THE NEF CONE OF THE MODULI SPACE OF SHEAVES AND STRONG BOGOMOLOV INEQUALITIES

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1 THE NEF CONE OF THE MODULI SPACE OF SHEAVES AND STRONG BOGOMOLOV INEQUALITIES IZZET COSKUN AND JACK HUIZENGA Abstact. Let (X, H) be a polaized, smooth, complex pojective suface, and let v be a Chen chaacte on X with positive ank and sufficiently lage disciminant. In this pape, we compute the Gieseke wall fo v in a slice of the stability manifold of X. We constuct explicit cuves paameteizing non-isomophic Gieseke stable sheaves of chaacte v that become S-equivalent along the wall. As a coollay, we conclude that if thee ae no stictly semistable sheaves of chaacte v, the Baye-Macì diviso associated to the wall is a bounday nef diviso on the moduli space of sheaves M H (v). We ecove pevious esults fo P and K3 sufaces, and illustate applications to highe Picad ank sufaces with an example on P 1 P 1. Contents 1. Intoduction 1. Peliminaies 3 3. The destabilizing sequence and stong Bogomolov inequalities 7 4. Bounding the Gieseke wall Nesting walls 1 6. Othogonal cuves Examples 16 Refeences 0 1. Intoduction Let (X, H) be a polaized, smooth, complex pojective suface. Let D be a Q-diviso on X and let v K num (X) be the class of a stable sheaf with positive ank and sufficiently lage disciminant. The divisos H and D detemine a half-plane in the Bidgeland stability manifold Stab(X) called the (H, D)-slice (see.3 fo the pecise definition). Let M H,D (v) denote the moduli space paameteizing (H, D)-twisted Gieseke semistable sheaves with Chen chaacte v. The lage volume limit of the Bidgeland moduli spaces in the (H, D)-slice is M H,D (v). Let the Gieseke wall W be the lagest Bidgeland wall in the slice along which an (H, D)-semistable sheaf with Chen chaacte v is destabilized. In this pape, we compute the Gieseke wall W fo v in the (H, D)-slice. We also constuct an explicit cuve paameteizing non-isomophic (H, D)-twisted Gieseke stable sheaves that become S-equivalent fo the Bidgeland stability conditions along W. As a coollay, we conclude that if Date: Mach 9, Mathematics Subject Classification. Pimay: 14J60. Seconday: 14E30, 14J9, 14C05. Key wods and phases. Moduli spaces of sheaves, ample cone, Bidgeland stability. Duing the pepaation of this aticle the fist autho was patially suppoted by the NSF CAREER gant DMS and NSF gant DMS , and the second autho was patially suppoted by a National Science Foundation Mathematical Sciences Postdoctoal Reseach Fellowship. 1

2 I. COSKUN AND J. HUIZENGA M H,D (v) has no stictly semistable sheaves, then the Baye-Macì diviso constucted in [BM14a, Lemma 3.3] is a nef diviso which lies in the bounday of Nef(M H,D (v)). When Pic(X) = Z, we show that the poblem of computing Gieseke walls in (H, D)-slices fo lage disciminants is equivalent to the poblem of classifying stable Chen chaactes. Ou computations ecove pevious esults fo P [CH15] and K3 sufaces [BM14a, BM14b]. We also exploe new applications in the setting of P 1 P 1, sufaces in P 3 and double coves of P. Let Y be a pojective vaiety. The ample cone Amp(Y ) N 1 (Y ) is the open convex cone in the Néon-Sevei space spanned by the classes of ample divisos. It encodes embeddings of Y in pojective space and is among the most impotant invaiants of Y. The closue of Amp(Y ) is the nef cone Nef(Y ) N 1 (Y ) spanned by the classes of divisos that have nonnegative intesection with evey integal cuve on Y. By definition, Nef(Y ) is dual to the Moi cone of cuves (see [La04]). Computing Nef(Y ) equies finding nef divisos to geneate a subcone of Nef(Y ) and dually finding integal cuves on Y to bound Nef(Y ) fom above. In this pape, we cay out this stategy fo Nef(M H,D (v)) when M H,D (v) contains only stable sheaves. We now explain ou stategy in geate detail. Given a Chen chaacte v, we define an extemal Chen chaacte w. Intuitively, w is chosen to make the numeical wall W (w, v) that it detemines as lage as possible, subject to natual estictions which will ensue that the wall is an actual wall whee a semistable sheaf is destabilized. See Definition 3.1 fo the pecise conditions defining w. If sufficiently stong Bogomolov-type esults ae known about the Chen chaactes of stable sheaves on X, then w can be computed explicitly. Theoem 1.1. Assume that the disciminant Δ H,D (v) 0. Then the Gieseke wall fo v in the (H, D)-slice is given by the wall W (w, v). Thoughout the pape we will conside (v), c 1 (v), X, H, and D as fixed, and Δ H,D (v) as vaiable. Thus, we wite Δ H,D (v) 0 to mean that Δ H,D (v) > C fo some constant C depending on (v), c 1 (v), X, H, and D. A numeical check, using inequalities discoveed in [ABCH13], [CH15] and futhe elaboated in [Bo15], confims that the Gieseke wall is contained in W (w, v). Convesely, we need to constuct a family of non-isomophic Gieseke stable sheaves that become S-equivalent fo the Bidgeland stability conditions on W (w, v). We constuct the necessay sheaves inductively. The key is to show that the Gieseke walls coesponding to w and the quotient Chen chaacte u = v w ae both nested inside W (w, v). One can then inductively constuct the equied cuves (see 6). Given two Chen chaactes z and v, define the incidence vaiety Z(z) := {(E, F ) M H,D (z) M H,D (v) Hom(E, F ) 0}. In geneal, descibing the geometic popeties of Z(z), even detemining when they ae nonempty, can be vey challenging. Ou esults in 6 imply that Z(w) is nonempty fo the extemal Chen chaacte w povided the disciminant of v is sufficiently lage. Applications to biational geomety. Fo the est of the intoduction, we additionally assume that v is a Chen chaacte such that M H,D (v) contains only stable sheaves. In this case, Baye and Macì [BM14a] associate a nef diviso to the Gieseke wall. Coollay 1.. If M H,D (v) contains only stable sheaves and Δ H,D (v) 0, then the Baye-Macì diviso associated to W (w, v) is a nef diviso lying in the bounday of Nef(M H,D (v)). Computing the Gieseke wall is closely elated to Bogomolov-type inequalities fo (H, D)- semistable sheaves on X. The coespondence is tightest when Pic(X) = ZH with H effective, so we focus on this case. Then the deteminantal line bundles on M H (v) span a -dimensional subspace of N 1 (M H (v)). The intesection of Nef(M H (v)) with this deteminantal subspace is a cone spanned by two classes. One of these classes L 1 coesponds to the Donaldson-Uhlenbeck-Yau

