Equivariant Embeddings of Algebraic Groups

2.1 Toric varieties as group embeddings
Let T = G r m be a split algebraic torus defined over k. A toric variety X is a normal T -variety which
contains T as a dense open subvariety so that the action of T on X extends the regular action of
T on itself by multiplication. As the rest of this paper hinges upon the ideas originally used to
classify toric varieties in [28], we present a brief overview of those ideas here. Other references for
this subject are [12], [33], [17]. We borrow much from their treatment of the topic and refer to
them for proofs.
Let T be an algebraic torus and X a toric variety for T , both defined over an algebraically
closed field k. By a theorem of Sumihiro [42], there is a covering of X by T -stable, affine, open
subvarieties which are therefore also toric varieties.
Hence every toric variety may be obtained
by gluing together affine toric varieties, so the classification of toric varieties reduces to the two
problems of classifying affine toric varieties and describing how affine toric varieties may be glued
together. We review the solutions to these problems in the following subsections.
2.1.1 Affine toric varieties and one-parameter subgroups
Suppose X is an affine toric variety, say X = Spec A. The open embedding T → X corresponds to
an injective homomorphism A → k[T ] ∼ = k[x 1 , x −1
1 , . . . , x r, x −1
r ]. The action of T on X induces an
action of T on A as well via t · f : x ↦→ f(t −1 x) for all f ∈ A.
Let X ∗ (T ) denote the set of one-parameter subgroups of T , X ∗ (T ) = Hom(G m , T ). This is a
free abelian group, which is dual to the group of characters of T , X ∗ (T ), with respect to the perfect
pairing 〈·, ·〉 : X ∗ (T ) × X ∗ (T ) → Z determined by
χ(γ(a)) = a 〈χ,γ〉 (2.1)
for all χ ∈ X ∗ (T ), γ ∈ X ∗ (T ), a ∈ k × .
Let S be a semi-group in X ∗ (T ), which is finitely generated. Set k[S] equal to the k-subalgebra
of k[T ] = k[X ∗ (T )] generated by characters χ ∈ S. Since S is finitely generated, k[S] is a k-algebra
of finite type. If S generates X ∗ (T ) as a group, then the inclusion k[S] ⊂ k[T ] induces an equivariant
embedding T ⊂ Spec k[S], and every equivariant T -embedding arises in this way. Therefore,
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