Geodesic
Classification of affine homogeneous spaces of complexity~one
Sbornik. Mathematics, Tome 195 (2004) no. 6, pp. 765-782

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The complexity of an action of a reductive algebraic group $G$ on an algebraic variety $X$ is the codimension of a generic $B$-orbit in $X$, where $B$ is a Borel subgroup of $G$. Affine homogeneous spaces $G/H$ of complexity 1 are classified in this paper. These results are the natural continuation of the earlier classification of spherical affine homogeneous spaces, that is, spaces of complexity 0. 
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     author = {I. V. Arzhantsev and O. V. Chuvashova},
     title = {Classification of affine homogeneous spaces of complexity~one},
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I. V. Arzhantsev; O. V. Chuvashova. Classification of affine homogeneous spaces of complexity~one. Sbornik. Mathematics, Tome 195 (2004) no. 6, pp. 765-782. http://geodesic.mathdoc.fr/item/SM_2004_195_6_a0/