Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces

TL;DR

This paper studies Severi varieties on Enriques surfaces and rational elliptic surfaces, proving their nonemptiness and providing examples of special superabundant logarithmic Severi varieties across various genera.

Contribution

It establishes the nonemptiness of certain Severi varieties on Enriques surfaces of base change type and introduces examples of superabundant logarithmic Severi varieties on rational elliptic surfaces.

Findings

Countable families of Severi varieties are nonempty on Enriques surfaces.

Members of these families split into nonlinearly equivalent curves in the K3 cover.

Examples of superabundant logarithmic Severi varieties are provided on rational elliptic surfaces.

Abstract

If an irreducible curve on the very general Enriques surface splits in the K3 cover, its preimage consists of two linearly equivalent irreducible curves. We prove the nonemptiness of countable families of Severi varieties of curves of any genus on Enriques surfaces of base change type, whose members split in nonlinearly equivalent curves in the K3 cover. Our machinery leads us to provide examples of special superabundant logarithmic Severi varieties of curves of any genus on rational elliptic surfaces.