Invariants for algebraic group actions

对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个问题)。20世纪这一理论的亮点之一是证明了作用于仿射簇的约化代数群的不变量环(坐标环)是有限生成的--这意味着在这种情况下有一个有用的商簇的概念。如果群作用不是约化的，有限生成一般是失败的，但通常可以恢复该理论的某些方面。这方面的一个例子是代数群动作的“分离不变量”的概念--这些是有限的不变量集，即使不变量的整个环不是有限地生成的，它们也可以像整个不变量环一样分离。这个项目的目的是研究非约化情况下的代数群作用的不变量，首先建立在Duresne和合著者关于加性群作用的不变量的工作基础上。

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.