Weak Brill-Noether for rational surfaces

TL;DR

This paper establishes conditions under which the weak Brill-Noether property holds for moduli spaces of sheaves on rational surfaces, extending known results from the projective plane to more general surfaces.

Contribution

It provides sufficient conditions and a complete characterization of Chern characters for weak Brill-Noether on Hirzebruch and del Pezzo surfaces.

Findings

Weak Brill-Noether holds on certain rational surfaces under specified conditions.

Complete characterization of Chern characters on Hirzebruch surfaces for weak Brill-Noether.

Weak Brill-Noether holds on del Pezzo surfaces of degree ≥4 with nef first Chern class.

Abstract

A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Goettsche and Hirschowitz prove that on the projective plane every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef.