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

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 $\operatorname{Spec}(A)$.