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 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 5 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.