仿射簇自同构群的Borel子群

中文摘要 AI 辅助

我们研究了仿射簇X的自同构群的Borel子群及其相关的李代数。我们证明了Aut(X)的Borel子群集合与Aut(X)的极大嵌套幂幺子群集合之间存在双射对应。我们证明，若X是维数至多为3的阶乘簇，则每个极大嵌套幂幺子群的李代数具有我们称之为“积分”的特殊性质；同时我们也证明，当X为维数大于3的仿射空间时，该性质不成立。作为我们结果的一个应用，我们证明了Nagata自同构包含在唯一的Borel子群中。

英文摘要

We study Borel subgroups of the automorphism group of an affine variety X and their associated Lie algebras. We prove a bijective correspondence between the set of Borel subgroups of Aut(X) and the set of maximal nested unipotent subgroups of Aut(X). We show that if X is factorial of dimension at most 3, then the Lie algebra of every maximal nested unipotent subgroup has the special property of being what we call "integral"; we also show that this fails for X = affine space of dimension > 3. As an application of our results, we show that Nagata's automorphism is contained in a unique Borel subgroup.

发表机构

• University of Ottawa（渥太华大学）
• Università di Padova（帕多瓦大学）

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