Comment on: On the irreducibility of the Severi variety of nodal curves in a smooth surface, by E. Ballico

Abstract

In this short note, I point out that results of Ballico and Kool–Shende–Thomas together imply that on K3, Enriques, and Abelian surfaces, if L is a very ample and $$(2p_a(L)-2g-1)$$-spanned line bundle, then the equigeneric Severi variety $$V_{g}(L)$$ of all curves in |L| having genus g is non-empty, irreducible, of the expected dimension, and its general member is a $$(p_a(L)-g)$$-nodal curve.