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: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=524245
• 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.

代数几何中的一个经典但又非常困难的问题是将代数变体分类到双同构-代数变体集合上的一种自然等价关系。在这种等价关系下，有理代数变体形成了一个特殊的类别。已知最简单的有理变数是线性变数和代数环面。有限群在线性变异和代数环面上的自然作用来源于群的表示。有人可能会问，在有限群作用下，线性变量或代数环面的轨道空间何时是有理的？这个问题最初是由艾米·诺特在研究伽罗瓦反问题时提出的。人们也可以将一个有有限群作用的代数环面与一个有有限群作用的相关线性变化进行比较，通过问它们何时是双同构的，而双同构对于有限群作用是等变的。等变双族线性化问题涉及到寻找经典Cremona群——射影空间的双族同构群中有限子群的共轭类问题。它还与确定代数群是否为Cayley或与其李代数是否等价双同构的经典问题有关，这是Cayley首先研究的问题。我之前曾与Vladimir Popov和Zinovy Reichstein共同研究过这个问题。在其他方面，我们确定了代数闭域上的简单代数群的集合，它们是Cayley。我研究有限群作用下代数环面合理性问题的类似和推广，以及等变两族线性化问题。