3 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 3 compactification by slope-semistable sheaves. When Δ H,D (v) 0, then thee ae singula sheaves in M H,D (v) and the map to the twisted Donaldson-Uhlenbeck-Yau space is not an isomophism ([HL10],[GRT15]). Let L span the othe extemal ay. Given a ank and slope μ = c 1 /, let δ(, μ) denote the minimal disciminant of a semistable sheaf of slope μ and ank at most. Then the inequality Δ δ(, μ) holds fo any semistable sheaf with invaiants (, μ, Δ). This efines the odinay Bogomolov inequality Δ 0. Coollay 1.3. Assume that Pic(X) = ZH with H effective. The computation of L fo all chaactes v with Δ(v) 0 is equivalent to the computation of the function δ(, μ) fo all > 0, μ Q with μ Z. In the highe Picad ank case, global infomation about the nef cone can fequently be obtained by vaying the twisting diviso D. If v is a Chen chaacte such that H-Gieseke semistability and μ H -slope stability coincide, then both of these notions ae also equivalent to (H, D)-Gieseke stability fo any choice of twisting diviso D. Vaying the twisting diviso, the vaious Bidgeland (H, D)-slices coespond to ays in N 1 (M H (v)) by the Baye-Macì constuction. The coesponding extemal nef divisos vay as well. This method was used in [Bo15] to compute the nef cone of the Hilbet scheme of points on a del Pezzo suface of degee 1. In this pape we will illustate this method by studying a moduli space of sheaves on P 1 P 1 in detail. Bidgeland stability has been successfully used by many authos to study the biational geomety of moduli spaces of sheaves on sufaces. We efe the eade to [ABCH13], [BMW14], [CH15], [CHW14], [Oh10], [LZ13] fo P, [BC13] fo Hizebuch and del Pezzo sufaces, [BM14a], [BM14b], [MYY1], [MYY14] fo K3 sufaces, [Nu14] fo Eniques sufaces, and [MM13], [YY14], [Y1] fo abelian sufaces. The ample cones of Hilbet schemes of points on sufaces wee studied in [ Bo15]. This pape genealizes and unifies the techniques in these papes fo computing the ample cones to moduli spaces of sheaves on abitay sufaces. In a paallel development, the papes [ BM15] and [Y15] study Thaddeus flips esulting fom change of polaization in tems of Bidgeland stability. Oganization of the pape. In, we intoduce the necessay backgound on M H,D (v) and Bidgeland stability. In 3, we intoduce the extemal Chen chaacte w associated to v and state ou main esult. In 4, we show that the Gieseke wall is no lage than W (w, v) if the disciminant of v is sufficiently lage. In 5, we show that the Gieseke walls coesponding to w and u = v w ae nested in W (w, v). Using the nesting esult, in 6, we constuct cuves of Gieseke stable sheaves which become S-equivalent along W (w, v). In 7, we study the nef cone of moduli spaces of ank sheaves seveal families of sufaces. Acknowledgements. We would like to thank Aend Baye, Aaon Betam, Emanuele Macì, Benjamin Schmidt and Matthew Woolf fo many enlightening discussions on Bidgeland stability. Pat of this wok was caied out duing the Algebaic Geomety Summe Reseach Institute in Utah. We thank the Ameican Mathematical Society, the Univesity of Utah and the oganizes fo poviding us with ideal woking conditions.. Peliminaies In this section, we eview basic facts concening moduli spaces of Gieseke semistable sheaves and Bidgeland stability conditions.

4 4 I. COSKUN AND J. HUIZENGA.1. Basic definitions. We efe the eade to [HL10] and [MW97] fo an in depth teatment of (twisted) Gieseke semistability. Let X be a smooth pojective suface ove C. A sheaf on X will always mean a coheent sheaf of pue dimension. Fix an ample diviso H Pic(X). Fo any Q-diviso D on X, define the twisted Chen chaacte ch D = e D ch, with expansion ch D 0 = ch 0, ch D 1 = ch 1 D ch 0, ch D = ch D ch 1 + D ch 0. We will find it convenient to wok with coodinates povided by the slope and the disciminant. Fo a sheaf E with k(e) > 0, define the (H, D)-slope μ H,D and (H, D)-disciminant Δ H,D by μ H,D = H chd 1 H ch D 0, Δ H,D = 1 μ H,D chd H ch D 0 A sheaf E of positive ank is μ H,D -(semi)stable if fo evey nonzeo pope subsheaf F E, μ H,D (F ) < μ H,D (E). ( ) Note that μ H,D only diffes fom μ H := μ H,0 by a constant, so μ H,D -(semi)stability and μ H - (semi)stability coincide. A slightly modified vesion of these invaiants will often be moe useful. We define ch D = ch D+ 1 K X, and define modified slopes μ H,D and disciminants Δ H,D using this modified Chen chaacte; thus μ H,D = μ H,D+ 1 K X and Δ H,D = Δ H,D+ 1 K X. The additional twist by 1 K X (especially in the disciminant) geatly simplifies computations with twisted Gieseke semistability, which we now discuss. The educed twisted Hilbet polynomial of a positive ank sheaf E is defined by p E H,D(m) = χ(e O X(mH D)), k(e) whee the Eule chaacteistic is computed fomally. (semi)stable if fo evey nonzeo pope subsheaf F E, p F H,D(m) < p E H,D(m) ( ). A sheaf E is (H, D)-twisted Gieseke fo all m 0. When D = 0, we ecove usual H-Gieseke semistability. When efeing to H- Gieseke semistability, we will omit D fom ou notation. A simple Riemann-Roch computation shows that a sheaf E is (H, D)-twisted Gieseke (semi)stable if and only if (1) E is μ H,D -semistable, and () if F E with μ H,D (F ) = μ H,D (E), then Δ H,D (F ) > Δ H,D (E). ( ) Note that this equivalence is not typically coect when using the odinay disciminant Δ H,D instead of Δ H,D unless K X is paallel to H. Remak.1. Note that when (v), c 1 (v), X, H, and D ae fixed, we have Δ H,D (v) 0 if and only if Δ H,D (v) 0. Thus the esults in the intoduction could also be stated in tems of the disciminant Δ H,D (v) instead of Δ H,D (v).

