Invariants for algebraic group actions

不变量环的研究有着悠久而辉煌的历史（例如，参见希尔伯特的第十四问题）。该理论在20世纪的亮点之一是证明了作用于仿射变量的（坐标环）上的约化代数群的不变量环是有限生成的——这意味着在这种情况下有一个有用的商变量的概念。如果群体行为不是约化的，有限生成通常会失败，但通常可以检索理论的某些方面。其中一个例子是代数群作用的“分离不变量”概念——这些是有限的不变量集合，它们可以分离整个不变量环，即使整个不变量环不是有限生成的。本项目的目的是研究代数群作用在非约化情况下的不变量，从Dufresne及其合作者关于加性群作用的不变量的工作开始。

The study of rings of invariants has a long and illustrious history (see, for example, Hilbert's fourteenth problem). One of the highlights of the theory in the twentieth century was the proof that rings of invariants for reductive algebraic groups acting on (the coordinate rings of) affine varieties are finitely generated -- this implies that there is a useful notion of quotient variety in this situation. Finite generation fails in general if the group acting is not reductive, but it is often possible to retrieve some aspects of the theory. One example of this is the notion of "separating invariants" for algebraic group actions -- these are finite sets of invariants which can separate as much as the whole ring of invariants, even when the full ring of invariants is not finitely generated. The aim of this project is to study invariants for algebraic group actions in non-reductive cases, starting by building on work of Dufresne and co-authors for invariants of additive group actions.