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

代数几何中一个经典但出了名的困难问题是对代数簇进行分类，直到二元同构-代数簇集合上的一种自然等价关系。在这种等价关系下，有理代数簇形成了一个独特的类。已知的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。这个问题最早是由艾米·诺特在研究伽罗瓦逆问题时提出的。人们也可以通过询问具有有限群作用的代数环面和具有有限群作用的相关线性簇何时与关于有限群作用等变的二元同构进行比较。这个等变双子线性化问题涉及到在经典Cremona群-射影空间的双子同构群中寻找有限子群的共轭类的问题。它还与确定一个代数群是否为Cayley--或与其李代数同构--的经典问题有关，这是由Cayley最先研究的问题。我之前曾与弗拉基米尔·波波夫和齐诺维·赖希斯坦就这个问题进行过联合研究。其中，我们确定了代数闭域上的Cayley单代数群的集合。我研究了有限群作用下代数环面的合理性问题以及等变双线性问题的类比和推广。

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.