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.
Categories: math.AG