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

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。 我还对基本维度进行了研究。 这是对定义在基域上的代数或几何对象的复杂程度的度量。 在联合工作与Ajneet Dhillon，我们正在努力确定界限的基本尺寸的模栈的主要G-丛的曲线，一个推广的原始定义的基本尺寸的代数群由于布勒和Reichstein。

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.