Proper $G_a$-actions on $\C^4$ preserving a coordinate

TL;DR

This paper proves that certain algebraic group actions on four-dimensional complex affine space are translations and explores the structure of quotients under these actions, revealing conditions for their affineness.

Contribution

It establishes that specific $G_a$-actions on $ ext{C}^4$ are translations and characterizes the geometric structure of their quotients.

Findings

All proper $G_a$-actions on $ ext{C}^4$ preserving a coordinate are translations.

Certain $G_a$-actions on affine fourfolds have automatically affine quotients.

The geometric structure of these quotients is explicitly described.

Abstract

We prove that the actions mentioned in the title are translations. We show also that for certain $G_a$-actions on affine fourfolds the quotient of the action is automatically affine and describe the geometric structure of such quotients.