Diagonal quotient surfaces

Let X be a smooth projective curve which admits a finite group G of automorphisms. The purpose of this paper is to analyze the geometry of the associated diagonal quotient surface Z(X,G,alpha) = Delta(alpha)\(X x X), where Delta(alpha) less than or equal to G x G denotes the diagonal subgroup twisted by an automorphism alpha is an element of Aut(G). In particular, we calculate some of the numerical invariants of its minimal desingularization (Z) over tilde(X,G,alpha) such as its Betti and Chern numbers and establish some criteria for determining its place in the Enriques-Kodaira classification table.