The classification of surfaces with p_g = q = 1 isogenous to a product of curves

Abstract

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 ...

The classification of isotrivially fibred surfaces with p g = q = 2

“…If |G| = 16 then g(C) = 5, and the branching data for F and C are (2) and (2). Looking at the table in [14] one sees that among the 14 groups of order 16 only Z 4 (Z 2 ) 2 , D 4,4,−1 and D 2,8,5 are (1 | 2)-generated. Moreover with a computer computation using the program of the Appendix, one sees that in these cases the action of the groups on the product C × F cannot be free.…”

“…Among the surfaces which admit an isotrivial fibration one can find examples of surfaces with χ(O S ) = 1. Since [15] appeared several authors studied intensively standard isotrivially fibred surfaces, and eventually classified all those, which are minimal with p g = q = 0 [5,7,8] and with p g = q = 1 [14,30,34].…”

“…While Chen and Hacon [19] constructed a first example of a surface with K 2 S = 5. The classification of the surfaces isogenous to a product involves techniques from both geometry and combinatorial group theory developed in [4,5,14,33]. In the second section of this paper we recall the relation between coverings of Riemann surfaces and orbifold surface groups, which enables us to transform the geometric problem of classification into an algebraic one.…”

“…Since it is written in great generality we hope it can be used for other tasks. For example we were able to finish the classification given in [14] adding the number of connected components of the moduli spaces. We tested the program also on the cases with p g = q = 0 given in [5] and p q = q = 1 given in [33].…”

On threefolds isogenous to a product of curves

“…Since then, a considerable amount of literature appeared, especially in the case of surfaces. In particular surfaces S isogenous to a product of curves with χ(O S ) = 1 (equivalently p g (S) = q(S)) are completely classified, see [BCG08,CP09,Pen11,CCML98,Pir02,HP02,Bea82].…”