Beauville Surfaces, Moduli Spaces and Finite Groups

Abstract

In this article we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.

Key Words:

• Beauville surfaces
• Finite groups
• Moduli spaces
• Surface of general type

2010 Mathematics Subject Classification:

• 14J10
• 14J29
• 20D06
• 20H10
• 30F99

ACKNOWLEDGEMENT

The authors are grateful to Fritz Grunewald for inspiring and motivating us to work on this problem together. He is deeply missed.

The authors would like to thank Ingrid Bauer and Fabrizio Catanese for suggesting the problems, for many useful discussions and for their helpful suggestions. We are grateful to Moshe Jarden and Martin Kassabov for interesting discussions. We would also like to thank G. Jones, G. Gonzales-Diez, and D. Torres-Teigell for pointing out some subtleties in our first draft.

The authors acknowledge the support of the DFG Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds.” The first author acknowledges the support of the European Post-Doctoral Institute and the Max-Planck-Institute for Mathematics in Bonn.

Notes

Communicated by L. Ein.