Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
G · D ρ1 = D = G · D ρ2 , from the discussion above. Hence Γ 1 (X, x 0 )/ ∼ → {D 1 , D 2 , . . . , D r } is<br />
well-defined. To prove that it is a bijection, we use the following<br />
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<br />
is the valuation <strong>of</strong> G · D ρ .<br />
Pro<strong>of</strong>. 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<br />
m vγρ = {f ∈ k(G) : v γρ (f) > 0} is the maximal ideal <strong>of</strong> the valuation ring O vγρ = {f ∈ k(G) :<br />
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,<br />
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<br />
,<br />
since v γρ (f) > 0. Therefore k[X] ∩ m vγρ<br />
⊂ Γ(X, O(G · D ρ )), where both are prime ideals in k[X]<br />
and the latter is <strong>of</strong> height one. Hence they are equal, so the Lemma is proven.<br />
We return to the pro<strong>of</strong> <strong>of</strong> the Proposition. Suppose ρ 1 , ρ 2 ∈ Γ 1 (X, x 0 ) such that γ ρ1 ≁ γ ρ2 .<br />
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<br />
is also surjective, as any <strong>of</strong> the prime divisors D i <strong>of</strong> X are closed G-subvarieties, and thus contain<br />
the limit point <strong>of</strong> some one-parameter subgroup γ ∈ Γ(X, x 0 ) by Theorem 9. This γ is a oneparameter<br />
subgroup <strong>of</strong> some maximal torus T <strong>of</strong> G, and so T ∩ D i ≠ ∅ is a T -stable divisor <strong>of</strong> T .<br />
By Lemma 2, there is a prime divisor D ρ <strong>of</strong> T corresponding to a ray ρ ∈ Γ(X, x 0 ) ∩ X ∗ (T ) such<br />
that D ρ ⊂ T ∩ D i . Thus G · D ρ ⊂ D i , from which we conclude G · D ρ = D i as before. Therefore,<br />
Γ 1 (X, x 0 )/ ∼ → {D 1 , D 2 , . . . , D r } is a bijection.<br />
3.2.2 States and strongly convex lattice cones<br />
Our sets Γ(X, x 0 ) <strong>of</strong> one-parameter subgroups associated to an affine G-embedding are identical to<br />
sets arising in geometric invariant theory as studied by Mumford [32], Kempf [27] and Rousseau [39].<br />
In [27], Kempf studies affine G-schemes X and one-parameter subgroups γ <strong>of</strong> G with a specialization<br />
in X. In his paper, Kempf describes such sets in terms <strong>of</strong> bounded, admissible states as follows.<br />
Definition 9. A state Ξ is an assignment <strong>of</strong> a nonempty subset Ξ(R) ⊂ X ∗ (R) to each torus R <strong>of</strong><br />
G so that the image <strong>of</strong> Ξ(R 2 ) in X ∗ (R 1 ) under the restriction map X ∗ (R 2 ) → X ∗ (R 1 ) is equal to<br />
Ξ(R 1 ) whenever R 1 ⊂ R 2 are tori <strong>of</strong> G.<br />
48