Proper triangular $\mathbb{G}_{a}$-actions on $\mathbb{A}^{4}$ are translations

Abstract

We describe the structure of geometric quotients for proper locally triangulable ${\mathbb{G}}_a$-actions on locally trivial ${\mathbb{A}}^3$-bundles over a nœtherian normal base scheme $X$ defined over a field of characteristic $0$. In the case where $dimX=1$, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank $2$ over $X$. As a consequence, every proper triangulable ${\mathbb{G}}_a$-action on the affine four space ${\mathbb{A}}_k^4$ over a field of characteristic $0$ is a translation with geometric quotient isomorphic to ${\mathbb{A}}_k^3$.