Equivariant Embeddings of Algebraic Groups

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