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

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.