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: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=478761
• 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.

代数几何形状中的一个经典但臭名昭著的问题是将代数品种分类为生育同构，这是代数品种集的自然对等关系。 在这种对等关系下，有理代数品种形成了杰出的类别。最简单的已知理性品种是线性品种和代数托里。有限基团对线性品种和代数托里的自然作用来自组的代表。 可能会询问有限群体行动下线性品种或代数托里的轨道空间是合理的。 这个问题首先是艾米·诺瑟（Emmy Noether）提出的，当时她正在处理逆向加洛伊斯问题。 一个人还可以通过询问两者何时在有限群体作用方面具有同性恋的同构同构，将代数曲与有限的群体作用与有限的线性变化与有限的群体作用进行比较。 这个模棱两可的异性线性化问题与在经典的Cremona组中找到有限亚组的共轭类别的问题有关，这是投影空间的生育同构。 这也与确定代数群是凯利（Cayley）或与其谎言代数同构的经典问题有关 - 卡利（Cayley）最初研究的问题。 我之前曾与弗拉基米尔·波波夫（Vladimir Popov）和Zinovy Reichstein进行联合研究。 除其他事项外，我们确定了一组简单的代数群体，这是Cayley的代数封闭场。我研究了在有限的群体行动以及均等的异性线性化问题下，代数托里的合理性问题的类似物和概括。