ON SURFACES OF GENERAL TYPE WITH pg = q = 1 ISOGENOUS TO A PRODUCT OF CURVES

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F )/G. In this paper we classify the surfaces of general type with pg = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV . The moduli spaces MI , MII , MIV are irreducible, whereas MIII is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples. 0. Introduction The problem of classification of surfaces of general type is of exponential computational complexity, see [Ca92], [Ch96], [Man97]; nevertheless, one can hope to classify at least those with small numerical invariants. It is well-known that the first example of surface of general type with pg = q = 0 was given by Godeaux in [Go31]; later on, many other examples were discovered. On the other hand, any surface S of general type verifies χ(OS) > 0, hence q(S) > 0 implies pg(S) > 0. It follows that the surfaces with pg = q = 1 are the irregular ones with the lowest geometric genus, hence it would be important to achieve their complete classification; so far, this has been obtained only in the cases K2 S = 2, 3 (see [Ca81], [CaCi91], [CaCi93], [Pol05], [CaPi05]). As the title suggests, this paper considers surfaces of general type with pg = q = 1 which are isogenous to a product. This means that there exist two smooth curves C, F and a finite group G, acting freely on their product, so that S = (C × F )/G. We have two cases: the mixed case, where the action of G exchanges the two factors (and then C and F are isomorphic) and the unmixed case, where G acts diagonally. In the unmixed case G acts separately on C and F , and the two projections πC : C ×F −→ C, πF : C ×F −→ F induce two isotrivial fibrations α : S −→ C/G, β : S −→ F/G, whose smooth fibres are isomorphic to F and C, respectively. If S is isogenous to a product, there exists a unique realization S = (C × F )/G such that the genera g(C), g(F ) are minimal ([Ca00], Proposition 3.13); we will always work with minimal realizations. Surfaces of general type with pg = q = 0 isogenous to a product appear in [Be96], [Par03] and [BaCa03]; their complete classification has been finally obtained in [BaCaGr06]. Some unmixed examples with pg = q = 1 have been given in [Pol06]; so it seemed natural to attack the following Main Problem. Classify all surfaces of general type with pg = q = 1 isogenous to a product, and describe the corresponding irreducible components of the moduli space. In this paper we fully solve the Main Problem in the unmixed case assuming that the group G is abelian. Our results are the following: Theorem A (see Theorem 4.1). If the group G is abelian, then there exist exactly four families of surfaces of general type with pg = q = 1 isogenous to an unmixed product. In every case g(F ) = 3, whereas the occurrences for g(C) and G are Date: August 3, 2008. 1991 Mathematics Subject Classification. 14J29 (primary), 14J10, 20F65.