Geometric Invariant Theory (Fall 2020)

Geometric invariant theory is an important tool in the study of moduli spaces in algebraic geometry. In particular, GIT is used to construct coarse moduli spaces. Some classical references are:

The relationship between GIT and symplectic reduction is further expanded on in

Some more constructions of moduli are given in sections 7 and 8 of

Classically, invariant theory focused on studying invariants of rings under group actions, for example

Schedule

Sep 11
Organizational Meeting
Sep 18 Sep 25
Anna Abasheva
Three approaches to GIT: overview and examples
GIT rests on three pillars: invariant theory, the Hilbert-Mumford criterion, and symplectic reduction. They all serve one purpose, which is to construct the quotient of a variety by a group action. My goal is to give a short overview of the three approaches without giving complete proofs or even without giving them at all (we’ll be able to discuss all the proofs in detail later during the seminar). I’ll present several useful exampes like toric varieties and moduli of finite sets of n points in \mathbb{P}^1 for small n. They are nice to have in mind later when dealing with abstract concepts of GIT. During my talk, they will give us a taste of how the three pillars are related and in which cases their methods may be used in practice.
Sep 25 Oct 02
Caleb Ji
Reference: [MFK], Chapter 0