Unipotent group actions on affine varieties

Details Unipotent group actions on affine varieties Unipotent group actions on affine varieties

2010-02-22

arxiv.org/abs/1002.4181v1

Mathematics

Algebraic Geometry

10 pages. This submission comes out of an older submission ("A commuting derivations theorem on UFDs") and contains part of it

Scientific paper

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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