). Since the moduli space has dimension 4|Z|−12, the expected

dimension of BP ,L (v) is |Z| + 12. Observe that we obtain unexpectedly large dimensional loci in BP ,L (v) if |Z| > 10. One can also easily check that the two components of BP ,L (w) yield two distinct components of BP ,L (v).

Similar constructions using Example 7.3 produce reducible Brill-Noether loci on more general

surfaces. Subbundles with a large space of sections is another source of components of Brill- Noether loci. Example 7.5. [GLL22, Example 5.11] Let vk be a a Chern character on P with µL (vk ) = 3 and ∆(vk ) = 5 + k . Then one can consider the following two types of bundles in BP ,L (vk ). (1) Let C be a smooth conic, D be a divisor of degree 1 − k and V be a general extension of the form

0 → OP⊕3 → V → OC (D) → 0.

(2) Let TP be the tangent bundle of P , Z be a zero-dimensional scheme of length k + 1 and

V be defined by a nonsplit extension of the form

0 → TP (−1) → V → IZ (1) → 0.

One can check that in both cases V is stable and give distinct irreducible components of BP ,L (vk ) when k ≫ 0. Using a systematic analysis of the ideas in these examples, Gould, Lee and Liu prove the following structure theorem on Brill-Noether loci in moduli spaces of sheaves on P . Theorem 7.6. [GLL22, Theorem 1.1] Let MP ,L (v) be a nonempty moduli space of sheaves on P with µL (v) > 0. (1) For any V ∈ MP ,L (v),   0 2 1 2 3

h (P , V) ≤ max r(V), P (c (V)) = c (V) + c (V) + 1 . 2 2 In particular, B k (v) = ∅ if k > max(r(v), P (c (v))). (2) The Brill-Noether locus B r (v) is nonempty. (3) If ch (v) = L, then all the nonempty Brill-Noether loci are irreducible of the expected dimension. (4) If µL (v) > 2 and is not an integer and ∆(v) ≫ 0, then B r (v) is reducible and contains components of both the expected and larger than expected dimension. The irreducibility in part (3) of the theorem also holds for more general surfaces. Let (X, H)

be a polarized surface. Let v = (r, C, d) be a Chern character on X with C · H > 0. Assume that there does not exist a class D on X with 0 < D · H < C · H. On P , the class of a line L satisfies this property. More generally, if X has Picard rank 1, then the ample generator of the Picard group has this property. In this case, the Brill-Noether loci are better behaved. The linear system |C| cannot have any reducible or nonreduced elements (it can however be empty if C is not effective). Given a degree κ, let

γκ := max{h (B, L)|B ∈ |C|, deg(L) = κ}. We have the easy bound γκ ≤ κ + 1, hence γκ = 0 when κ < 0. Combining the Riemann-Roch Theorem with Clifford’s Theorem, one can obtain better bounds on γκ .

Proposition 7.7. Let (X, H) be a polarized surface. Let v = (r, C, d) be a Chern character on X with C · H > 0. Assume that there does not exists a divisor class D with 0 < D · H < C · H. Set κ = d − C . Then h (X, V) ≤ r + γκ . In particular, if κ < 0, then h (V) ≤ r for every V ∈ MX,H (v) and BX,H k (v) = ∅ for k > r. r Conversely, if C is effective and ∆(v) ≫ 0 (equivalently, d ≪ 0), then BX,H (v) is nonempty.

Proof. Let V ∈ MX,H (v) be a stable sheaf such that h (X, V) ≥ r. Choosing r linearly independent ⊕r sections induces a map ϕ : OX → V. We claim that ϕ is an injective sheaf map. Suppose the image of ϕ is a sheaf W of rank s < r. Since h (X, W) ≥ r, we conclude that µH (W) > 0. Since C is the minimal class with 0 < C · H, we have that c (W) · H ≥ C · H. Since s < r, this contradicts the stability of V. We conclude that the image of ϕ must have rank r, hence its kernel is zero. So

