Equivariant Embeddings of Algebraic Groups

x i ≠ 0} for i = 0, 1, 2, the one dimensional cones correspond to the T -stable affine subvarieties
U ij = {[x 0 : x 1 : x 2 ] : x i , x j ≠ 0}, and the zero dimensional cone corresponds to T itself, viewed as
the subvariety {[t 0 : t 1 : t 2 ] : t 0 , t 1 , t 2 ≠ 0}.
Toric varieties play an important role both in algebraic and combinatorial geometry due to the
nature of their classification. In particular, toric varieties are useful for the calculation of the number
of lattice points in polytopes in combinatorial geometry. Their applications in algebraic geometry
are numerous as well, especially as a “source of interesting, yet manageable examples” [17]. Toric
varieties also play a critical role in mirror symmetry [11].
Due to the relationship between algebraic geometry and combinatorial geometry afforded by
the classification of toric varieties, many have recently preferred to introduce and even define toric
varieties not in terms of the torus within, but instead in terms of their combinatorial classification.
However, this separation from the roots of the subject has had negative consequences. By studying
toric varieties as varieties whose transition functions are monomials or via Cox’s homogeneous coordinate
ring [10], as many now do, the role toric varieties play as part of the equivariant classification
program has been diminished.
In the introduction to [12], after motivating the importance of toric varieties and their relationship
with combinatorial geometry, Danilov returns to their original definition and proposes “An
extension of this theory would seem possible, in which the torus T is replaced by an arbitrary
reductive group G” ([12], page 101).
1.2 Luna–Vust theory and spherical varieties
In [30], Luna and Vust began a program to describe up to G-isomorphism varieties admitting an
action of an algebraic group G. A birational classification of G-varieties (i.e., classifying how G acts
on fields) may be obtained using Galois cohomology [38]. The second part of the problem requires
one to describe all G-models of a given function field K with G acting birationally on K.
Let K be a field of finite type over k. A model of K is a variety X with k(X) = K. We remark
that all models of K may be glued together to form a large (non-Noetherian, non-separated) scheme
X = X(K/k). The group G acts on X in a natural way, but this is not an action in the category of
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