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

Thomas Dedieu

paper

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

arXiv:1905.11469

Abstract

In this short note, I point out that results of Ballico and Kool--Shende--Thomas together imply that on $K 3$ , Enriques, and Abelian surfaces, if $L$ is a very ample and $(2 p_{a} (L) - 2 g - 1)$ -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.

Comment on arXiv:1905.07116. Final version, to appear in Archiv der Mathematik