Equivariant Embeddings of Algebraic Groups

Theorem 3 ([28], Theorem I.1 ′ ). The correspondence σ ↦→ Spec k[σ ∨ ∩ X ∗ (T )] = X σ defines
a bijection between the set of strongly convex, rational, polyhedral cones in X ∗ (T ) R and the set of
affine toric varieties of T . Moreover, for γ ∈ X ∗ (T ), γ ∈ σ if and only if lim t→0 γ(t) exists in X σ .
Therefore, affine toric varieties X are all of the form X σ , where
σ = {γ ∈ X ∗ (T ) : lim
t→0
γ(t) exists in X}
is the intersection of a strongly convex, rational, polyhedral cone with X ∗ (T ).
2.1.2 Toric varieties from fans
In this subsection, we explain how affine toric varieties are glued together to form new toric varieties
using the classification of affine toric varieties given above. As we will only be considering strongly
convex, rational, polyhedral cones in the remainder of this chapter, we shall simply use the term
“cone” and suppress the adjectives.
Recall σ ∨ ⊂ X ∗ (R) R is the set of u ∈ X ∗ (T ) R such that 〈u, v〉 ≥ 0 for all v ∈ σ. For each
χ ∈ σ ∨ ∩ X ∗ (T ), define χ ⊥ = {x ∈ X ∗ (T ) R : 〈χ, x〉 = 0}. We call τ = σ ∩ χ ⊥ a face of σ and write
τ < σ.
Definition 4. A fan Σ in X ∗ (T ) R is a finite collection of (strongly convex, rational, polyhedral)
cones σ such that
1. every face of a cone of Σ is itself a cone of Σ;
2. if σ and σ ′ are cones in Σ, then σ ∩ σ ′ is a common face of σ and σ ′ .
Toric varieties for T are classified by fans as follows. Let Σ be a fan in X ∗ (T ) R . For each
σ ∈ Σ, we have an affine toric variety X σ as constructed above. We may naturally glue X σ and
X σ ′ together along their common open toric subvariety X σ∩σ ′, since if τ ≤ σ, then k[τ ∨ ∩ X ∗ (T )]
is the localization of k[σ ∨ ∩ X ∗ (T )] at χ, where τ = σ ∩ χ ⊥ , so X τ is an open subvariety of X σ . In
this way, we may form the toric variety associated to the fan Σ,
X Σ = ⋃ σ∈Σ
X σ ,
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