Equivariant Embeddings of Algebraic Groups

G · D ρ1 = D = G · D ρ2 , from the discussion above. Hence Γ 1 (X, x 0 )/ ∼ → {D 1 , D 2 , . . . , D r } is
well-defined. To prove that it is a bijection, we use the following
Lemma 10. Let ρ ∈ Γ 1 (X, x 0 ) and let γ ρ denote the first lattice point in X ∗ (G) along ρ. Then v γρ
is the valuation of G · D ρ .
Proof. The ideal Γ(X, O(G · D ρ )) = {f ∈ k[X] : f = 0 on G · D ρ } is equal to k[X] ∩ m vγρ , where
m vγρ = {f ∈ k(G) : v γρ (f) > 0} is the maximal ideal of the valuation ring O vγρ = {f ∈ k(G) :
v γρ (f) ≥ 0}. For D ρ = T · z ρ , where z ρ = lim t→0 γ ρ (t)x 0 ∈ T , so that G · D ρ = G · z ρ as well. Thus,
if f ∈ k[X] ∩ m vγρ then f(z ρ ) = f(lim t→0 γ ρ (t)x 0 ) = lim t→0 t vγρ (f) u = 0, where u ∈ k[X] ∩ O v × γρ
,
since v γρ (f) > 0. Therefore k[X] ∩ m vγρ
⊂ Γ(X, O(G · D ρ )), where both are prime ideals in k[X]
and the latter is of height one. Hence they are equal, so the Lemma is proven.
We return to the proof of the Proposition. Suppose ρ 1 , ρ 2 ∈ Γ 1 (X, x 0 ) such that γ ρ1 ≁ γ ρ2 .
Then v γρ1 ≠ v γρ2 , so G · D ρ1 ≠ G · D ρ2 . Hence Γ 1 (X, x 0 )/ ∼ → {D 1 , D 2 , . . . , D r } is injective. It
is also surjective, as any of the prime divisors D i of X are closed G-subvarieties, and thus contain
the limit point of some one-parameter subgroup γ ∈ Γ(X, x 0 ) by Theorem 9. This γ is a oneparameter
subgroup of some maximal torus T of G, and so T ∩ D i ≠ ∅ is a T -stable divisor of T .
By Lemma 2, there is a prime divisor D ρ of T corresponding to a ray ρ ∈ Γ(X, x 0 ) ∩ X ∗ (T ) such
that D ρ ⊂ T ∩ D i . Thus G · D ρ ⊂ D i , from which we conclude G · D ρ = D i as before. Therefore,
Γ 1 (X, x 0 )/ ∼ → {D 1 , D 2 , . . . , D r } is a bijection.
3.2.2 States and strongly convex lattice cones
Our sets Γ(X, x 0 ) of one-parameter subgroups associated to an affine G-embedding are identical to
sets arising in geometric invariant theory as studied by Mumford [32], Kempf [27] and Rousseau [39].
In [27], Kempf studies affine G-schemes X and one-parameter subgroups γ of G with a specialization
in X. In his paper, Kempf describes such sets in terms of bounded, admissible states as follows.
Definition 9. A state Ξ is an assignment of a nonempty subset Ξ(R) ⊂ X ∗ (R) to each torus R of
G so that the image of Ξ(R 2 ) in X ∗ (R 1 ) under the restriction map X ∗ (R 2 ) → X ∗ (R 1 ) is equal to
Ξ(R 1 ) whenever R 1 ⊂ R 2 are tori of G.
48