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arXiv · 1002.4181

Unipotent group actions on affine varieties

Harm Derksen·Arno van den Essen·David R. Finston·Stefan Maubach
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Abstract

Algebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$ ($k$ an algebraically closed field of characteristic 0) for which the algebraic quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial, $O(X)^{\ast}=k^{\ast},$ and $X//U$ is one-dimensional, it is shown that $O(X)^{U}$=$k[f]$, and if some point in $X$ has trivial isotropy, then $X$ is $U$ equivariantly isomorphic to $U\times A^{1}(k).$ The main results are given distinct geometric and algebraic proofs. Links to the Abhyankar-Sathaye conjecture and a new equivalent formulation of the Sathaye conjecture are made.

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Harm Derksen, Arno van den Essen, David R. Finston, Stefan Maubach. 2010-02-22. Unipotent group actions on affine varieties. https://arxiv.org/abs/1002.4181

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