  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
$g(L)-k$.
