Irreducibility of the moduli space of vector bundles on surfaces and Brill-Noether theory on singular curves

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We prove the irreducibility of the moduli space of rank 2 semistable torsion free sheaves (with a generic polarization and any value of c₂) on a K3 or a del Pezzo surface. In the case of a K3 surface, we need to prove a result on the connectivity of the Brill-Noether locus for singular curves on the surface. In the case of a del Pezzo surface, we reduce the problem to the case of P² by first relating the moduli spaces of the plane and the blown-up plane, and then studying how the moduli space changes when we change the polarization.