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

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