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 $ \operatorname{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 $ \operatorname{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 \operatorname{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.