On semi†isogenous mixed surfaces

Let C be a smooth projective curve and G be a finite subgroup of Aut (C)2⋊Z2 whose action is mixed, i.e. there are elements in G exchanging the two isotrivial fibrations of C×C. Let G0◃G be the index two subgroup G∩ Aut (C)2. If G0 acts freely, then X:=(C×C)/G is smooth and we call it semi†isogenous mixed surface. In this paper we give an algorithm to determine semi†isogenous mixed surfaces with given geometric genus, irregularity and self†intersection of the canonical class. As an application we classify irregular semi†isogenous mixed surfaces with K2>0 and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with χ=1. We provide an example of a minimal surface of general type with K2=7 and pg=q=2.