we have a short exact sequence ⊕r

(4) 0 → OX → V → OC (D) → 0,

where OC (D) is a pure rank one sheaf on a curve of class C since the quotient of a torsion-free sheaf by a locally free sheaf cannot have torsion supported in dimension zero. [HL10, Example 1.1.16]. By the long exact sequence on cohomology we conclude that h (X, V) ≤ r + h (X, OC (D)) ≤ r + γκ . In particular, if κ < 0, then h (X, V) ≤ r. If C is effective and ∆ ≫ 0 (or equivalently, d ≪ 0), then there exists nonsplit extensions of the form (4). It suffices to take ∆ large enough so that χ(OC (D), OX ) < 0. Note that V is torsion-free

(see [GLL22, Lemma 4.5]), stable and h (X, V) ≥ r. Hence, BX,H r (v) is nonempty. □ Remark 7.8. When c (V) is not minimal, one can use the Grauert-Mülich Theorem and restriction to curves to give a bound on maxV∈MX,H (v)s (h (X, V)) (see [HL10, §3.3]). Theorem 7.9. Let (X, H) be a polarized surface such that KX · H ≤ 0. Let v = (r, C, d) be a Chern character on X with C ·H > 0 and ∆(v) ≫ 0. Assume that h (X, L) > 0 and H i (X, L) = 0 for i ≥ 1 for every L with c (L) = C. Further assume that there does not exist a divisor class D

k with 0 < D · H < C · H. Then BX,H (v) is empty if k > r and nonempty, irreducible and of the expected dimension if k ≤ r. k Proof. We prove the theorem by induction on the rank. We already know that BX,H (v) = ∅ if k > r by Proposition 7.7. It is also easy to see this when r = 1. An element of MX,H (v) is of the form IZ (L), where L is a line bundle with class C. Since ∆(v) ≫ 0, we have |Z| ≫ 0. We only need that |Z| > C 2 . By Bézout’s Theorem, if there exists two independent curves C , C in | L |

containing Z, then they must have a common component. This is a contradiction since there are no reducible or nonreduced curves in the linear system | L | by the minimality of C. On the other hand, if k = 1, then Z is a zero-dimensional scheme contained in an element of some linear system | L |. Since every curve in | L | is reduced and irreducible and X is smooth, the Hilbert scheme on the curve is irreducible of dimension |Z|. We conclude that BX,H (v) is irreducible of dimension

|Z| + h (X, C) + dim(Pic (X)) − 1 = dim(MX,H (v) − 1 + χ(v) as expected.

Suppose that the theorem holds for rank less than r. By Proposition 7.7, we may assume that k ⊕k k ≤ r. For V ∈ BX,H (v), consider the evaluation map ϕ : OX → V. Then the saturation W of the image of ϕ has rank s ≤ k. If s < k, then W is a subsheaf of V with µH (W) > 0. Since C · H is the minimal possible positive value and r(W) < r, we get a contradiction to the stability of V.

We conclude that the evaluation map is injective. Hence, we obtain an exact sequence

0 → OX → V → V ′ → 0.

As above, V ′ cannot have torsion supported in dimension zero as the quotient of a torsion-free sheaf by a locally free one. In fact, when k < r, V ′ is a µH -stable sheaf. Indeed, if V ′ had either one dimensional torsion or a destabilizing subsheaf, then since the H-degree of V ′ is the minimal, taking the quotient by the torsion or destabilizing subsheaf would yield a torsion free sheaf Q with µH (Q) ≤ 0. Since Q would also be a quotient of V, this would contradict the stability of V. So

we conclude that when k < r any V ∈ BX,H k (v) is a nonsplit extension of a µH -stable V ′ of Chern ⊕k character v − kv(OX ) by OX .

Conversely, consider nonsplit extensions of the form

0 → OX → V → V ′ → 0.

We have hom(V ′ , OX ) = 0 by stability. We split the proof into two cases, KX · H < 0 and

KX · H = 0.

