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

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