Interpolation and moduli spaces of vector bundles on very general blowups of P²

In this paper, we study certain moduli spaces of vector bundles on the blowup of P² in at least ten 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 P² 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.