  In this paper, we study certain moduli spaces of vector bundles on the blowup
of the projective plane in at least 10 very general points. Moduli spaces of
sheaves on general type surfaces may be nonreduced, reducible and even
disconnected. In contrast, moduli spaces of sheaves on minimal rational
surfaces and certain del Pezzo surfaces are irreducible and smooth along the
locus of stable bundles. We find examples of moduli spaces of vector bundles on
more general blowups of the projective plane that are disconnected and have
components of different dimensions. In fact, assuming the SHGH Conjecture, we
can find moduli spaces with arbitrarily many components of arbitrarily large
dimension.
