The classification of minimal product-quotient surfaces with pg=0

The present article is the fourth in a series of papers (cf. [BC04], [BCG08], [BCGP08]), where the goal is to contribute to the classification problem of surfaces of general type by giving a systematic way to construct and distinguish algebraic surfaces. We will use the basic notations from the classification theory of complex projective surfaces, in particular the basic numerical invariants K S, pg := h (S,ΩS), q(S) := h(S,OS); the reader unfamiliar with these may consult e.g. [Be83]. The methods we introduced in the above cited articles, and substantially develop and refine in the present paper are in principle applicable to many more situations. Still we restrict ourselves to the case of surfaces of general type with geometric genus pg = 0. By Gieseker’s theorem (cf. [Gie77]) and standard inequalities (cf. [BCP10, thm. 2.3 and the following discussion]) minimal surfaces of general type with pg = 0 yield a finite number of irreducible components of the moduli space of surfaces of general type. Although it is theoretically possible to describe all irreducible components of the moduli space corresponding to surfaces of general type with pg = 0, this ultimate goal is far out of reach, even if there has been a substantial progress in the study of these surfaces especially in the last five years. We refer to to [BCGP08] and [BCP10] for a historical account and recent update on what is known about surfaces of general type with pg = 0. We study the following situation: let G be a finite group acting on two compact Riemann surfaces C1, C2 of respective genera at least 2. We shall consider the