Abstract

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also partial derivative, be a homogeneous locally nilpotent derivation on the normal affine Z(n)-graded domain A, so that partial derivative, generates a k(+)-action on X that is normalized by the T-action. We provide a complete classiffication of pairs (X, partial derivative,) in two cases: for toric varieties (n = dim X) and in the case where n = dim X - 1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we compute the homogeneous Makar-Limanov invariant of such varieties. In particular, we exhibit a family of nonrational varieties with trivial Makar-Limanov invariant.