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摘要翻译：
1973年，V.L.Popov对仿射SL(2)-嵌入进行了分类。他证明了正规仿射三维簇X上的局部传递SL(2)-作用是由一对(P/Q，r)唯一决定的，其中0<P/Q<=1是一个未取消的分数，r是一个正整数。这里r是一个泛型点的稳定器的阶。本文证明了簇X是环面的，即允许代数环面的局部传递作用，当且仅当r可被Q-P整除。为此，我们证明了仿射G/H-嵌入是toric的以下充要条件。假定X是正规仿射簇，G是正则作用于X的单连通半单代数群，H是G的闭子群，使得特征群$\Mathfrak{X}(H)$是有限的，G/H->X是稠密开等变嵌入。则X是toric当且仅当存在一个拟点T和一个$(G\乘以T)$-模V，使得$X\stackrel{G}{\cong}V//T$。在证明中起关键作用的是D.考克斯的构造。
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英文标题：
《Affine Toric SL(2)-embeddings》
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作者：
Sergey A. Gaifullin
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
In 1973 V.L.Popov classified affine SL(2)-embeddings. He proved that a locally transitive SL(2)-action on a normal affine three-dimensional variety X is uniquely determined by a pair (p/q, r), where 0<p/q<=1 is an uncancelled fraction and r is a positive integer. Here r is the order of the stabilizer of a generic point. In this paper we show that the variety X is toric, i.e. admits a locally transitive action of an algebraic torus, if and only if r is divisible by q-p. To do this we prove the following necessary and sufficient condition for an affine G/H-embedding to be toric. Suppose X is a normal affine variety, G is a simply connected semisimple algebraic group acting regularly on X, H is a closed subgroup of G such that the character group $\mathfrak{X}(H)$ is finite and G/H -> X is a dense open equivariant embedding. Then X is toric if and only if there exist a quasitorus T and a $(G\times T)$-module V such that $X\stackrel{G}{\cong} V//T$. The key role in the proof plays D. Cox's construction.
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PDF链接：
https://arxiv.org/pdf/0801.0162
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