Физико-математический клуб при ПОМИ и СПбГУ

Gareth Jones (University of Southampton, UK) Mini-course "Beauville surfaces" (2 lectures)


I shall give a survey of recent progress on Beauville surfaces, an area in
which group theory (both finite and infinite) has applications in algebraic
geometry. Specifically, a Beauville surface S is a complex surface (that
is, a 2-dimensional projective algebraic variety) which is isogenous to a
higher product; this means that S is the quotient of the cartesian product
of two quasiplatonic curves (equivalently, two regular dessins) of genus at
least 2 by some finite group G acting freely on the product. This
construction allows properties of finite groups G, and their realisations
as quotients of triangle groups, to be applied to the study of the surfaces
S.

I shall consider the following problems. Which groups G (and in particular
which abelian or simple groups) can be used in this construction? What can
be said about the automorphism group of S? What is the fundamental group of
S, and to what extent does it determine S (rigidity)? Belyi's Theorem
implies that Beauville surfaces are defined over the field of algebraic
numbers, so how does the absolute Galois group (the automorphism group of
this field) act on them?

The lectures will be aimed at those who are not necessarily specialists in
either algebraic geometry or group theory. Some acquaintance with
Grothendieck's theory of dessins d'enfants will be useful, though I will
start with a brief summary and some basic examples.