Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
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Proposition 3 ([28], Proposition I.1). The correspondence S ↦→ Spec k[S] defines a bijection<br />
between the set <strong>of</strong> finitely generated semi-groups S ⊂ X ∗ (T ) which generate X ∗ (T ) as a group and<br />
the set <strong>of</strong> isomorphism classes <strong>of</strong> equivariant affine embeddings <strong>of</strong> T . Moreover, the morphisms<br />
<strong>of</strong> equivariant affine embeddings correspond (in a contravariant way) to the inclusions between<br />
semi-groups contained in X ∗ (T ).<br />
We say a semi-group S ⊂ X ∗ (T ) is saturated if χ n ∈ S for some positive n ≥ 1 and χ ∈ X ∗ (T )<br />
implies that χ ∈ S. This saturated condition corresponds to the normality <strong>of</strong> the associated torus<br />
embedding, so that<br />
Theorem 2 ([28], Theorem I.1). The correspondence S ↦→ Spec k[S] defines a bijection between<br />
the set <strong>of</strong> finitely generated semi-groups S ⊂ X ∗ (T ) which generate X ∗ (T ) as a group and are<br />
saturated and the set <strong>of</strong> equivariant normal affine embeddings <strong>of</strong> T .<br />
Hence, affine toric varieties correspond to saturated semi-groups S <strong>of</strong> X ∗ (T ) which generate<br />
X ∗ (T ) as a group. Now we classify all such saturated semi-groups <strong>of</strong> X ∗ (T ).<br />
Regard X ∗ (T ) as a Z-lattice in the real vector space X ∗ (T ) R = X ∗ (T ) ⊗ Z R.<br />
Definition 3. Let σ ⊂ X ∗ (T ) R . We call σ a convex rational polyhedral cone if σ = { ∑ N<br />
i=1 λ ix i :<br />
λ i ∈ R ≥0 , for all i} for some finite collection <strong>of</strong> elements x i ∈ X ∗ (T ).<br />
We say σ is a strongly<br />
convex rational polyhedral cone if it is a convex rational polyhedral cone and it does not contain<br />
any non-zero linear subspace <strong>of</strong> X ∗ (T ) R .<br />
We associate to a strongly convex rational polyhedral cone σ its dual cone, σ ∨ ⊂ X ∗ (T ) R , which<br />
is defined as σ ∨ = {u ∈ X ∗ (T ) R : 〈u, v〉 ≥ 0 for all v ∈ σ}. Then σ ∨ is a convex, rational, polyhedral<br />
cone in X ∗ (T ) R and (σ ∨ ) ∨ = σ. The correspondence σ ↦→ σ ∨ ∩ X ∗ (T ) defines a bijection between<br />
the set <strong>of</strong> strongly convex rational polyhedral cones and the set <strong>of</strong> finitely generated semi-groups<br />
in X ∗ (T ) which are saturated and generate X ∗ (T ) as a group by<br />
Lemma 1 ([28], Gordan’s Lemma). Given a finite set <strong>of</strong> homogeneous linear integral inequalities,<br />
the semi-group <strong>of</strong> integral solutions is finitely generated.<br />
Therefore, we obtain the classification <strong>of</strong> affine toric varieties below.<br />
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