TY - JOUR

T1 - Existence of semistable sheaves on Hirzebruch surfaces

AU - Coskun, Izzet

AU - Huizenga, Jack

N1 - Funding Information:
During the preparation of this article the first author was partially supported by the NSF grant DMS-1500031 and NSF FRG grant DMS 1664296 and the second author was partially supported by the NSA Young Investigator Grant H98230-16-1-0306 and NSF FRG grant DMS 1664303 .

PY - 2021/4/16

Y1 - 2021/4/16

N2 - Let Fe denote the Hirzebruch surface P(OP1⊕OP1(e)), and let H be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves MFe,H(r,c1,c2) is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one. We then study sharp Bogomolov inequalities Δ≥δH(c1/r) for the discriminants of stable sheaves which take the polarization and slope into account; these inequalities essentially completely describe the characters of stable sheaves. The function δH(c1/r) can be computed to arbitrary precision by a limiting procedure. In the case of an anticanonically polarized del Pezzo surface, exceptional bundles are always stable and δH(c1/r) is computed by exceptional bundles. More generally, we show that for an arbitrary polarization there are further necessary conditions for the existence of stable sheaves beyond those provided by stable exceptional bundles. We compute δH(c1/r) exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of MFe,H(v).

AB - Let Fe denote the Hirzebruch surface P(OP1⊕OP1(e)), and let H be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves MFe,H(r,c1,c2) is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one. We then study sharp Bogomolov inequalities Δ≥δH(c1/r) for the discriminants of stable sheaves which take the polarization and slope into account; these inequalities essentially completely describe the characters of stable sheaves. The function δH(c1/r) can be computed to arbitrary precision by a limiting procedure. In the case of an anticanonically polarized del Pezzo surface, exceptional bundles are always stable and δH(c1/r) is computed by exceptional bundles. More generally, we show that for an arbitrary polarization there are further necessary conditions for the existence of stable sheaves beyond those provided by stable exceptional bundles. We compute δH(c1/r) exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of MFe,H(v).

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U2 - 10.1016/j.aim.2021.107636

DO - 10.1016/j.aim.2021.107636

M3 - Article

AN - SCOPUS:85100797918

VL - 381

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107636

ER -