Recent work on Beauville surfaces, structures and groups

Summary

Abstract

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. In this survey we discuss the groups that may be used in this way. En route we discuss several open problems, questions and conjectures.

Introduction

Roughly speaking (precise definitions will be given in the next section), a Beauville surface is a complex surface S defined by taking a pair of complex curves, i.e., Riemann surfaces, C₁ and C₂ and letting a finite group G act freely on their product to define S as a quotient (C₁×C₂)/G. These surfaces have a wide variety of attractive geometric properties: they are surfaces of general type; their automorphism groups [50] and fundamental groups [20] are relatively easy to compute (being closely related to G — see Section 7.2 and 7.3); these surfaces are rigid surfaces in the sense of admitting no nontrivial deformations [10] and thus correspond to isolated points in the moduli space of surfaces of general type [37].

Much of this good behaviour stems from the fact that the surface (C₁× C₂)/G is uniquely determined by a particular pair of generating sets of G known as a ‘Beauville structure’. This converts the study of Beauville surfaces to the study of groups with Beauville structures, i.e., Beauville groups.

Beauville surfaces were first defined by Catanese in [20] as a generalisation of an earlier example of Beauville [14, Exercise X.13(4)] (native English speakers may find the English translation [15] somewhat easier to read and get hold of) in which C = C’ and the curves are both the Fermat curve defined by the equation X⁵ +Y⁵ +Z⁵ = 0 being acted on by the group (Z/5Z)×(Z/5Z) (this choice of group may seem somewhat odd at first, but the reason will become clear later). Bauer, Catanese and Grunewald went on to use these surfaces to construct examples of smooth regular surfaces with vanishing geometric genus [11]. Early motivation came from the consideration of the ‘Friedman-Morgan speculation’ — a technical conjecture concerning when two algebraic surfaces are diffeomorphic which Beauville surfaces provide counterexamples to.