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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - D. A. Timashev
TI  - Classification of $G$-varieties of complexity~1
JO  - Izvestiya. Mathematics 
PY  - 1997
SP  - 363
EP  - 397
VL  - 61
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1997_61_2_a6/
LA  - en
ID  - IM2_1997_61_2_a6
ER  - 
%0 Journal Article
%A D. A. Timashev
%T Classification of $G$-varieties of complexity~1
%J Izvestiya. Mathematics 
%D 1997
%P 363-397
%V 61
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1997_61_2_a6/
%G en
%F IM2_1997_61_2_a6
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/

[1] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[2] Vinberg E. B., Popov V. L., “Teoriya invariantov”, Itogi nauki i tekhn. Sovr. probl. mat. Fund. napr., 55, VINITI, M., 1989, 137–309 | MR

[3] Danilov V. I., “Geometriya toricheskikh mnogoobrazii”, UMN, 33:2 (1978), 85–134 | MR | Zbl

[4] Popov V. L., “Kvaziodnorodnye affinnye algebraicheskie mnogoobraziya gruppy $\operatorname {SL}(2)$”, Izv. AN SSSR. Ser. matem., 37:4 (1973), 792–832 | MR | Zbl

[5] Springer T. A., “Lineinye algebraicheskie gruppy”, Itogi nauki i tekhn. Sovr. probl. mat. Fund. napr., 55, VINITI, M., 1989, 6–136

[6] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR | Zbl

[7] Brion M., “Parametrization and embeddings of a class of homogeneous spaces”, Contemporary Mathematics, 131:3 (1992), 353–360 | MR | Zbl

[8] Hartshorn R., “Ample vector bundles”, Publ. Math. IHES, 1966, no. 29, 63–94 | MR | Zbl

[9] Knop F., Kraft H., Luna D., Vust Th., “Local properties of algebraic group actions”, Algebraische Transformationsgruppen und Invariantentheorie, DMV Seminar 13, eds. H. Kraft, P. Slodowy, T. A. Springer, Birkhäuser, Basel–Boston–Berlin, 1989, 77–88 | MR

[10] Kempf G., Knudson F., Mumford D., Saint-Donat B., Toroidal embeddings, I, Lect. Notes Math., 339, 1973, 209 pp. | MR | Zbl

[11] Knop F., “The Luna–Vust theory of spherical embeddings”, Proc. of the Hyderabad Conf. on Algebraic Groups, Manoj Prakashan, Madras, 1991, 225–249 | MR | Zbl

[12] Knop F., “Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind”, Math. Ann., 295 (1993), 333–363 | DOI | MR | Zbl

[13] Knop F., “Über Hilbert vierzehntes problem für Varietäten mit Kompliziertheit eins”, Math. Zeit., 213 (1993), 33–36 | DOI | MR | Zbl

[14] Luna D., Vust Th., “Plongements d'espaces homogenes”, Comment. Math. Helv., 58 (1983), 186–245 | DOI | MR | Zbl

[15] Moser-Jauslin L., Normal embeddings of $\operatorname {SL}(2)/\Gamma $, Thesis, Geneva, 1987

[16] Oda T., Convex bodies and algebraic geometry: an introduction to the theory of toric varieties, Springer-Verlag, Berlin, 1988, 212 pp. | MR | Zbl

[17] Sumihiro H., “Equivariant completion”, J. Math. Kyoto Univ., 14:1 (1974), 1–28 | MR | Zbl

[18] Wargane B., Détermination des valuations invariantes de $\operatorname {SL}(3)/T$, Thèse, Grenoble, 1982, 30 pp.