Student seminar

## Goal

This series is intended to be an introduction to the subject. We will be working mostly with Mukai's excellent introduction [2] to the subject, and in some sense the first three weeks could be considered a reading seminar based on this book (although we will cover far from everything). So whenever things are unclear, expositions didn't cover a proof, or you wish to read about related things, consult this book.

Besides this book, there are also Dolgachev's lecture notes [4], which are a down-to-earth introduction to the subject, Tevelev's notes [5], which emphasise the moduli point of view and discuss weighted projective spaces explicitly.

The underlying goal of the seminar is to make Mumford's classic [GIT] a less daunting read, but no-one will be forced to read this wonderful, yet hard-to-crack classic.

After this we will discuss some applications of geometric invariant theory.

## Schedule

- February 28
- Motivation for geometric invariant theory
- Algebraic groups and actions
- by Dennis Presotto and Theo Raedschelders, in G.017, at 10h00
- March 7
- Quotients of affine varieties, and projective quotients
- By Kevin De Laet, in G.017, at 10h00
- March 14
- The Hilbert–Mumford criterion
- Grassmannians
- by Pieter Belmans and Julia Ramos González, in G.017, at 10h00
- March 21
- no lecture
- March 28
- Application: McKay correspondence (see [6,7])
- by Kevin De Laet, in G.017, at 10h00
- April 4
- Application: Moduli of quiver representations
- by Theo Raedschelders, in G.017, at 10h00

## References

- [1]
- Schmitt, Alexander
*Geometric invariant theory and decorated principal bundles*
- European Mathematical Society,
**2008**
- [2]
- Mukai, Shigeru
*An introduction to invariants and moduli*
- Cambridge University Press,
**2003**(81)
- [3]
- Derksen, Harm and Kemper, Gregor
*Computational invariant theory*
- Springer,
**2002**(130)
- [4]
- Dolgachev, Igor
*Lectures on invariant theory* (online)
- Cambridge University Press,
**2003**(296)
- [5]
- Tevelev, Jenia
*Moduli spaces and invariant theory* (online)
- [6]
- Le Bruyn, Lieven
*Quotient singularities and the conifold algebra* (online)
- [7]
- Bocklandt, Raf
*Kleinian singularities* online)
- [GIT]
- Mumford, David
*Geometric invariant theory*
- Springer-Verlag,
**1965**(34), pp. vi+145

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