On the irreducibility of the Severi variety of nodal curves in a smooth surface

TL;DR

This paper proves the irreducibility of certain Severi varieties of nodal, irreducible curves on smooth projective surfaces under a specific spannedness condition of the line bundle.

Contribution

It establishes the irreducibility of the subset of nodal, irreducible curves with fixed nodes in a linear system on a smooth surface, under a new spannedness condition.

Findings

Proves irreducibility of $ ilde{V}_k(L)$ for $(2k-1)$-spanned line bundles.

Shows $ ilde{V}_k(L)$ is an open subset of the Severi variety.

Provides conditions under which the Severi variety is irreducible.

Abstract

Let $X$ be a smooth projective surface and $L\in \mathrm{Pic}(X)$. We prove that if $L$ is $(2k-1)$-spanned, then the set $\tilde{V}_k(L)$ of all nodal and irreducible $D\in |L|$ with exactly $k$ nodes is irreducible. The set $\tilde{V}_k(L)$ is an open subset of a Severi variety of $|L|$, the full Severi variety parametrizing all integral $D\in |L|$ with geometric genus $g(L)-k$.