Proper triangular Ga-actions on A^4 are translations

TL;DR

This paper characterizes proper triangular Ga-actions on A^4, proving they are always translations with geometric quotients isomorphic to A^3, extending understanding of additive group actions on affine spaces.

Contribution

It demonstrates that all proper triangular Ga-actions on A^4 are translations, providing a complete classification in characteristic zero.

Findings

Proper triangulable Ga-actions on A^4 are translations.

Geometric quotients of such actions are isomorphic to A^3.

In the case dim X=1, actions are translations with quotients as vector bundles.

Abstract

We describe the structure of geometric quotients for proper locally triangulable additve group actions on locally trivial A^3-bundles over a noetherian normal base scheme X defined over a field of characteristic 0. In the case where dim X=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 Ga-action on the affine four space A^4 over a field of characteristic 0 is a translation with geometric quotient isomorphic to A^3.