Suppose first that KX · H < 0. If k < r − 1, then ext (V ′ , OX ) = hom(OX , V ′ (KX )) = 0. The latter follows from the fact that µH (V ′ (KX )) < 0 since r(V ′ ) ≥ 2 and c (V ′ ) has minimal positive possible H-degree. If k = r − 1 and KX · H < 0, then ext (V ′ , OX ) = hom(OX , V ′ (KX )) = 0 unless V ′ (KX ) = OX . The latter cannot happen since ∆(v) ≫ 0. We conclude that ext (V ′ , OX ⊕k ) is ′ ′ ′ constant for every V ∈ MX,H (v ) of dimension −kχ(V , OX ). Since ∆(v) ≫ 0, we must also have

∆(v′ ) ≫ 0. By O’Grady’s Theorem, the moduli space MX,H (v′ ) is irreducible of the expected k dimension. We conclude that BX,H (v) is the image of a projective bundle over an irreducible k space. Hence, BX,H (v) is irreducible and a dimension count shows that k dim(BX,H (v)) = dim(MX,H (v′ )) − kχ(V ′ , OX ) − k 2 . We have v′ = (r − k, c, d). Since ∆(v), ∆(v′ ) ≫ 0, we have that dim(MX,H (v)) = 2r ∆(v) − (r − 1)χ(OX ) and dim(MX,H (v′ )) = 2r ∆(v′ ) − ((r − k)2 − 1)χ(OX )

by O’Grady’s Theorem. Hence, by Riemann-Roch k k dim(BX,H (v)) = C 2 + (k − 2d)r + (kr − r + 1)χ(OX ) − C · KX − k 2 . k

On the other hand, the expected dimension of BX,H (v) is given by

dim(MX,H (v)) − k(k − χ(v)) = 2r ∆(v) − (r − 1)χ(OX ) − k(k − χ(v)) which by Riemann-Roch is equal to k C 2 + (k − 2d)r + (kr − r + 1)χ(OX ) − C · KX − k 2 . k Hence, BX,H (v) has the expected dimension. When k = r, then we get an extension of the form (4) and the irreducibility follows by a similar argument using the irreducibility of the compactified Jacobian.

Finally, if KX ·H = 0, then hom(OX , V ′ (KX )) = s along the Brill-Noether locus BX,H s (v′ ) which by induction on the rank has codimension s(s − χ(v′ ⊗ KX )). Since the latter is very large when ∆ ≫ 0, these loci do not contribute new components. We conclude that the same estimate holds k in this case as well. We conclude that the Brill-Noether locus BX,H (v) is irreducible, nonempty and of the expected dimension if k ≤ r and ∆(v) ≫ 0. □

Remark 7.10. Under further assumptions on X, one can give more explicit bounds on ∆ in Theo- rem 7.9. For example, in addition to P [GLL22], the Brill-Noether loci where c (v) is a minimal effective class have been studied for K surfaces by Leyenson [Ley06, Ley12] and Bayer [Bay18] and for abelian surfaces by Bayer and Li [BL17] (see also [CN23]).

References

[Ant18] V. Antonelli, Characterization of Ulrich bundles on Hirzebruch surfaces, Rev. Mat. Complut. 3 no. 1 (2021), 43–74. [Ati57] M.F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc., 7 (1957), 414–452. [AGO17] M. Aprodu, G. Farkas and A. Ortega, Minimal resolutions, Chow forms and Ulrich bundles on K surfaces, J. Reine Angew. Math. 7 (2017), 225–249. [AB13] D. Arcara, A. Bertram. Bridgeland-stable moduli spaces for K-trivial surfaces, with an appendix by

Max Lieblich, J. Eur. Math. Soc., 1 (2013), 1–38. [ABCH13] D. Arcara, A. Bertram, I. Coskun and J. Huizenga, The birational geometry of the Hilbert Scheme of Points on the plane and Bridgeland stability conditions, Adv. Math., 2 (2013), 580–626. [BaHU91] J. Herzog, B. Ulrich and J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 7 (1991), no. 2-3, 187–202. [Bay18] A. Bayer, Wall-crossing implies Brill-Noether: applications of stability conditi