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

In this paper, we study the Brill-Noether theory of the normalizations of singular, irreducible curves on a $K3$ surface. We introduce a {\em singular} Brill-Noether number $\rho_{sing}$ and show that if the Picard group of the K3 surface is ${\mathbb Z} [L]$, there are no $g^r_d$'s 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 {\em hyperelliptic normalizations}, and classify linear systems $|L|$ containing irreducible nodal curves with hyperelliptic normalizations, for $\rho_{sing}<0$, without any assumption on its Picard group.