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

Abstract

We study mixed surfaces, the minimal resolution S of the singularities of a quotient $$(C \times C)/G$$ of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors. We study them through the further quotients $$(C \times C)/G'$$ where $$G' \supset G$$ . 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 well determined further quotient $$(C \times C)/G'$$ 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.