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On homogeneity of two-orbit varieties with respect to the full group of automorphisms. (Russian) Zbl 0698.14054
Vopr. Teor. Grupp Gomologicheskoj Algebry 8, 141-154 (1988).
Theorem: Let X be a complete non-singular complex algebraic variety, and let G be a connected linear algebraic group which acts regularly on X. Let X be the union of two orbits of group G and let the closed orbit A have codimension 1. Then the following conditions are equivalent:
(a) $$H^ 0(A,{\mathbb{N}}_{X| A})\neq 0$$, where $${\mathbb{N}}_{X| A}$$ is the normal bundle to A in X;
(b) X is homogeneous under Aut(X);
(c) $$X=X_ 1\times X_ 2$$ and the image of G in Aut(X) is a direct product $$G_ 1\times G_ 2$$, where $$G_ 1$$ acts transitively on $$X_ 1$$, $$G_ 2$$ acts transitively on $$X_ 2$$ and some alternative formulated in the paper takes place.
Reviewer: N.I.Osetinski

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations
##### Keywords:
action of linear group; automorphism group