Equivariant Embeddings of Algebraic Groups

schemes. However, there is a largest open subset X G ⊆ X such that G acts on X G as an algebraic
group.
In the paper [30], G is assumed to be reductive and of simply connected type (that is, G is the
direct product of a torus and a simply connected semisimple group). Furthermore, all G-models
are assumed to be normal.
The following theorem of Sumihiro is the basic tool for the local study of G-varieties.
Theorem 1 ([42], Lemma 8). Let G be a connected linear algebraic group and let X be a normal
G-variety. Then any point x ∈ X has a G-stable neighborhood X 0 ⊆ X admitting an equivariant
locally closed embedding in a projective space.
Therefore, every regular action of a connected linear algebraic group on a normal variety is
obtained by patching finitely many linear actions on normal quasi-projective varieties. Thus, if we
are interested in local properties of a G-model X in a neighborhood of a point x or a subvariety Y ,
we may assume that X is quasi-projective.
Let X be a quasihomogeneous G-variety with open orbit G/H and k(X) = k(G/H) = K. Let
B be a Borel subgroup of G. We introduce the following terminology.
The complexity of the variety X or of the field K, denoted c(X) or c(K) respectively, is the
transcendence degree of K B .
Hence c(X) is equal to the codimension of a B-orbit in general
position on X.
The set of G-stable discrete Q-valued valuations of K/k that are geometric (that is, correspond
to prime divisors on a suitable model of K) is denoted V = V(K). Elements of V are called G-
valuations. Let D = D(G/H) be the set of irreducible subvarieties of G/H of codimension one that
are not G-stable. The valuation corresponding to a divisor D ∈ D is denoted by v D . Prime divisors
that are B-stable but not G-stable are called B-divisors. The set of B-divisors is denoted D B .
Suppose X is a G-model of K, so k(X) = K. Then the complexity measures the codimension
of a B-orbit in general position on X. When the complexity is ≤ 1, D B is particularly simple. If
c(X) = 0, G has an open orbit in X, say G/H. In this case, D B is the finite set of the irreducible
components of the complement of the open orbit of B in G/H. Suppose c(X) = 1. Let G/H be a
sufficiently generic orbit and let U be the open subset of G/H formed of B-orbits of codimension
1. Then D ∈ D B implies either D is the closure of a B-orbit in U or D is a component of the
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