Geodesic
Classification of $G$-varieties of complexity~1
Izvestiya. Mathematics , Tome 61 (1997) no. 2, pp. 363-397

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We consider the problem of finding a combinatorial description of the algebraic varieties in a given birational class that admit an action of a reductive group $G$. This is a direct generalization of the theory of toric varieties. A general approach to this problem is described, and the solution is given for varieties in which the orbits in general position of a Borel subgroup $G$ have codimension 1 (varieties of complexity 1). 
@article{IM2_1997_61_2_a6,
     author = {D. A. Timashev},
     title = {Classification of $G$-varieties of complexity~1},
     journal = {Izvestiya. Mathematics },
     pages = {363--397},
     publisher = {mathdoc},
     volume = {61},
     number = {2},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_2_a6/}
}
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D. A. Timashev. Classification of $G$-varieties of complexity~1. Izvestiya. Mathematics , Tome 61 (1997) no. 2, pp. 363-397. http://geodesic.mathdoc.fr/item/IM2_1997_61_2_a6/