Equivariant Embeddings of Algebraic Groups

1.1 Toric varieties
The theory of toric varieties is a well-established and valuable class of examples for the study
of equivariant algebraic geometry. Toric varieties were introduced by Demazure [13] in the early
1970s, in which he classified all smooth toric varieties. The classification of all toric varieties was
accomplished three years later by Kempf–Knudsen–Mumford–Saint-Donat [28] using fans in the
vector space associated to the set of one-parameter subgroups of a torus.
We begin by recalling the definition of toric varieties. Suppose T is an algebraic torus. Since
we are assuming that our ground field k is algebraically closed, T ∼ = G r m for some r, where G m =
Spec k[x, x −1 ] is the multiplicative group scheme of k.
Definition 1. A toric variety is a normal variety X containing an algebraic torus T as an open
subset such that the translations of T on itself extend to give an action of T on X.
Example 1 (A toric variety and its fan). Consider T = G 2 m. Then X = P 2 is a toric variety for
T as it is normal, contains T as the open subvariety {[t 0 : t 1 : t 2 ] : t 0 , t 1 , t 2 ≠ 0}, and has T -action
given by [t 0 : t 1 : t 2 ] · [x 0 : x 1 : x 2 ] = [t 0 x 0 : t 1 x 1 : t 2 x 2 ]. There are seven open affine T -subvarieties
of X, which are glued together along common open affine subvarieties within X. This is recorded
by the “fan” below, consisting of seven cones: 3 are two dimensional (cones 1 , 2 , 3 ), 3 are one
dimensional (cones 4 , 5 , 6 ) and one is zero dimensional (cone 7 ).
• • 5 • 1
2 • • • •
• • 7 • 4
• • • • •
6 • • 3 •
The two dimensional cones each correspond to T -stable affine spaces in X, U i = {[x 0 : x 1 : x 2 ] :
2