Basics of GIT, part I -- Lloyd West, March 4, 2011

<p>
Geometric invariant theory is a technique for taking quotients by group actions in algebraic geometry. It was 
invented by Mumford in the course of his construction of the moduli space of curves. Despite a reputation for 
being technical, the theory is in fact very natural and has may applications apart from moduli problems; toric 
varieties are GIT quotients, for example. In this talk, I will introduce the basic definitions and 
constructions. I will follow the books by <a 
href="http://books.google.com/books?id=B9S_0iPvfrUC&lpg=PP1&dq=Dolgachev&pg=PP1#v=onepage&q&f=false">Dolgachev</a>
and <a 
href="http://books.google.com/books?id=dFlv3zn_2-gC&lpg=PP1&dq=David%20Mumford&pg=PP1#v=onepage&q&f=false">Mumford-Fogarty-Kirwan</a>.

</p>

<p> Topics to be covered (this week and next): 
<ol>
<li>Motivation: Invariants and Moduli; 
<li>Definition of 'quotient' in algebraic geometry
<li>Affine quotients
<li>Linearisations
<li>The projective quotient; stability
<li>Criteria for stability: Hilbert-Mumford Criterion
</ol>
</p>