k-gonal loci in Severi varieties of curves on K3 surfaces

This is a report on joint work with C. Ciliberto. Let $(S,H)$ be a general primitively polarized complex K3 surface $(S,H)$ of genus p. It is well-known that the Severi varieties of $\delta$-nodal curves in $|H|$, for $0 \leq \delta \leq p$, are smooth and nonempty of dimension $p-\delta$. We consider the subloci of curves with k-gonal normalizations and prove necessary and sufficient conditions in terms of $p$, $\delta$ and $k$ for these to be nonempty. In contrast to the case of smooth curves, the Severi varieties contain proper subloci of gonalities lower than the maximal gonality given by Brill-Noether theory. Besides its intrinsic interest for Brill-Noether theory and moduli problems, the subject is related to Mori theory of the 2k-dimensional hyperk\"ahler manifold $Hilb^k(S)$ parametrizing 0-dimensional length k-subschemes of S, since curves with k-gonal normalizations on S give rise to rational curves in $Hilb^k(S)$. I will discuss connections with recent works of Bayer and Macri and talk about some of our expectations.