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

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 $g(L)-k$.

References in corpus (1)

• On the irreducibility of Severi varieties on K3 surfaces

Cited by in corpus (1)

• Dependent subsets of embedded projective varieties