Existence of semistable sheaves on Hirzebruch surfaces

Abstract

Let $X$ be a Hirzebruch surface, and let $H$ be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves $M_{X,H}(r,c_1,c_2)$ 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 $Δ\geq δ_H(c_1/r)$ 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 $δ_H(c_1/r)$ 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 $δ_H(c_1/r)$ 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 $δ_H(c_1/r)$ exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of moduli spaces of sheaves.