Brill–Noether and existence of semistable sheaves for del Pezzo surfaces

Let X m be a del Pezzo surface of degree 9-m. When m≤5, we compute the cohomology of a general sheaf in M(v), the moduli space of Gieseker semistable sheaves with Chern character v. We also classify the Chern characters for which the general sheaf in M(v) 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 m≤6, we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-canonical polarization.