5 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 5.. Popeties of the moduli space. Recall that two semistable sheaves ae S-equivalent with espect to a notion of stability if they have the same Jodan-Hölde factos fo that stability condition. Fix the Chen chaacte v of an (H, D)-twisted Gieseke semistable sheaf. Matsuki and Wentwoth [MW97] pove that thee ae pojective moduli spaces M H,D (v) paameteizing S-equivalence classes of (H, D)-twisted Gieseke semistable sheaves on X with invaiants v. Recall the following fundamental theoems of O Gady fo the odinay Gieseke moduli space. Theoem. ([O G96, Theoems B, D], [HL10, Theoems 5..5, 9.3.3, 9.4.3]). Let (X, H) be a smooth polaized suface, and let v K(X) with (v) > 0. If Δ H (v) 0 (depending on (v), X, and H), then the moduli space M H (v) is nomal, geneically smooth, ieducible, and nonempty of the expected dimension. Futhemoe, the slope stable sheaves ae dense in M H (v). As we emaked ealie, the two slopes μ H and μ H,D diffe only by a constant. Consequently, μ H - and μ H,D -stability coincide. Since μ H,D -stability is an open condition, Theoem. povides a nonempty Zaiski open set in M H,D (v) paameteizing μ H,D -stable sheaves..3. Bidgeland stability. In this subsection, we eview key facts concening Bidgeland stability on sufaces. We efe the eade to [AB13], [ABCH13], [BM14a], [Bo15], [B07], [B08], [CH14] and [CH15] fo moe details. Let D b (X) denote the bounded deived categoy of coheent sheaves on X. A Bidgeland stability condition on D b (X) is a pai σ = (Z, A), whee A is the heat of a bounded t-stuctue on D b (X) and Z : K 0 (X) C is a goup homomophism mapping A to the extended uppe half plane and satisfying the Hade-Naasimhan and Suppot Popeties [B07], [BM14a]. The set Stab(X) of Bidgeland stability conditions on X is a complex manifold [B07]. Bidgeland [B08] and Acaa, Betam [AB13] constucted Bidgeland stability conditions on sufaces. Given an ample diviso H, an abitay R-diviso D and β R, define two subcategoies of the categoy of coheent sheaves Coh(X) by T β = {E Coh(X) : μ H,D (G) > β fo evey quotient G of E} F β = {E Coh(X) : μ H,D (F ) β fo evey subsheaf F of E}. The pai (T β, F β ) foms a tosion pai in Coh(X). Tilting Coh(X) with espect to this tosion pai yields the heat A β of a new bounded t-stuctue on D b (X) defined by A β = {E D b (X) : H 1 (E ) F β, H 0 (E ) T β, H i (E ) = 0, i 1, 0}. Let α be a positive eal numbe. Define the cental chage Z β,α = ch D+βH + α H ch D+βH 0 + ihch D+βH 1. Then the pai σ β,α = (Z β,α, A β ) is a Bidgeland stability condition fo α, β R, α > 0 [AB13]. These stability conditions span a half-plane in Stab(X) which we call the (H, D)-slice. The σ β,α - slope of an object with invaiants > 0, μ H,D and Δ H,D is given by ν σβ,α = RZ β,α = (μ H,D β) α Δ H,D. IZ β,α μ H,D β.3.1. Bidgeland Walls. Fix an invaiant v K num (X). Assume that w K num (X) is an invaiant that does not have the same σ β,α -slope as v eveywhee in the (H, D)-slice. Then the numeical wall W (w, v) is the set of points (β, α) such that v and w have the same σ β,α -slope. A numeical wall is an actual wall if thee exists a point (β, α) W (w, v) and an exact sequence 0 F E G 0

6 6 I. COSKUN AND J. HUIZENGA in A β with ch(f ) = w, ch(e) = v such that E, F, G ae σ β,α -semistable. We will efe to the sequence as the destabilizing sequence. We will fequently use the following facts about the Bidgeland walls [Bo15], [CH14], [Ma14]: (1) The numeical walls W (w, v) in the (H, D)-slice ae disjoint. Let v and w have positive ank. If μ H,D (v) = μ H,D (w), then W (w, v) is the vetical wall β = μ H,D (v). If μ H,D (v) = μ H,D (w), then W (w, v) is the semicicula wall with cente (s, 0) and adius ρ, whee s = 1 (μ H,D(v) + μ H,D (w)) Δ H,D(v) Δ H,D (w) μ H,D (v) μ H,D (w), ρ = (s μ H,D (v)) Δ H,D (v). If ρ is negative, then the wall is empty. () Let W 1, W be two numeical walls to the left of β = μ H,D (v) with centes (s 1, 0), (s, 0). Then W 1 is nested inside W if and only if s 1 > s. (3) Let W (w, v) be an actual wall. If 0 F E G 0 is a destabilizing sequence at a point (β, α) W (w, v), then it is a destabilizing sequence fo evey point of W (w, v). Define the Gieseke wall fo v to be the lagest actual semicicula wall to the left of β = μ H,D (v) whee a Gieseke semistable sheaf is destabilized. In this pape, we will be concened with computing the Gieseke wall..3.. Lage volume limit. Let v K num (X) have positive ank, and conside stability conditions σ β,α with β < μ H,D (v) and α 0. Maciocia [Ma14] shows that any σ β,α -semistable object of chaacte v is a μ H -semistable (equivalently, μ H,D -semistable) sheaf. Obseve that if a μ H - semistable sheaf E is σ β,α -(semi)stable fo any β, α with β < μ H,D (E), then it is also (H, D)- Gieseke (semi)stable. Indeed, if F E is an (H, D)-Gieseke stable subsheaf with μ H,D (F ) = μ H,D (E), then ν σβ,α (F ) < ν σβ,α (E) ( ) if and only if Δ H,D (F ) > Δ H,D (E) ( ) by ou explicit fomula fo the slope ν σβ,α. Thus fo α 0 the moduli space M σβ,α (v) coincides with M H,D (v). If we only know that σ β,α lies above the Gieseke wall, it a pioi may be the case that M σβ,α (v) is lage than M H,D (v). Howeve, it is still tue that evey (H, D)-Gieseke semistable sheaf is σ β,α -semistable. This is all we will need to apply the Positivity Lemma The Positivity Lemma. Ou main tool fo constucting nef divisos on the moduli space is the Positivity Lemma of Baye and Macì. Let σ = (Z, A) be a Bidgeland stability condition on X, v K num (X) and S a pope algebaic space of finite type ove C. Let E D b (X S) be a flat family of σ-semistable objects. Denote the two pojections on X S by p and q, espectively. Then Baye and Macì define a numeical class on D σ,e N 1 (S) by setting ( Z(p (E q ) O C )) D σ,e C = I Z(v) fo evey integal cuve C on S. Theoem.3 (Positivity Lemma [BM14a, Lemma 3.3]). The diviso D σ,e is nef on S. A pojective, integal cuve C S satisfies D σ,e C = 0 if and only if objects paameteized by C ae geneically S-equivalent with espect to σ.

7 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 7 Fom now on we assume that M H,D (v) contains only stable sheaves. Then M H,D (v) admits a quasiunivesal family E (see [Mu84, Theoem A.5] o [HL10] in the case of odinay Gieseke stability). Fo (β, α) in the egion bounded by the Gieseke wall and the vetical wall, E is a family of σ β,α -semistable objects. Hence, the Positivity Lemma povides a nef diviso on M H,D (v) associated to the Gieseke wall. By [BM14a, 4], the nef diviso may be identified with the deteminantal class coesponding to the unique vecto α satisfying I(Z( )) = χ(α ), whee χ denotes the Eule paiing. Thus, the constuction of Baye and Macì can only give deteminantal classes. When the iegulaity q(x) = 0, it is natual to guess that the deteminant line bundles span NS(M(v)) if Δ(v) 0. This is known when k(v) = by esults of Jun Li [Li96], but open in geneal. 3. The destabilizing sequence and stong Bogomolov inequalities Fix divisos (H, D) giving a slice of Stab(X). Let e > 0 be a geneato of the subgoup H Pic(X) Z. We define the educed slope of a class v of positive ank by μ H (v) = H e μ H(v) = c 1(v) H (v)e. The set of educed slopes of stable vecto bundles on X of ank at most is pecisely the set of ational numbes with denominato at most. The educed slope detemines and is detemined by the odinay slope μ H o any of the twisted slopes μ H,D, μ H,D Extemal chaactes. Conside a vaiable Chen chaacte v whee (v) > 0 and c 1 (v) ae fixed but Δ H,D (v) is vaiable, subject to the estiction that v is (H, D)-stable. By Theoem., v will be (H, D)-stable so long as Δ H,D (v) is sufficiently lage and v is integal. In this section, we descibe the Gieseke wall fo M H,D (v) in the (H, D)-slice unde the assumption that Δ H,D (v) is sufficiently lage. Fist, we descibe the numeical invaiants of the destabilizing subobject. Definition 3.1. An extemal chaacte w fo v is any Chen chaacte satisfying the following defining popeties. (E1) We have 0 < (w) (v), and if (w) = (v), then c 1 (v) c 1 (w) is effective. (E) We have μ H (w) < μ H (v), and μ H (w) is as close to μ H (v) as possible subject to (E1). (E3) The moduli space M H,D (w) is nonempty. (E4) The disciminant Δ H,D (w) is as small as possible, subject to (E1)-(E3). (E5) The ank (w) is as lage as possible, subject to (E1)-(E4). Note that conditions (E1) and (E) uniquely detemine μ H (w). Popety (E4) uniquely detemines Δ H,D (w) (note that the Bogomolov inequality and the bound on the ank guaantee that a minimum actually exists), and popety (E5) uniquely detemines (w). Futhemoe, the disciminant Δ H,D (v) plays no ole in the detemination of w. Thus the tiple ((w), μ H (w), Δ H,D (w)) is uniquely detemined by (v) and c 1 (v); on the othe hand, thee may be seveal possible choices fo c 1 (w). The equiement that Δ H,D (w) is as small as possible may estict which fist Chen classes c 1 (w) ae pemissible. Ou main esult in this pape is the following.

8 8 I. COSKUN AND J. HUIZENGA Theoem 3.. If Δ H,D (v) 0, then the Gieseke wall fo M H,D (v) in the (H, D)-slice is W (w, v). Thee ae cuves in M H,D (v) paameteizing sheaves which become S-equivalent along this wall. Thee ae two main steps to the poof of Theoem 3.. Fist, we show that no actual wall fo M H,D (v) is lage than the wall W (w, v) given by w. This step is not too difficult; it will follow fom a bound on highe ank walls and an asymptotic study of walls. Next, we pove this wall is the Gieseke wall and that the coesponding nef diviso lies on the bounday of the nef cone. Put u = v w. We show that thee ae sheaves F M H,D (w) and Q M H,D (u) and cuves in Ext 1 (Q, F ) such that the coesponding family of sheaves E fitting as extensions 0 F E Q 0 ae geneically (H, D)-Gieseke stable and vay in moduli. Then the wall W (w, v) is the Gieseke wall since such sheaves E ae destabilized along it. Futhemoe, the coesponding cuves in M H,D (v) ae othogonal to the nef diviso given by the Gieseke wall, so the diviso is on the bounday of the nef cone. This second pat of the poof is faily delicate, and pimaily depends on computing the Gieseke wall fo M H,D (u) by induction on the ank. Remak 3.3. Note that if w is any chaacte satisfying popeties (E1)-(E4) in Definition 3.1 (but not necessaily popety (E5)), then the walls W (w, v) and W (w, v) will coincide. Popety (E5) has been imposed to make the constuction of othogonal cuves to the nef diviso as easy as possible. 3.. The quotient chaacte. The definition of the extemal chaacte w ensues that the moduli space M H,D (w) is nonempty. In the pevious discussion we needed to know that the moduli space M H,D (u) coesponding to the quotient chaacte u = v w is also nonempty. We now addess this point, and study u moe closely. Thee ae two cases to conside, based on whethe (u) > 0 o (u) = 0. Fist assume (u) > 0. Note that (u) and μ H (u) depend only on (v) and c 1 (v). The class c 1 (u) depends on the choice of c 1 (w). The elationship between Δ H,D (u) and the othe invaiants is encoded in the identity (v)δ H,D (v) = (w)δ H,D (w) + (u)δ H,D (u) (w)(u) (μ (v) H,D (w) μ H,D (u)). In paticula, as Δ H,D (v) we find Δ H,D (u). Theefoe, if Δ H,D (v) 0, Theoem. applies to the moduli space M H (u). Fo instance, thee ae μ H,D -stable sheaves of chaacte u. On the othe hand, if (u) = 0, then by (E1) we find that c 1 (u) is effective. Let C be an effective cuve epesenting this class. By (E), the line bundle O X (C) has the smallest possible educed slope μ H (O X (C)) among all effective line bundles on X. Theefoe C is educed and ieducible, and the moduli space M H,D (u) contains sheaves which ae line bundles of the appopiate degee suppoted on C. In eithe case, M H,D (u) is nonempty and contains well-behaved points Backgound on Faey sequences. The aguments in this pape ely on undestanding the numbe theoy which detemines the slope μ H (w) of the exceptional chaacte w. Recall that the (unesticted) Faey sequence F n of ode n consists of the odeed list of educed factions with denominato at most n. Fo example, F 6 = {..., 16, 01, 16, 15, 14, 13, 5, 1, 35, 3, 34, 45, 56, 11, 76 },...

9 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 9 Thus the elements of F ae pecisely the educed slopes of stable vecto bundles on X of ank at most. Suppose that a b < c d ae Faey neighbos, i.e. that they ae adjacent tems in the Faey sequence F max{b,d}. Then bc ad = 1. The mediant of two (educed) ational numbes a b and c d is the ational numbe a + c b + d. If a b < c d ae Faey neighbos, then the mediant is aleady witten in lowest tems. Futhemoe, the mediant is the unique ational numbe in the inteval ( a b, c d ) with denominato at most b + d. That is, the thee tems a b, a + c b + d, c d ae adjacent in the Faey sequence F b+d. Remak 3.4. Hee we explain how to use the Faey sequence to compute μ H (w) moe explicitly. Let d = μ H (L) be the minimum educed slope of an effective line bundle L on X. If (v) = 1, then μ H (w) = μ H (v) d. If (v), let α be the numbe in F (v) immediately peceding μ H (v). If μ H (v) is not an intege, then the denominato of α is stictly less than (v). Theefoe { α if μh (v) / Z o d = 1 μ H (w) = μ H (v) 1 (v) 1 if μ H (v) Z and d > Bogomolov inequalities. The extemal chaacte w associated to v can be computed given the classification of stable Chen chaactes on X. Fo example, on P it is easy to compute w fom the Dézet-Le Potie classification (see [CH15] and [LP97]). In paticula, when X = P, Theoem 3. specializes to the main theoem of [CH15]. Convesely, suppose that Pic(X) = ZH with H effective. In this case we can expess Chen chaactes in tems of thei ank, slope μ = μ H, and disciminant Δ = Δ H,0. Fix a ank > 0 and slope μ Q with μ Z. Let δ(, μ) be the minimal disciminant of a stable bundle E such that μ(e) = μ and k(e). Then the inequality Δ δ(, μ) is valid fo any stable bundle with invaiants (, μ, Δ), and typically impoves the odinay Bogomolov inequality. Coollay 3.5. Suppose Pic(X) = ZH with H effective. Computing the Gieseke wall fo all v with sufficiently lage disciminant is equivalent to computing the function δ(, μ) fo all > 0 and μ Q with μ Z. Poof. If δ(, μ) is known, then it is staightfowad to detemine the chaacte w fom v using Remak 3.4. Convesely, suppose the computation of the Gieseke wall is known. Let > 0 and μ Q with c 1 := μ Z. Wite μ = b b s as a educed faction, so = ks fo some intege k > 0. Define s Q by equiing b s < b s to be Faey neighbos in the Faey sequence F. Then the mediant μ := b + b s + s has denominato := s + s >. Let v = (, μ, Δ ). Then the extemal Chen chaacte w to v has slope μ.

