Irreducibility of Severi varieties on K3 surfaces

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 $\mathrm{\mid }L\mathrm{\mid }$ with exactly $\delta$ nodes as singularities. We prove that its closure in $\mathrm{\mid }L\mathrm{\mid }$ 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.