By D.A. Timashev

Homogeneous areas of linear algebraic teams lie on the crossroads of algebraic geometry, thought of algebraic teams, classical projective and enumerative geometry, harmonic research, and illustration conception. through regular purposes of algebraic geometry, so one can clear up quite a few difficulties on a homogeneous house, it really is ordinary and precious to compactify it whereas keeping an eye on the crowd motion, i.e., to contemplate equivariant completions or, extra in most cases, open embeddings of a given homogeneous area. Such equivariant embeddings are the topic of this booklet. We concentrate on the category of equivariant embeddings by way of yes facts of "combinatorial" nature (the Luna-Vust idea) and outline of assorted geometric and representation-theoretic homes of those kinds in line with those info. the category of round kinds, intensively studied over the past 3 a long time, is of distinctive curiosity within the scope of this publication. round types comprise many classical examples, resembling Grassmannians, flag types, and types of quadrics, in addition to famous toric forms. we've tried to hide lots of the vital concerns, together with the new giant development received in and round the idea of round varieties.

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**Example text**

9. Let X = P(S2 kn∗ ) be the space of quadrics in Pn−1 , where char k = 2. Then G = GLn (k) acts on X by linear variable changes with the orbits O1 , . . , On , where Or is the set of quadrics of rank r, and O1 ⊂ · · · ⊂ On = X. Choose the standard Borel subgroup B ⊆ G of upper-triangular matrices and the standard maximal torus T ⊆ B of diagonal matrices. (1) Put Y = O1 , the unique closed G-orbit in X, which consists of double hyperplanes. 6, we have V = S2 kn∗ v = x12 , V ∗ = S2 kn ω = e21 .

Affine), then G/K is also complete (resp. affine). 9 below. 12(4). 2 Projective Homogeneous Spaces. 3. G/H is projective if and only if H is parabolic. Proof. 2]. Hence H ⊇ g−1 Bg is parabolic. To prove the converse, consider a faithful representation G : V . The induced action of G on the variety of complete flags in V has a closed orbit. Its stabilizer B is solvable, and we may assume that B ⊆ H. 1, G/B is complete, and hence G/H is complete. 3 Affine Homogeneous Spaces. A group-theoretical characterization of affine homogeneous spaces is not known at the moment.

Let X be a locally linearizable irreducible G-variety and let Y ⊆ X be a G-stable irreducible subvariety. 6. (2) Y˚ = Y ∩ X˚ = 0, / and the kernel L0 of the natural action L = P/Pu : Y˚ /Pu contains L . (3) The quotient torus A = L/L0 acts on Y˚ /Pu freely, so that Y˚ /Pu A ×C, with A acting on C trivially. In characteristic zero, Y ∩ Z A ×C. In particular, let P = P(X) be the smallest stabilizer of a B-stable divisor in X. Then there exists a T -stable (L-stable if char k = 0) locally closed affine subset Z ⊆ X such ˚ Z → X/P ˚ u that X˚ = PZ is an open affine subset of X, the natural maps Pu × Z → X, ˚ are finite and surjective (are isomorphisms if char k = 0), and X/Pu A ×C, where A = L/L0 is a quotient torus of L acting on C trivially.