Abstract:
Given an algebraic variety X with an action of an algebraic group G, is there a reasonable notion of what the ‘quotient variety X/G’ should be?
The main motivation for this question come from moduli theory: many moduli spaces classifying objects in algebraic geometry and elsewhere can naturally be realised as quotients of some space of parameters by a group action.
The answer to this question, at least if G is reductive, was given by David Mumford in the 1960’s and goes by the name of Geometric Invariant Theory (GIT). After giving an introduction to the main main features of GIT, I will give some examples of its use in practice to construct some famous moduli spaces. Time permitting, I will then say a few words about what goes wrong in the non-reductive case, and what we can do to fix it.