STANDARD ISOTRIVIAL FIBRATIONS WITH pg = q = 1

A smooth, projective surface S of general type is said to be a standard isotrivial fibration if there exist a finite group G acting faithfully on two smooth projective curves C and F so that S is isomorphic to the minimal desingularization of T := (C × F)/G. If T is smooth then S = T is called a quasi-bundle. In this paper we classify the standard isotrivial fibrations with pg = q = 1 which are not quasi-bundles, assuming that all the singularities of T are rational double points. As a by-product, we provide several new examples of minimal surfaces of general type with pg = q = 1 and K 2 S = 4, 6. 0. Introduction Recently, there has been considerable interest in understanding the geometry of complex projective surfaces with small birational invariants, and in particular of surfaces with p g = q = 1. 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 of general type 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 K 2 is any surface with q = 1, its Albanese map α : S −→ E is a fibration over an elliptic curve E; we denote by g alb the genus of the general fibre of α. The universal property of the Albanese morphism implies that α is the unique fibration on S with irrational base. As the title suggests, this paper considers surfaces with p g = q = 1 which are standard isotrivial fibrations. This means that there exists a finite group G which acts faithfully on two smooth projective curves C and F so that S is isomorphic to the minimal desingularization of T := (C × F)/G. If T is smooth then S = T is called a quasi-bundle or a surface isogenous to an unmixed product (see [Se90], [Se96], [Ca00]). Quasi-bundles of general type with p g = q = 1 are classified in [Pol08] and [CarPol]. In the present work we consider the case where all the singularities of T are rational double points (RDPs). Our classification procedure combines ideas from [Pol08] and combinatorial methods of finite group theory. …