Birational invariants of algebraic groups and algebraic tori with finite group actions

具有有限群作用的代数群和代数环的双有理不变量

• 批准号: 229820-2010
• 负责人: Lemire, Nicole
• 金额: $ 1.75万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508766
• 关键词: Birational; invariants; algebraic; groups; tori

A classical but notoriously difficult problem in algebraic geometry is to classify algebraic varieties up to birational isomorphism - a natural equivalence relation on the set of algebraic varieties. Rational algebraic varieties form a distinguished class under this equivalence relation. The simplest known rational varieties are the linear varieties and the algebraic tori. There are natural actions of finite groups on linear varieties and algebraic tori coming from representations of the groups. One may ask when the orbit space of a linear variety or algebraic tori under a finite group action is rational. This question was first posed by Emmy Noether while she was doing work on the inverse Galois problem. One may also compare an algebraic torus with a finite group action to the associated linear variety with a finite group action by asking when the two are birationally isomorphic with a birational isomorphism which is equivariant with respect to the finite group action. This question of equivariant birational linearisation is related to the question of finding conjugacy classes of finite subgroups in the classical Cremona group - the group of birational isomorphisms of projective space. It is also related to the classical problem of determining whether an algebraic group is Cayley - or equivariantly birationally isomorphic to its Lie algebra - a problem first studied by Cayley. I previously did joint work on this problem with Vladimir Popov and Zinovy Reichstein. Among other things, we determined the set of simple algebraic groups over an algebraically closed field which are Cayley. I study analogues and generalisations of the rationality problem for algebraic tori under finite group actions and also the equivariant birational linearisation problem.

代数几何中一个经典但众所周知的难题是将代数簇分类到双有理同构-代数簇集合上的一种自然等价关系。 在这个等价关系下，有理代数簇形成了一个特殊的类。已知最简单的有理簇是线性簇和代数环面。有限群在线性簇和代数环面上的自然作用来自于群的表示。 人们可能会问，在有限群作用下，线性簇或代数环面的轨道空间何时是有理的。 这个问题首先提出的埃米诺特，而她正在做工作的逆伽罗瓦问题。 人们也可以比较代数环面与有限群作用相关的线性簇与有限群作用，问当两者是双有理同构与双有理同构，这是关于有限群作用等变。 这个问题的等变双有理线性化是有关的问题找到共轭类的有限子群在经典克雷莫纳组-该组的双有理同构的射影空间。 它也涉及到经典的问题，确定是否一个代数群是凯莱-或equivariantly双有理同构的李代数-一个问题首先研究凯莱。 我以前曾与弗拉基米尔波波夫和齐诺维·赖希施泰因共同研究过这个问题。 除其他事项外，我们确定了一套简单的代数群代数闭域是凯莱。我研究类似物和概括的合理性问题，代数环面有限群行动下，也是等变双有理线性化问题。