Unipotent group actions on affine varieties

TL;DR

This paper investigates algebraic actions of unipotent groups on affine varieties with small-dimensional quotients, providing structural results and links to longstanding conjectures in algebraic geometry.

Contribution

It offers new geometric and algebraic proofs of properties of unipotent group actions and relates these to the Abhyankar-Sathaye and Sathaye conjectures.

Findings

When $X$ is factorial and $X//U$ is one-dimensional, $O(X)^U$ is a polynomial ring in one variable.

If a point in $X$ has trivial isotropy, then $X$ is $U$-equivariantly isomorphic to $U imes A^1(k).

Connections are established between the actions studied and key conjectures in affine algebraic geometry.

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.