On the irreducibility of Severi varieties on K3 surfaces

TL;DR

This paper proves the irreducibility of Severi varieties of nodal curves on certain polarized K3 surfaces when the genus and number of nodes satisfy specific inequalities, advancing understanding of algebraic curve moduli.

Contribution

It establishes the irreducibility of Severi varieties on polarized K3 surfaces under new genus and nodal constraints, extending previous results in algebraic geometry.

Findings

Severi varieties are irreducible for genus p ≥ 4δ - 3

Applicable to K3 surfaces with Picard group generated by L

Results hold for δ-nodal curves in |L|

Abstract

Let $(S,L)$ be a polarized $K3$ surface of genus $p \geqslant 11$ such that $\mathrm{Pic}(S)=\mathbf{Z}[L]$, and $\delta$ a non-negative integer. We prove that if $p\geqslant 4\delta-3$, then the Severi variety of $\delta$-nodal curves in $|L|$ is irreducible.