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 $$g(L)-k$$ .