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

Authors: Adrien Dubouloz and David R. Finston
Journal: Proc. Amer. Math. Soc. 142 (2014), 1513-1526
MSC (2010): Primary 14R20, 14L30
Published electronically: February 13, 2014
MathSciNet review: 3168459
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Abstract: An additive group action on an affine $ 3$-space over a complex Dedekind domain $ A$ is said to be twin-triangular if it is generated by a locally nilpotent derivation of $ A[y,z_{1},z_{2}]$ of the form $ r\partial _{y}+p_{1}(y)\partial _{z_{1}}+p_{2}(y)\partial _{z_{2}}$, where $ r\in A$ and $ p_{1},p_{2}\in A[y]$. We show that these actions are translations if and only if they are proper. Our approach avoids the computation of rings of invariants and focuses more on the nature of geometric quotients for such actions.

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Adrien Dubouloz
Affiliation: Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary - BP 47870, 21078 Dijon cedex, France

David R. Finston
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Address at time of publication: Department of Mathematics, Brooklyn College CUNY, 2900 Bedford Avenue, Brooklyn, New York 11210

Received by editor(s): October 14, 2011
Received by editor(s) in revised form: June 8, 2012
Published electronically: February 13, 2014
Additional Notes: This research was supported in part by NSF Grant OISE-0936691 and ANR Grant 08-JCJC-0130-01
Dedicated: Dedicated to Jim Deveney on the occasion of his retirement
Communicated by: Harm Derksen
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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