#### Vol. 12, No. 2, 2018

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

### Shulim Kaliman

Vol. 12 (2018), No. 2, 227–258
##### 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 categorical quotient of the action is automatically an affine algebraic variety and describe the geometric structure of such quotients.

 Dedicated to my teachers, Vladimir Yakovlevich Lin and Evgeniy Alekseevich Gorin
##### Keywords
proper $G_a$-action on affine 4-space
##### Mathematical Subject Classification 2010
Primary: 14R20
Secondary: 14L30, 32M17
##### Milestones
Revised: 18 November 2016
Accepted: 9 August 2017
Published: 13 May 2018
##### Authors
 Shulim Kaliman Department of Mathematics University of Miami Coral Gables, FL United States