Equivariant Embeddings of Algebraic Groups

Therefore, the classification of affine G-embeddings is equivalent to the characterization of such
subsets of X ∗ (G) that are obtained from affine G-embeddings. In order to classify admissible subsets
Γ, we will explore the properties of the sets Γ(X, x 0 ) for an arbitrary affine G-embedding X with
choice of base point x 0 in the next section. We record one such result here.
Recall from Section 2.1.4 that if σ is a strongly convex rational polyhedral cone in X ∗ (T ) R , then
σ(1) denotes the set of rays of σ, σ(1) = {τ < σ : dim τ = 1}. This is called the one-skeleton of
the cone. By Proposition 7, for each maximal torus T of G, Γ(X, x 0 ) ∩ X ∗ (T ) is a strongly convex
rational polyhedral cone in X ∗ (T ). So we may define the one-skeleton of the set Γ(X, x 0 ) to be
Γ 1 (X, x 0 ) = ⋃ T
[Γ(X, x 0 ) ∩ X ∗ (T )](1), (3.11)
which is the set of extremal rays of Γ(X, x 0 ).
Proposition 8. There is a bijection between Γ 1 (X, x 0 )/ ∼ and the finite set of prime divisors of
X contained in ∂X.
Proof. Write X = Ω ∪ D 1 ∪ D 2 ∪ · · · ∪ D r , where D 1 , . . . , D r are the finitely many G-stable prime
divisors of X in ∂X.
Suppose that ρ ∈ Γ 1 (X, x 0 ). Then there is a maximal torus T of G such that ρ ∈ (Γ(X, x 0 ) ∩
X ∗ (T ))(1) is a ray of the strongly convex rational polyhedral cone Γ(X, x 0 ) ∩ X ∗ (T ) corresponding
to T ⊂ X. From the description of T -stable divisors in toric varieties in Section 2.1.4, the ray ρ
corresponds to a prime divisor D ρ in T . Thus D ρ is an irreducible T -stable subvariety of T , so G · D ρ
is an irreducible G-stable subvariety of X, as both G and D ρ are irreducible. Moreover, G · D ρ is
contained in ∂X, so there is a G-stable prime divisor D i of X such that G · D ρ ⊆ D i ⊂ X. However,
codim X (G · D ρ ) = dim X − dim G · D ρ = dim G − (dim G + dim D ρ − dim T ) = dim T − dim D ρ =
codim T
(D ρ ) = 1. Thus, as G · D ρ is irreducible and of codimension one in X, and G · D ρ ⊂ D i X,
where D i is also irreducible and of codimension one, we conclude that G · D ρ = D i . Therefore,
there is a map Γ 1 (X, x 0 ) → {D 1 , D 2 , . . . , D r } given by ρ ↦→ G · D ρ .
Furthermore, this map respects the equivalence relation ∼ described in (3.4), for if γ ρ denotes the
first lattice point in X ∗ (T ) along ρ and γ ρ1 ∼ γ ρ2 , then lim t→0 γ ρ1 (t)x 0 and lim t→0 γ ρ2 (t)x 0 belong to
the same G-orbit in X, and hence to the same prime divisor D of X, as D is G-stable. Therefore,
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