Abstract. 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. IntroductionThe 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 p g = 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(O S ) > 0, hence q(S) > 0 implies p g (S) > 0. It follows that the surfaces with p g = 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 CaPi05]). As the title suggests, this paper considers surfaces of general type with p g = 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 π C

Abstract. We classify minimal surfaces of general type with p g = q = 2 and K 2 = 6 whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components M Ia , M Ib , M II of dimension 4, 4, 3, respectively.

Abstract. A smooth, projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C × F so that S = (C × F )/G. In this paper we classify all surfaces with pg = q = 1 which are isogenous to a product. IntroductionThe classification of smooth, complex surfaces S of general type with small birational invariants is quite a natural problem in the framework of algebraic geometry. For instance, one may want to understand the case where the Euler characteristic χ(O S ) is 1, that is, when the geometric genus p g (S) is equal to the irregularity q(S). All surfaces of general type with these invariants satisfy p g ≤ 4. In addition, if p g = q = 4 then the self-intersection K 2 S of the canonical class of S is equal to 8 and S is the product of two genus 2 curves, whereas if p g = q = 3 then K 2 S = 6 or 8 and both cases are completely described. On the other hand, surfaces of general type with p g = q = 0, 1, 2 are still far from being classified. We refer the reader to the survey paper [BaCaPi06] for a recent account on this topic and a comprehensive list of references. A natural way of producing interesting examples of algebraic surfaces is to construct them as quotients of known ones by the action of a finite group. For instance Godeaux constructed in [Go31] the first example of surface of general type with vanishing geometric genus taking the quotient of a general quintic surface of P 3 by a free action of Z 5 . In line with this Beauville proposed in [Be96, p. 118] the construction of a surface of general type with p g = q = 0, K 2 S = 8 as the quotient of a product of two curves C and F by the free action of a finite group G whose order is related to the genera g(C) and g(F ) by the equality |G| = (g(C) − 1)(g(F ) − 1). Generalizing Beauville's example we say that a surface S is isogenous to a product if S = (C × F )/G, for C and F smooth curves and G a finite group acting freely on C × F . A systematic study of these surfaces has been carried out in [Ca00]. They are of general type if and only if both g(C) and g(F ) are greater than or equal to 2 and in this case S admits a unique minimal realization where they are as small as possible. From now on, we tacitly assume that such a realization is chosen, so that the genera of the curves and the group G are invariants of S. The action of G can be seen to respect the product structure on C × F . This means that such actions fall in two cases: the mixed one, where there exists some element in G exchanging the two factors (in this situation C and F must be isomorphic) and the unmixed one, where G acts faithfully on both C and F and diagonally on their product. After [Be96], examples of surfaces isogenous to a product with p g = q = 0 appeared in [Par03] and [BaCa03], and their complete classification was obtained in [BaCaGr06]. The next natural step is therefore the analysis of the case p g = q = 1. Surfaces of general type with these invariants are the irregular ones with the lowest geometric genus ...

Abstract. We classify the minimal algebraic surfaces of general type with p g = q = 1, K 2 = 8 and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, i.e. if S is such a surface, then there exist two smooth curves C, F and a finite group G acting freely on C × F such that S = (C × F )/G. We describe the C, F and G that occur. In particular the curve C is a hyperelliptic-bielliptic curve of genus 3, and the bicanonical map φ of S is composed with the involution σ induced on S by τ × id : C × F −→ C × F , where τ is the hyperelliptic involution of C. In this way we obtain three families of surfaces with p g = q = 1, K 2 = 8 which yield the first-known examples of surfaces with these invariants. We compute their dimension and we show that they are three generically smooth, irreducible components of the moduli space M of surfaces with p g = q = 1, K 2 = 8. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val. IntroductionIn [Par03] R. Pardini classified the minimal surfaces S of general type with p g = q = 0, K 2 S = 8 and a rational involution, i.e. an involution σ : S −→ S such that the quotient T := S/σ is a rational surface. All the examples constructed by Pardini are isogenous to a product, i.e. there exist two smooth curves C, F and a finite group G acting faithfully on C, F and whose diagonal action is free on the product C × F , in such a way that S = (C × F )/G. Pardini's classification contains five families of such surfaces; in particular, four of them are irreducible components of the moduli space of surfaces with p g = q = 0, K 2 S = 8, and represent the surfaces with the above invariants and non-birational bicanonical map.In this paper we deal with the irregular case, in fact we study the case p g = q = 1, K 2 S = 8. Surfaces with p g = q = 1 are the minimal irregular surfaces of general type with the lowest geometric genus, therefore it would be very interesting to obtain their complete classification; for such a reason, they are currently an active topic of research. However, such surfaces are still quite mysterious, and only a few families have been hitherto discovered. If S is a surface with p g = q = 1, then 2 ≤ K 2 S ≤ 9; the case K

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