Moduli of sheaves on Hirzebruch surfaces

Let X be a Hirzebruch surface.  Moduli spaces of semistable sheaves on X with fixed numerical invariants are always irreducible by a theorem of Walter.  On the other hand, many other basic properties of sheaves on Hirzebruch surfaces are unknown.  I will discuss two different problems on this topic.  First, what is the cohomology of a general sheaf on X with fixed numerical invariants?  Second, when is the moduli space
actually nonempty? The latter question should have an answer reminiscient of the Drezet-Le Potier classification of semistable sheaves on the projective plane; in particular, there is a fractal-like hypersurface in the space of numerical invariants which bounds the invariants of semistable sheaves.  This is joint work with Izzet Coskun.