Geodesic
Affine toric $\operatorname{SL}(2)$-embeddings
Sbornik. Mathematics, Tome 199 (2008) no. 3, pp. 319-339 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the theory of affine $\operatorname{SL}(2)$-embeddings, which was constructed in 1973 by Popov, a locally transitive action of the group $\operatorname{SL}(2)$ on a normal affine three-dimensional variety $X$ is determined by a pair $(p/q,r)$, where $0

is a rational number written as an irreducible fraction and called the height of the action, while $r$ is a positive integer that is the order of the stabilizer of a generic point. In the present paper it is shown that the variety $X$ is toric, that is, it admits a locally transitive action of an algebraic torus if and only if the number $r$ is divisible by $q-p$. For that, the following criterion for an affine $G/H$-embedding to be toric is proved. Let $X$ be a normal affine variety, $G$ a simply connected semisimple group acting regularly on $X$, and $H\subset G$ a closed subgroup such that the character group $\mathfrak X(H)$ of the group $H$ is finite. If an open equivariant embedding $G/H\hookrightarrow X$ is defined, then $X$ is toric if and only if there exist a quasitorus $\widehat T$ and a $(G\times\widehat T)$-module $V$ such that $X\stackrel G\cong V/\!/\widehat T$. In the substantiation of this result a key role is played by Cox's construction in toric geometry. Bibliography: 12 titles. 

@article{SM_2008_199_3_a0,
     author = {S. A. Gaifullin},
     title = {Affine toric $\operatorname{SL}(2)$-embeddings},
     journal = {Sbornik. Mathematics},
     pages = {319--339},
     year = {2008},
     volume = {199},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_3_a0/}
}
TY  - JOUR
AU  - S. A. Gaifullin
TI  - Affine toric $\operatorname{SL}(2)$-embeddings
JO  - Sbornik. Mathematics
PY  - 2008
SP  - 319
EP  - 339
VL  - 199
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2008_199_3_a0/
LA  - en
ID  - SM_2008_199_3_a0
ER  - 
%0 Journal Article
%A S. A. Gaifullin
%T Affine toric $\operatorname{SL}(2)$-embeddings
%J Sbornik. Mathematics
%D 2008
%P 319-339
%V 199
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2008_199_3_a0/
%G en
%F SM_2008_199_3_a0
S. A. Gaifullin. Affine toric $\operatorname{SL}(2)$-embeddings. Sbornik. Mathematics, Tome 199 (2008) no. 3, pp. 319-339. http://geodesic.mathdoc.fr/item/SM_2008_199_3_a0/

[1] W. Fulton, Introduction to toric varieties. The William H. Roever lectures in geometry, Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[2] D. A. Cox, “The homogeneous coordinate ring of a toric variety”, J. Algebraic Geom., 4:1 (1995), 17–50 | MR | Zbl

[3] H. Kraft, V. L. Popov, “Semisimple group actions on the three dimensional affine space are linear”, Comment. Math. Helv., 60:1 (1985), 466–479 | DOI | MR | Zbl

[4] V. L. Popov, “Quasihomogeneous affine algebraic varieties of the group $SL(2)$”, Math. USSR-Izv., 7:4 (1973), 793–831 | DOI | MR | Zbl | Zbl

[5] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects Math., D1, Vieweg, Braunschweig, 1984 | MR | MR | Zbl | Zbl

[6] D. I. Panyushev, “The canonical module of a quasihomogeneous normal affine $SL_2$-variety”, Math. USSR-Sb., 73:2 (1992), 569–578 | DOI | MR | Zbl

[7] F. Berchtold, J. Hausen, “Homogeneous coordinates for algebraic varieties”, J. Algebra, 266:2 (2003), 636–670 | DOI | MR | Zbl

[8] Y. Hu, S. Keel, “Mori dream spaces and GIT”, Michigan Math. J., 48:1 (2000), 331–348 | DOI | MR | Zbl

[9] F. Knop, H. Kraft, Th. Vust, “The Picard group of a $G$-variety”, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., 13, Birkhäuzer, 1989, 77–87 | MR | Zbl

[10] V. L. Popov, “Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles”, Math. USSR-Izv., 8:2 (1974), 301–327 | DOI | MR | Zbl | Zbl

[11] V. Alexeev, M. Brion, “Toric degenerations of spherical varieties”, Selecta Math. (N.S.), 10:4 (2004), 453–478 | DOI | MR | Zbl

[12] V. L. Popov, “Contraction of the actions of reductive algebraic groups”, Math. USSR-Sb., 58:2 (1987), 311–335 | DOI | MR | Zbl | Zbl