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.