Equivariant Embeddings of Algebraic Groups

Chapter 1
Introduction
A basic problem in algebraic geometry is the study of algebraic actions. Even actions with a dense
orbit are not well understood. A quasihomogeneous variety is a normal G-variety X possessing an
open orbit isomorphic to G/H, for some closed subgroup H of G. Toric varieties are an important
family of quasihomogeneous varieties, where G is an algebraic torus and H is its trivial subgroup.
A partial classification of quasihomogeneous varieties was obtained in the important paper of Luna
and Vust [30], written twenty years ago.
Their classification seeks to generalize that of toric
varieties, but it is only feasible in special cases.
The goal of this dissertation is to solve the
equivariant classification problem for one case not covered in [30], namely the classification of
quasihomogeneous varieties in which H is trivial.
In this introduction, we review the history of and some of the results that are essential to our
problem. As mentioned above, toric varieties are the natural place to begin, so we discuss them
in the first section. After that, we present a short overview of Luna and Vust’s paper [30] and
some of the work that has been done in an attempt to exploit the Luna–Vust classification in cases
that initially appear to be outside the scope of their original work. This chapter concludes with an
overview of our results.
First, we establish a few conventions that will be used throughout this thesis and notation
that we use freely hereafter. We will always work over a ground field k, which we assume to be
algebraically closed and of characteristic zero. The term variety will refer to a separated, integral
scheme of finite type over k. All algebraic groups are assumed to be linear and defined over k, and
will be denoted by letters such as G and H. In particular, G will refer to a connected reductive
linear algebraic group defined over k, unless otherwise stipulated.
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