Equivariant Embeddings of Algebraic Groups

and every toric variety for T arises in this way for some fan Σ.
Moreover, we note that since T ∼ = X {0}
⊂ X σ for all σ ∈ Σ (since {0} is a face of every
cone σ), the torus T is naturally an open dense subset of X Σ . Thus we have an open embedding
i Σ : T → X Σ . Therefore,
Theorem 4 ([28], Theorem I.6). The correspondence Σ ↦→ X Σ defines a bijection between fans
in X ∗ (T ) R and isomorphism classes of equivariant normal embeddings of T .
Many of the algebro-geometric features of a toric variety may be described by corresponding
features of its fan. For instance, the affine toric variety X σ is smooth if and only if the cone σ is
generated by part of a basis for the vector space X ∗ (T ) R . In this case, we say that the cone σ is
smooth. Thus any toric variety X Σ is smooth if and only if each cone σ ∈ Σ is smooth in this sense.
Additionally, we may detect when X Σ is a complete variety using the support of the fan Σ. The
support of Σ, denoted |Σ|, is defined to be
|Σ| = ⋃ σ∈Σ
σ.
We call a fan Σ complete if its support is the entire vector space X ∗ (T ) R . Complete fans are so
named because they are the ones that correspond to complete toric varieties.
2.1.3 The torus action
The definition of toric varieties we gave at the beginning of this section required them to be T -
varieties, so we now show how the torus acts on its toric varieties in terms of the cones and fan
which classify the embedding.
First, we give a coordinate-free description of the action, making use of the isomorphism of
functors Spec ∼ = Hom k ( , k). Then T = Spec k[X ∗ (T )] = Hom k (k[X ∗ (T )], k) can be thought of as
the collection of homomorphisms Hom(X ∗ (T ), G m ), while X σ = Spec k[σ ∨ ∩X ∗ (T )] = Hom k (k[σ ∨ ∩
X ∗ (T )], k) consists of k-algebra homomorphisms from k[σ ∨ ∩ X ∗ (T )] to k. The multiplication in k
gives us the ability to multiply such homomorphisms, and this determines the action of T on X σ ,
for
(t · x)(m) = t(m)x(m)
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