10 10 I. COSKUN AND J. HUIZENGA What is the ank of w? Since w has slope μ, we have (w) = ls fo some l > 0. By definition, s. If s =, then since denominatos of Faey neighbos ae copime we see that μ is an intege and δ(, μ) = 0. If instead s <, then (k + 1)s = + s > s + s =, and thus (w) ks =. It then follows that Δ(w) = δ(, μ), computing δ(, μ). Remak 3.6. Note that the chaacte v constucted in the poof of Coollay 3.5 has copime ank and fist Chen class. Thus the moduli space M H (v) caies a univesal family. We conclude Coollay 1.3 holds as well. 4. Bounding the Gieseke wall In this section we show that if Δ H,D (v) 0, then the Gieseke wall fo M H,D (v) in the (H, D)- slice is no lage than the wall W := W (w, v) defined by the extemal Chen chaacte w (see Definition 3.1). Suppose W is a semicicula wall in the (H, D)-slice lying left of the vetical wall such that W is at least as lage as W and some E M H,D (v) is destabilized along W. Let σ 0 be a stability condition on W, and let 0 F E Q 0 be an exact sequence of σ 0 -semistable objects of the same σ 0 -slope which defines the wall W. Let w = ch(f ); then W = W (w, v). We will show that if Δ H,D (v) 0, then μ H (w ) = μ H (w) and Δ H,D (w ) = Δ H,D (w). That is, the walls W and W actually coincide, and the Gieseke wall is no lage than W. A now-standad agument gives some initial estictions on F. Lemma 4.1. The object F is a nonzeo tosion-fee sheaf. We have μ H,D (w ) < μ H,D (v), and evey Hade-Naasimhan facto of F has (H, D)-slope at most μ H,D (v). Poof. Fix a categoy A β such that some point (β, α) is on the wall W. Taking cohomology sheaves, since H 1 (E) = 0 we find that F is a sheaf in T β ; it is nonzeo since the wall W is not the whole slice. Since K := H 1 (Q ) and E ae tosion fee, so is F. If F has an (H, D)-Gieseke stable subsheaf F 1 with μ H,D (F 1 ) > μ H,D (v), then F 1 is a subsheaf of K, which violates K F β since E T β. Finally, if μ H,D (w ) = μ H,D (v), then W contains the vetical wall, which again is a contadiction. Theefoe μ H,D (w ) < μ H,D (v). We next ecall a lemma which fist appeaed in [CH15] fo P and was late genealized in [Bo15]. Lemma 4. ([Bo15, Lemma 3.1]). With the notation and hypotheses of this section, if the map F E of sheaves is not injective, then the adius ρ W of the wall W satisfies ρ W (min{(w ) 1, (v)}) (w Δ H,D (v). ) The lemma allows us to show the map of sheaves F E is injective once Δ H,D (v) is sufficiently lage. This povides a estiction on the anks of subobjects. Poposition 4.3. If Δ H,D (v) 0, then the map F E of sheaves is injective. In paticula, 0 < (w ) (v). Futhemoe, in case (w ) = (v), the induced map on line bundles det F det E is an injection, and theefoe c 1 (v) c 1 (w ) is effective.

11 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 11 Poof. We compae the adius of W with the bound on ρ W in Lemma 4.. The cente (s W, 0) and adius ρ W of W satisfy s W = μ H,D(v) + μ H,D (w) ρ W = (μ H,D (v) s W ) Δ H,D (v). Δ H,D(v) Δ H,D (w) μ H,D (v) μ H,D (w) Theefoe ρ W gows quadatically as a function of Δ H,D(v). Let { (min{ 1, (v)}) } C = max : N >0. By Lemma 4., if the map F E is not injective, then ρ W is bounded by C Δ H,D (v). Since W is at least as lage as W, we conclude that if Δ H,D (v) is sufficiently lage, then F E is injective. Having esticted the ank (w ), we next tun to esticting the slope μ H (w ) and disciminant Δ H,D (w ). We begin with a simple obsevation. Lemma 4.4. Let (x W, 0) be the ight endpoint of the wall W, so that x W = s W + ρ W. Then x W is inceasing as a function of Δ H,D (v), and lim x W = μ H,D (w). Δ H,D Poof. The walls W = W (w, v) ae a family of numeical walls fo w, so they ae all nested. The fomula fo s W shows that the centes decease (and tend to ) as Δ H,D (v) inceases. Coespondingly, the walls become lage and x W inceases. As the walls become abitaily lage, they come abitaily close to the vetical wall β = μ H,D (w), and the limit follows. We now complete the poof of the main theoem in this section. Theoem 4.5. If Δ H,D (v) 0, then W = W. Thus the Gieseke wall fo M H,D (v) is no lage than W. Poof. Suppose Δ H,D (v) is lage enough that (1) Poposition 4.3 holds, and () x W is sufficiently close to μ H,D (w) that no numbe in the inteval (x W, μ H,D (w)) is the μ H,D -slope of a Chen chaacte of ank at most (v). Since W is at least as lage as W, the sheaf F lies in T xw. By Lemma 4.1, we have μ H,D (w ) (x W, μ H,D (v)). Moe pecisely, since (w ) (v) we actually have μ H,D (w ) [μ H,D (w), μ H,D (v)). Since we know that c 1 (v) c 1 (w ) is effective in case (w ) = (v), we conclude fom the definition of μ H,D (w) that μ H,D (w ) = μ H,D (w). The sheaf F is also μ H,D -semistable, fo if F has a quotient sheaf of smalle slope, then F is not in T xw by constuction. Since F is σ 0 -semistable, it is also (H, D)-Gieseke semistable by.3. The fomula fo the cente of a wall and the assumption that W is at least as lage as W implies Δ H,D (w ) Δ H,D (w). By the minimality of Δ H,D (w), we conclude Δ H,D (w ) = Δ H,D (w).

