Irreducibility of Severi varieties on K3 surfaces

Abstract

Let $(S,L)$ be a general primitively polarized $K3$ surface of genus $g$. For every $0\le \delta \le g$ we consider the Severi variety parametrizing integral curves in $|L|$ with exactly $\delta$ nodes as singularities. We prove that its closure in $|L|$ is connected as soon as $\delta \le g-1$. If $\delta \le g-4$, we obtain the stronger result that the Severi variety is irreducible, as predicted by a well-known conjecture. The results are obtained by degeneration to Halphen surfaces.

Major revision due to a mistake in Proposition 3.2 of the previous version communicated to us by Xi Chen and Adrian Zahariuc