Severi varieties on blow--ups of the symmetric square of an elliptic curve

TL;DR

This paper proves the non-emptiness and smoothness of certain Severi varieties of nodal curves on blow-ups of the symmetric square of a general elliptic curve, aiding the understanding of curves on Enriques surfaces.

Contribution

It establishes the non-emptiness and smoothness of Severi varieties on specific blow-ups, advancing the study of algebraic curves on complex surfaces.

Findings

Severi varieties are non-empty on blow-ups of symmetric squares of elliptic curves.

These Severi varieties are smooth and have the expected dimension.

Results serve as a step towards understanding curves on Enriques surfaces.

Abstract

We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are non-empty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces in arXiv:2109.10735.