Brill-Noether and existence of semistable sheaves on del Pezzo surfaces

Abstract

Let $X$ be a del Pezzo surface. When the degree of $X$ is at least 4, we compute the cohomology of a general sheaf in the moduli space of Gieseker semistable sheaves. We also classify the Chern characters for which the general sheaf in the moduli space is non-special, i.e. has at most one nonzero cohomology group. Our results hold for arbitrary polarizations, slope semistability, and semi-exceptional moduli spaces. When the degree of $X$ is at least 3, we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-canonical polarization.

update the email address of the second author, update the references, correct typos. To appear in Annales de Institut Fourier

References in corpus (1)

• Irreducibility of Moduli Spaces of Vector Bundles on Birationally Ruled Surfaces

Cited by in corpus (1)

• Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane