Irreducibility of Severi varieties on K3 surfaces

TL;DR

This paper proves the connectedness and irreducibility of Severi varieties on general K3 surfaces for certain numbers of nodes, confirming longstanding conjectures through degeneration techniques.

Contribution

It establishes the irreducibility of Severi varieties on K3 surfaces for $ ext{delta} ext{ up to } g-4$, advancing understanding of their geometric structure.

Findings

Severi varieties are connected for $ ext{delta} ext{ up to } g-1$.

Severi varieties are irreducible for $ ext{delta} ext{ up to } g-4$.

Results are achieved via degeneration to Halphen surfaces.

Abstract

Let $(S,L)$ be a general primitively polarized $K3$ surface of genus $g$. For every $0\leq \delta \leq g$ 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 $\delta\leq g-1$. If $\delta\leq g-4$, 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.