Proper twin-triangular 𝔾ₐ-actions on 𝔸⁴ are translations

Dubouloz, Adrien; Finston, David

doi:10.1090/S0002-9939-2014-11932-0

Proper twin-triangular $\mathbb {G}_{a}$-actions on $\mathbb {A}^{4}$ are translations
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by Adrien Dubouloz and David R. Finston

Proc. Amer. Math. Soc. 142 (2014), 1513-1526

DOI: https://doi.org/10.1090/S0002-9939-2014-11932-0

Published electronically: February 13, 2014

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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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