Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

TL;DR

This paper investigates mixed surfaces formed from quotients of the square of a curve by a finite group, describing their Albanese morphisms and minimality conditions, with applications to semi-isogenous mixed surfaces.

Contribution

It provides a complete description of the Albanese morphism for these surfaces and establishes minimality criteria based on irregularity, extending understanding of mixed surfaces with maximal Albanese dimension.

Findings

If irregularity ≥ 3, the surface is minimal.

The Albanese morphism is described via a specific further quotient.

Surfaces with irregularity ≥ 2 have maximal Albanese dimension.

Abstract

We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C x C)/G of the "square" of a curve by a finite group G of automorphisms that contains elements not preserving the factors. We study them through the "further quotients" by (C x C)/G' where G' contains G. As a first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp. The main result is a complete description of the Albanese morphism of S through a determined further quotient (C x C)/G' that is an \'etale cover of the symmetric square of a curve. In particular, if the irregularity of S is at least 2, then S has maximal Albanese dimension. We apply our result to all the "semi-isogenous" mixed surfaces of maximal Albanese dimension constructed by Cancian and Frapporti, relating them with the other constructions appearing in the literature of surfaces of general type having the same invariants.