On the Invariant Theory of ${\mathbb{G}}_a$-Actions from a Geometric Perspective

Abstract

In this paper we give a strict classification of ${\mathbb{G}}_a$-representations. This is done through the notion of a $c(t)$-pair. Namely if $\mathrm{Spec}(A)$ is a ${\mathbb{G}}_a$-variety with action $\beta$, then a $c(t)$-pair is a pair of elements $(g,h)$ such that $g(t_0\ast x)=g(x)+c(t_0)h(x)$. This allows us to describe exactly when an affine, ${\mathbb{G}}_a$-stable, sub-variety $D(h)$ is a trivial bundle over $D(h)//{\mathbb{G}}_a$. If $\mathrm{Spec}(A)$ is a ${\mathbb{G}}_a$-variety, we define the large pedestal ideal ${\mathfrak{P}}_g(A)$ and the pedestal ideal $\mathfrak{P}(A)$. If $\beta :{\mathbb{G}}_a\to \mathrm{GL}(\mathbf{V})$ is a ${\mathbb{G}}_a$-representation, then we classify such a representation on whether: a) the large pedestal ideal ${\mathfrak{P}}_g(S_k({\mathbf{V}}^{\ast }))$ is equal to zero. b) the large pedestal ideal is non-zero, but the pedestal ideal is equal to zero. or c) the pedestal ideal is non-zero.

33 pages