Unipotent group actions on affine varieties

Algebraic actions of unipotent groups U on affine k-varieties X (k is 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)=k⁎, and X//U is one-dimensional, it is shown that OU(X)=k[f], and if some point in X has trivial isotropy, then X is U equivariantly isomorphic to U×A1(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.