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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570377
• 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. I also do research on essential dimension. This is a measure of the degree of complexity of an algebraic or geometric object defined over a base field. In joint work with Ajneet Dhillon, we are working on determining bounds on the essential dimension of moduli stacks of principal G-bundles over a curve, a generalisation of the original definition of essential dimension of algebraic groups due to Buhler and Reichstein.

代数几何中一个经典但众所周知的困难问题是将代数簇分类为双有理同构——代数簇集合上的自然等价关系。 有理代数簇在这种等价关系下形成了一个特殊的类。最简单的已知有理簇是线性簇和代数环。有限群对线性簇和代数环的自然作用来自群的表示。 人们可能会问，有限群作用下线性簇或代数环面的轨道空间何时是有理数。 这个问题是由艾米·诺特（Emmy Noether）在研究伽罗瓦反问题时首次提出的。 人们还可以通过询问两者何时与相对于有限群作用等变的双有理同构来将具有有限群作用的代数环面与具有有限群作用的相关线性簇进行比较。 等变双有理线性化问题与寻找经典克雷莫纳群（射影空间双有理同构群）中有限子群的共轭类的问题相关。 它还与确定代数群是否为凯莱群或与其李代数等变双有理同构的经典问题相关，这是凯莱首先研究的问题。 我之前曾与 Vladimir Popov 和 Zinovy Reichstein 共同研究过这个问题。 除此之外，我们确定了代数闭域上的一组简单代数群，即凯莱。我研究有限群作用下代数环面有理性问题的类比和推广，以及等变双有理线性化问题。 我也研究本质维度。 这是在基域上定义的代数或几何对象的复杂程度的度量。 在与 Ajneet Dhillon 的合作中，我们正在努力确定曲线上主 G 丛模堆的基本维数的界限，这是布勒和赖希斯坦对代数群基本维数原始定义的概括。