Singular curves on a K3 surface and linear series on their normalizations.

We study the Brill-Noether theory of the normalizations of singular,irreducible curves on a K3 surface. We introduce a singular Brill-Noether number rho_sing and show that if Pic(K3) = Z[L], there are no linear series of degree d and dimension r on the normalizations of irreducible curves in |L|, provided that rho_sing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for rho_sing < 0, without any assumption on the Picard group.