On the Invariant Theory of Unipotent Groups Actions from a Geometric Perspective

Abstract

This paper is a generalization of a previous paper by the author to connected unipotent linear algebraic groups. The notion of an $ \alpha $-pair answers when an open $ G $-stable, affine, sub-variety $ D(H) $ is a trivial bundle over $ G \backslash \backslash D(H) $ in the Zariski topology, or whether there is an fppf topology $ \phi: U \to D(H) $ such that $ U $ is a trivial bundle over $ G \backslash \backslash U $ and such that the fibres $ \phi^{-1}(x) \cong \ker(\alpha) $. We also describe a generalization of van den Essen's algorithm for quasi-principle $ G $-varieties $ \operatorname{Spec}(A) $.