Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
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for all t ∈ T and x ∈ X σ (where t(m) = m(t) and x(m) = m(x)).<br />
We may also describe this action in terms <strong>of</strong> coordinates for the affine toric variety X σ . Let<br />
(a 1 , . . . , a s ) be a system <strong>of</strong> generators for the monoid σ ∨ ∩ X ∗ (T ). Given a basis {x i = n i ⊗ 1 : i =<br />
1, . . . , r} for X ∗ (T ) R , each a i may be written in the form a i = (αi 1, . . . , αr i ) with αj i ∈ Z and t ∈ T<br />
is written as t = (t 1 , . . . , t r ) with t j ∈ G m (k). A point x ∈ X σ is written x = (x 1 , . . . , x s ). Then<br />
the action T × X σ → X σ <strong>of</strong> T on the affine subvariety X σ is<br />
(t, x) ↦→ (t a 1<br />
x 1 , . . . , t as x s )<br />
where t a i<br />
= t α1 i<br />
1 · · · tαr i<br />
r<br />
to give an action on the entire variety X Σ .<br />
∈ G m (k). The actions <strong>of</strong> T on each <strong>of</strong> the X σ , for σ ∈ Σ, clearly glue together<br />
As the action <strong>of</strong> T on a toric variety X Σ can be described in terms <strong>of</strong> the fan, the orbit structure<br />
<strong>of</strong> X Σ may also be characterized using Σ. In particular, for each cone σ ∈ Σ, there is a base point<br />
z σ ∈ X Σ which is the limit point <strong>of</strong> any <strong>of</strong> the one-parameter subgroups in the relative interior <strong>of</strong><br />
the cone σ. That is, z σ = lim t→0 γ(t) for any γ ∈ σ − ⋃ τ