On Surfaces of General Type with p_g = q = 1 Isogenous to a Product of Curves

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 article, we classify the surfaces of general type with p_g = 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 𝔐_I, 𝔐_II, 𝔐_IV are irreducible, whereas 𝔐_III 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.

Key Words:

• Actions of finite groups on curves
• Surfaces of general type

2000 Mathematics Subject Classification:

• Primary 14J29
• Secondary 14J10, 20F65

ACKNOWLEDGMENTS

Part of the present research was done during the author's visits to the Universities of Bayreuth, Cambridge, and Warwick, where he was supported by EU Research Training Network EAGER, no. HPRN-CT-2000-00099, and by a Marie Curie pre-doc fellowship. The author is grateful to C. Ciliberto for useful advice and constructive remarks during the preparation of this work and to F. Catanese and I. Bauer for suggesting the problem and sharing the ideas contained in Bauer and Catanese (2002). He also wishes to thank A. Corti, M. Reid, G. Infante for helpful conversations, and R. Pardini for pointing out some mistakes contained in the first version of this article. Finally, he is indebted with the referee for several detailed comments that considerably improved the presentation of these results.

To the memory of my colleague and friend Giulio Minervini.

Notes

Communicated by C. Pedrini.