Formula omitted]-actions of fiber type on affine [formula omitted]-varieties

Abstract

Let X be a normal affine T -variety, where T stands for the algebraic torus. We classify G a -actions on X arising from homogeneous locally nilpotent derivations of fiber type. We deduce that any variety with trivial Makar-Limanov (ML) invariant is birationally decomposable as Y × P 2 , for some Y. Conversely, given a variety Y, there exists an affine variety X with trivial ML invariant birational to Y × P 2 . Finally, we introduce a new version of the ML invariant, called the FML invariant. According to our conjecture, the triviality of the FML invariant implies rationality. We confirm this conjecture in dimension at most 3.