Abstract

This is a report on recent joint work with C. Ciliberto. Let $(S,H)$ be a general primitively polarized complex K3 surface of genus $p$ and consider the Severi varieties $V_{|H|,\delta}$ parametrizing $\delta$- nodal curves in the linear system $|H|$, for $0 \leq \delta \leq p$.

It is well-known that these are smooth and nonempty of dimension $p-\delta$. We consider the subloci V(k,|H|,\delta) of curves whose normalizations possess a g1k. We give necessary and sufficient conditions depending on $p$, $\delta$ and $k$ for these loci to be nonempty, and prove that, when nonempty, there is always an irreducible component of the expected dimension. In contrast to the case of smooth curves, the Severi varieties thus contain proper subloci of gonalities lower than the maximal gonality given by Brill-Noether theory.

I will also discuss applications to the study of the Mori cone of rational curves in the punctual Hilbert scheme Hilb(k,S)$, since the curves in V(k,|H|,\delta) naturally induce rational curves in \Hilb(k,S)$.