Среда 5 ноября в 14:00 в ауд. 413 (лаборатория Чебышева, 14-я линия В.О., 29):

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.