  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.