12 1 I. COSKUN AND J. HUIZENGA 5. Nesting walls Let w denote the extemal Chen chaacte fom Definition 3.1. In the next section we will pove that W = W (w, v) is actually the Gieseke wall fo M H,D (v) by poducing (H, D)-Gieseke stable sheaves which ae destabilized along W. The main ingedient in this constuction will be the inductive computation of the Gieseke wall fo the moduli space M H,D (u) coesponding to the quotients, which we addess hee. Recall that u = v w. Poposition 5.1. Assume Theoem 3. holds fo chaactes of (positive) ank less than (v). If Δ H,D (v) 0, then the Gieseke wall fo M H,D (u) in the (H, D)-slice is nested popely inside W. (If the ank (u) is zeo, it may happen that evey E M H,D (u) is semistable eveywhee in the (H, D)-slice. In this case we conside the Gieseke wall to be empty and the esult is vacuous.) The poof of Poposition 5.1 is diffeent based on whethe (u) > 0 o (u) = 0. We teat the moe inteesting positive ank case fist Positive ank quotients. Thoughout this subsection we assume (u) > 0, i.e. 0 < (w) < (v). In paticula, (v). We wite the inequality μ H (w) < μ H (v) < μ H (u) as a < a < a, whee the denominatos ae the anks of the coesponding chaactes. We begin with a couple useful lemmas. Wite the above factions in lowest tems as b s < b s < b s, with positive denominatos. Lemma 5.. The factions b s and b s ae Faey neighbos. Poof. This follows immediately fom Remak 3.4 since (v). Lemma 5.3. At least one of the factions a o a is aleady witten in lowest tems. Poof. Suppose not. Then s and s. The denominato s + s of the mediant satisfies b s < b + b s + s < b s s + s 1 ( + ). If s + s <, then condition (E) defining μ H (w) is violated. On the othe hand, if s + s =, then =, contay to ou assumption. We now elate an extemal chaacte fo u to w. Lemma 5.4. Let w be an extemal chaacte fo u. equality we have Δ H,D (w ) > Δ H,D (w). Then μ H (w ) μ H (w), and in case of Poof. Fist we show that μ H (w ) μ H (w). We wite μ H (w ) = a, and have a < a. It suffices to show that a / ( a, a ). We conside thee diffeent cases, depending on the elationship between a and a. Case 1: Suppose a ( a, a ). Since <, this contadicts the definition of μ H (w). and

13 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 13 Case : Suppose a ( a, a ). Let a = a a =. Then a ( a, a ), again contadicting the definition of μ H(w). Indeed, to pove this, we can view as a weighted mean in two ways: a a = a + a + = a + a +. Since a is close to a a than is and the weight a on in the second mean is smalle than the weight a on in the fist mean, it follows that a ( a, a ). Case 3: Suppose a = a. Since <, we find that the faction a is not aleady educed, and s and b s Thus theefoe a = b and = s by Lemma 5.3. We must have s + s, fo othewise the mediant of b a lies in (, a ) and has denominato less than, contadicting the definition of μ H(w). = s s. If this inequality is stict, then gives a contadiction. Suppose instead that = s and s + s =. Since b s and b s ae Faey neighbos by Lemma 5., thei mediant b + b s + s is witten in lowest tems and has denominato. Then b +b and b s ae consecutive tems in the Faey sequence of ode, so gcd(, s) = 1. Theefoe = s = 1, the slope μ H (v) = b is an intege, and μ H (u) = b + 1. Since μ H (w) b 1, thee is no effective line bundle on X of educed slope 1. This implies μ H (w ) b, a contadiction. This completes the poof that μ H (w ) μ H (w). Suppose μ H (w ) = μ H (w). Let w 0 be any chaacte satisfying conditions (E1)-(E4) of the definition of an extemal chaacte fo v, but such that the ank 0 of w 0 is as small as possible. We must have + 0 by condition (E5) in the definition of w, since othewise w + w 0 would satisfy (E1)-(E4) but have lage ank. Theefoe = 0. If the stict inequality < 0 holds, then by the definition of w 0 we have Δ H,D (w ) > Δ H,D (w). Instead assume = = 0. In this case we have Δ H,D(w ) Δ H,D (w), so assume Δ H,D (w ) = Δ H,D (w). Then w 0 and w have the same invaiants, except possibly thei fist Chen classes ae diffeent. But then w + w satisfies conditions (E1)-(E4) in the definition of w, since c 1 (v) (c 1 (w) + c 1 (w )) = c 1 (u) c 1 (w ) and c 1 (u) c 1 (w ) is effective. This contadicts the condition (E5) in the definition of w. Theefoe Δ H,D (w ) > Δ H,D (w) holds in this case as well. The lemma immediately allows us to teat the positive ank quotient case of Poposition 5.1. Poof of Poposition 5.1 when (u) > 0. By Lemma 5.4, we have μ H,D (w ) μ H,D (w), and in case of equality Δ H,D (w ) > Δ H,D (w). We must compae the walls W = W (w, v) = W (w, u) and W := W (w, u); note that both walls ae numeical walls fo u, so they ae disjoint, and it suffices to compae thei ight endpoints. Recall that Δ H,D (u) is an inceasing function of Δ H,D (v), and Δ H,D (u) as Δ H,D (v).

14 14 I. COSKUN AND J. HUIZENGA Fist suppose μ H,D (w ) < μ H,D (w). Then the ight endpoints x W and x W of W and W ae inceasing functions of Δ H,D (v). Futhemoe, as Δ H,D (v), we have x W μ H,D (w) and x W μ H,D (w ). Theefoe if Δ H,D (v) is sufficiently lage, x W < x W, and W is nested in W. Next suppose μ H,D (w ) = μ H,D (w) and Δ H,D (w ) > Δ H,D (w). Compaing the fomulas fo the centes of W (w, u) and W (w, u) immediately poves the esult; we don t even need to incease Δ H,D (v). 5.. Rank zeo quotients. In this case things ae consideably easie. Poof of Poposition 5.1 when (u) = 0. Recod the chaacte u in tems of its fist Chen class c 1 (u) and Eule chaacteistic χ(u), which depends on Δ H,D (v). By the constuction of w, c 1 (u) is effective, so the moduli spaces M H,D (u) ae all nonempty. Since tensoing by O X (H) gives an isomophism M H,D (c 1 (u), χ(u)) = M H,D (c 1 (u), χ(u) + H c 1 (u)), thee ae only finitely many isomophism types of spaces M H,D (u) as Δ H,D (v) vaies. Tensoing by O X (H) also peseves the adius of the Gieseke wall, assuming the Gieseke wall of eithe space is nonempty. Theefoe, thee is a univesal bound on the adii of the Gieseke walls of the spaces M H,D (u). Recall that the numeical walls fo u ae nested semicicles with a common cente that foliate the entie (H, D)-slice [CH14, Ma14]. Since W = W (w, v) = W (w, u) is also a numeical wall fo fo u, a numeical wall fo u is nested inside W if and only if its adius is smalle than the adius of W. Since W is abitaily lage fo Δ H,D (v) 0, this completes the poof. 6. Othogonal cuves In this section we pove that W is actually the Gieseke wall by poducing cuves of objects in M H,D (v) which ae destabilized along W. If M H,D (v) contains only stable sheaves, ou cuves will futhemoe be othogonal to the nef diviso given by W. We fist ecall some algebaic peliminaies Extensions. The basis fo ou constuction of stable sheaves is the following mild genealization of [BM14b, Lemma 6.9]. Recall that a simple object in an abelian categoy is an object with no pope subobjects, and a semisimple object is a (finite) diect sum of simple objects. In what follows we wite A = A n i i i with the A i simple and nonisomophic. Then evey subobject o quotient object of A is isomophic to an object A m i i fo some integes m i with 0 m i n i. In paticula, evey quotient of A is also a subobject of A. Lemma 6.1. Let A be an abelian categoy, and let A, B A with A semisimple and B simple. If E is any extension of the fom 0 A E B 0 with Hom(E, A) = 0, then any subobject of E is a subobject of A. Poof. Let S E be a subobject. Conside the composition φ : S E B. Since B is simple, φ is eithe sujective o zeo. If φ is zeo, then S is a subobject of A. Suppose instead that φ is sujective; in this case we will obtain a contadiction. Let C be the cokenel of the inclusion 0 S E C 0.

