### Geometric Invariant Theory (Fall 2020)

- Organizers: Patrick Lei, Anna Abasheva
- When: Friday 11am-12:30pm EDT
- Where: Zoom (please email Patrick for the link)

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:

**[MFK]** Mumford, Fogarty, Kirwan, *Geometric Invariant Theory*
**[D]** Dolgachev, *Lectures on Invariant Theory*

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

**[K]** Kirwan, *Cohomology of Quotients in Symplectic and Algebraic Geometry*

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

**[H]** Hoskins, *Moduli Problems and Geometric Invariant Theory*

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

**[N]** Nagata, *On the 14th Problem of Hilbert*

#### 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