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 Mat. Sb., 2008, Volume 199, Number 3, Pages 3–24 (Mi msb3836)

Affine toric $\operatorname{SL}(2)$-embeddings

S. A. Gaifullin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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 $0<p/q\le1$ 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.

DOI: https://doi.org/10.4213/sm3836

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English version:
Sbornik: Mathematics, 2008, 199:3, 319–339

Bibliographic databases:

UDC: 512.745.2
MSC: Primary 14M25; Secondary 14L30, 14M17, 52B20

Citation: S. A. Gaifullin, “Affine toric $\operatorname{SL}(2)$-embeddings”, Mat. Sb., 199:3 (2008), 3–24; Sb. Math., 199:3 (2008), 319–339

Citation in format AMSBIB
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This publication is cited in the following articles:
1. I. V. Arzhantsev, S. A. Gaifullin, “Cox rings, semigroups and automorphisms of affine algebraic varieties”, Sb. Math., 201:1 (2010), 1–21
2. Arzhantsev I., Gaifullin S., “Homogeneous toric varieties”, J. Lie Theory, 20:2 (2010), 283–293
3. Arzhantsev I., Liendo A., “Polyhedral divisors and SL$_2$-actions on affine T-varieties”, Mich. Math. J., 61:4 (2012), 731–762
4. Arzhantsev I. Flenner H. Kaliman S. Kutzschebauch F. Zaidenberg M., “Flexible Varieties and Automorphism Groups”, Duke Math. J., 162:4 (2013), 767–823
5. A. M. Meirmanov, A. A. Gerus, S. A. Gritsenko, “Usrednennye modeli izotermicheskoi akustiki v konfiguratsii uprugoe telo–porouprugaya sreda”, Matem. modelirovanie, 28:12 (2016), 3–19
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