15 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 15 Then the composition A E C is sujective, so C is a quotient of A. Since A is semisimple, C is also isomophic to a subobject of A, but this contadicts Hom(E, A) = 0. The next lemma gives a citeion fo the vanishing Hom(E, A) = 0 needed to apply Lemma 6.1. Recall that by Schu s lemma, if A, B ae simple objects in a C-linea abelian categoy, then Hom(A, B) = 0 unless A, B ae isomophic, and Hom(A, A) = C. Lemma 6.. Let A be a C-linea abelian categoy, and let A, B A with A semisimple and B simple. Assume B is not a simple facto of A. Conside an extension 0 A E B 0 given by an extension class e Ext 1 (B, A) = i Ext1 (B, A i ) n i. Fo each i, wite e i,1,..., e i,ni fo the n i components of e unde this isomophism. Then Hom(E, A) = 0 if and only if e i,1,..., e i,ni ae linealy independent in Ext 1 (B, A i ) fo all i. In paticula, if E is a geneal extension as above, then Hom(E, A) = 0 if and only if ext 1 (B, A i ) n i fo all i. Poof. Obseve that Hom(E, A) = 0 if and only if Hom(E, A i ) = 0 fo all i. Applying Hom(, A i ) to the sequence defining E and using that B = Ai, we get an exact sequence 0 Hom(E, A i ) Hom(A i, A i ) n i Ext 1 (B, A i ). The map C n i = Hom(Ai, A i ) n i Ext 1 (B, A i ) caies the identity map in the jth component of Hom(A i, A i ) n i to e i,j, so this map is injective if and only if the e i,j ae linealy independent. Finally, we study when two extensions ae isomophic. Lemma 6.3. Let A be a C-linea abelian categoy, and let A, B A with A semisimple, B simple, and B not a simple facto of A. If 0 A E B 0 0 A E B 0 ae two extensions of B by A, then any isomophism E E is induced by an automophism of A. Theefoe, if ext 1 (B, A) > dim Aut A = i n i then two geneal extensions E, E as above ae non-isomophic. Poof. Since Hom(B, B) = C and Hom(A, B) = 0, we have Hom(E, B) = C. Thus up to scale the maps E B and E B ae canonically detemined by E, E. Thei kenels ae theefoe identified unde the isomophism E E. 6.. Constuction of cuves. We now bing togethe the esults of Sections 4, 5 and 6.1 to pove ou main esult. Theoem 6.4. If Δ H,D (v) 0, then W is the Gieseke wall fo M H,D (v). Futhemoe, thee ae cuves in M H,D (v) paameteizing non-isomophic (H, D)-twisted Gieseke stable sheaves that become S-equivalent fo Bidgeland stability conditions along W. Poof. Let σ 0 = (Z 0, A 0 ) be a stability condition on W. Choose a polystable sheaf F = i F n i i M H,D (w). Since W can be made abitaily lage by inceasing Δ H,D (v), we may assume that evey stable facto of F is σ 0 -stable. By Theoem., we may incease Δ H,D (v) so that thee ae stable sheaves in M H,D (u). We let Q be such a stable sheaf (if u has ank 0, we additionally assume Q is sufficiently nice; see Step below). Inceasing Δ H,D (v), Poposition 5.1 shows that Q is actually

16 16 I. COSKUN AND J. HUIZENGA σ 0 -stable. It is clea that Q is not one of the stable factos of F. Inceasing Δ H,D (v) deceases the Eule chaacteistics χ(u, F i ). Thus, if Δ H,D (v) is sufficiently lage we will have χ(q, F i ) n i and χ(q, F ) < i n i. Let P A 0 be the full subcategoy of σ 0 -semistable objects with the same σ 0 -slope as F and Q. Then F is a semisimple object of P and Q is a simple object of P. If E is a geneal extension of the fom 0 F E Q 0, then by Lemma 6. we have Hom(E, F ) = 0. Futhemoe, by Lemma 6.3 we can find cuves in Ext 1 (Q, F ) such that two geneal paameteized objects E ae nonisomophic. To complete the poof, we pove that E is (H, D)-Gieseke stable. Step 1: if σ + is a stability condition just above W, then E is σ + -stable. Suppose F E destabilizes E with espect to σ +, so μ σ+ (F ) μ σ+ (E). Since E is σ 0 -semistable, we have μ σ0 (F ) = μ σ0 (E), and thus by Lemma 6.1 F is a subobject of F in P. But then μ σ+ (F ) = μ σ+ (F ) < μ σ+ (E), a contadiction. Step : E is tosion-fee. If u has positive ank this is tivial, so assume that (u) = 0. By the discussion in 3., we may assume Q is a line bundle L suppoted on a educed and ieducible cuve C. Suppose E has a nonzeo tosion subsheaf T, and let E = E/T. Since F is tosion-fee, T must be a subsheaf of Q. Since Q has pue dimension 1, T must be anothe line bundle L suppoted on C. Then c 1 (T ) = c 1 (u), so c 1 (E ) = c 1 (v) c 1 (u) = c 1 (w). Futhemoe, the composition F E E is injective. Since the stable factos of F have minimal disciminant, this is only possible if F E is an isomophism. But then the composition E E F with the invese isomophism gives a nontivial homomophism E F, which is a contadiction. Theefoe E is tosion-fee. Step 3: E is μ H,D -semistable. Suppose that E C is a μ H,D -stable quotient of E with μ H,D (C) < μ H,D (E) and (C) < (E). By the definition of w, we have μ H,D (C) μ H,D (F ). If μ H,D (C) < μ H,D (F ), then the composition F E C is 0, so induces a map Q C which must be 0 since μ H,D (C) < μ H,D (Q); thus C is zeo, a contadiction. If instead μ H,D (C) = μ H,D (F ), since the stable factos of F have minimal disciminant the composition F E C is eithe 0 o identifies C with one of the stable factos F i of F. In the fist case we conclude as befoe, and in the second we obtain a nontivial homomophism E F, which again is a contadiction. Thus E is μ H,D -semistable. Finally, since E is μ H,D -semistable and σ + -stable, it is (H, D)-Gieseke stable by the discussion in.3.. This completes the poof. 7. Examples In this section, we give applications and examples of ou geneal theoy. We discuss the nef cones of cetain moduli spaces of vecto bundles of ank on seveal classes of sufaces. Let (X, H) be a polaized suface and conside the vecto v with (v) =, fixed ch 1 (v), and vaiable ch (v) 0, so that Δ H,D (v) 0. In the cases we conside, μ H -semistability and μ H -stability will coincide, so the moduli space M H (v) caies a quasiunivesal family. Additionally, we will have q(x) = 0. Unde these assumptions, by O Gady s theoem [O G96] the moduli space M H (v) is ieducible. Witing v fo the othogonal complement of v in K num (X) with espect to the Eule paiing (v, w) = χ(v w), the Donaldson homomophism λ : v N 1 (M H (v))

