Abstracts Category : Other

Birational models of geometric invariant theory quotients

by Elliot Cheung

In this thesis, we study the problem of finding birational models of projective G-varieties with tame stabilizers. This is done with linearizations, so that each birational model may be considered as a (modular) compactification of an orbit space (of properly stable points).The main portion of the thesis is a re-working of a result in Kirwan's paper "Partial Desingularisations of Quotients of Nonsingular Varieties and their Betti Numbers", written in a purely algebro-geometric language. As such, the proofs are novel and require the Luna Slice Theorem as their primary tool.Chapter 1 is devoted to preliminary material on Geometric Invariant Theory and the Luna Slice Theorem.In Chapter 2, we present and prove a version of "Kirwan's procedure". This chapter concludes with an outline of some differences between the current thesis and Kirwan's original paper.In Chapter 3, we combine the results from Chapter 2 and a result from a paper by Reichstein and Youssin to provide another type of birational model with tame stabilizers (again, with respect to an original linearization).