17 THE NEF CONE OF THE MODULI SPACE OF SHEAVES 17 is an isomophism by a theoem of Jun Li [Li96]. Thus we can specify diviso classes by giving elements of v. By [Bo15, Poposition 3.8], if σ is a stability condition on a wall W in the (H, D)- slice of stability conditions, then the Baye-Macì diviso class associated to σ coesponds to a multiple of the class ( 1, s W H + D, m) v, whee the second Chen chaacte m is detemined by the equiement ( 1, s W H + D, m) v. Additionally, when Pic(X) = ZH, one extemal ay of the nef cone coesponds to the Jun Li mophism to the Donaldson-Uhlenbeck-Yau compactification [HL10, 8] and is given by λ(0, H, n), whee n is chosen by the equiement that (0, H, n) v. Hence, we only need to compute the othe extemal ay Sufaces in P 3. Let X be a vey geneal suface of degee d 4 in P 3. By the Noethe- Lefschetz theoem, Pic(X) = ZH, whee H is the class of a hypeplane section of X. Let v be the Chen chaacte with (v) =, c 1 (v) = H and vaiable second Chen chaacte ch (v). The educed slope is μ H (v) = 1. Since thee ae no line bundles with this educed slope, evey μ H- semistable sheaf of chaacte v is μ H -stable. We use the twisting diviso D = 0, and omit it fom ou notation. The extemal chaacte w must have educed slope μ H (w) = 0 and ank at most. The line bundle O X has this slope. By the Bogomolov inequality, we conclude that the extemal chaacte w is the chaacte of O X O X. If Δ H (v) is sufficiently lage, then the Gieseke wall is given by W (v, w). A computation shows that the wall W (v, w) has cente s W = K X H H + ch (v) H = d 4 + ch (v). d The inteesting extemal ay of the nef cone coesponds to the class ( 1, s W H, m) v. 7.. Double coves of P. Let X be the cyclic double cove of P banched along a vey geneal cuve of degee d 6. Let H be the pullback of O P (1). By the Noethe-Lefschetz theoem (see fo example [RS09]), Pic(X) = ZH. Let v be the Chen chaacte with (v) =, c 1 (v) = H and vaiable second Chen chaacte ch. The educed slope is μ H (v) = 1. Since thee ae no line bundles with this educed slope, evey μ H -semistable sheaf of chaacte v is μ H -stable. We again use the twisting diviso D = 0. The extemal chaacte w must have educed slope μ H (w) = 0 and ank at most. The line bundle O X has this slope. By the Bogomolov inequality, we conclude that the extemal chaacte is the chaacte of O X O X. If Δ H (v) is sufficiently lage, then the Gieseke wall is given by W (v, w). A computation shows that the wall W (v, w) has cente s W = K X H H + ch (v) H = d 3 One edge of the nef cone coesponds to a class ( 1, s W, H, m) v, + ch. and the othe edge coesponds to the Donaldson-Uhlenbeck-Yau compactification The quadic P 1 P 1. Let X = P 1 P 1. We wite classes in N 1 (X) Q = QH1 QH as (a, b), whee H 1 and H ae the two fibes on X. Fix the polaization H = (1, 1), and define a family D t = (t, t) of twisting divisos othogonal to H. We conside the vecto v with (v) =, c 1 (v) = (1, 0), and vaiable ch (v) 0. Since we vay the twisting diviso in this section, it is pefeable to view ch as vaying instead of the twisted disciminant as vaying, since the latte depends on the paticula twist.

18 18 I. COSKUN AND J. HUIZENGA Obseve that evey μ H -semistable sheaf of chaacte v is μ H -stable, since μ H (v) = 1 and no line bundle has this educed slope. Theefoe, the twisted moduli spaces M H,Dt (v) paameteize the same objects as the odinay Gieseke space M H (v), and M H (v) can be ealized as the lagevolume limit in the (H, D t )-slice fo any t Q. Fo any t Q, Coollay 1. allows us to detemine a bounday nef diviso on M H (v) so long as ch (v) 0; howeve, the equied bound on ch (v) depends on the paticula time t. As ch (v) becomes moe negative, we will find that the stuctue of the nef cone becomes inceasingly complicated; fo example, while any fixed space M H (v) is a Moi deam space and thus has a finite polyhedal nef cone [Ry16], the numbe of extemal ays of this cone becomes abitaily lage as ch (v) deceases. Let w t denote an extemal chaacte fo v in the (H, D t )-slice. In the next esult we compute w t in tems of t. Poposition 7.1. Let t Q and let n Z be an intege closest to t. (1) If n 0, then the chaacte of O X (n, n) is an extemal chaacte fo v in the (H, D t )-slice. () If n = 0, then the chaacte of O X is an extemal chaacte fo v in the (H, D t)-slice. Any extemal chaacte fo v in the (H, D t )-slice is obtained in this way. In paticula, the extemal chaacte is uniquely detemined if t is not a half-intege, and thee ae two choices of extemal chaacte if t is a half-intege. Poof. Let w t denote an extemal chaacte fo v in the (H, D t )-slice. Since μ H (w t ) = 0, the fist Chen class satisfies c 1 (w t ) = (n, n) fo some n Z. If w t has ank, then c 1 (v) c 1 (w t ) = (1, 0) (n, n) = (1 n, n) is effective, so eithe n = 0 o n = 1. In all othe cases, w t has ank 1. Next we minimize the disciminant of w t, viewing c 1 (w t ) = (n, n) and (w t ) as fixed. Fist ecall the odinay disciminant Δ and twisted disciminant Δ H,Dt ae c 1 Δ = 1 ch and Δ H,Dt = 1 μ H,D t chd t H. When the fomula fo Δ H,Dt is fully expanded using the definitions, thee ae seveal tems; howeve, only the tem ch /(H ) vaies when c 1 and ae held fixed and ch is vaied. Thus, the poblems of minimizing Δ and Δ H,Dt ae equivalent. If the class c 1 (w t )/(w t ) is integal, then Δ(w t ) must be zeo and w t is a diect sum of (w t ) copies of a line bundle. The only emaining possibility is that (w t ) = and n = 1, so that c 1 (w t ) = (1, 1). By Rudakov s classification of the numeical invaiants of semistable sheaves [ Ru94], the smallest disciminant of a semistable sheaf with this ank and fist Chen class is Δ(w t ) = 3 4. Thus w t will be the chaacte y = (, c 1, Δ) = (, (1, 1), 3 4 ). Howeve, we claim that fo each t Q, Δ H,Dt (y) > min n Z Δ H,D t (O X (n, n)), so that y is neve an extemal chaacte fo v in the (H, D t )-slice. To this end, a computation shows ( ) Δ H,Dt (O X (n, n)) = while Theefoe Δ H,Dt (y) 3 8 Δ H,Dt (O X (n, n)) 1 8. (n t) Δ H,Dt (y) = 1 (t 1)t +. fo all t. On the othe hand, if n is an intege closest to t, then