Affine T-varieties of complexity one and locally nilpotent derivations

TL;DR

This paper classifies certain affine varieties with torus actions and locally nilpotent derivations, extending known results and computing invariants, revealing non-rational examples with trivial Makar-Limanov invariant.

Contribution

It provides a complete classification of pairs (X,D) for affine varieties with torus actions of specific complexities, generalizing previous surface results.

Findings

Classified pairs (X,D) for toric and codimension-one cases.

Computed the homogeneous Makar-Limanov invariant for these varieties.

Identified non-rational varieties with trivial Makar-Limanov invariant.

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 D be a homogeneous locally nilpotent derivation on the normal affine Z^n-graded domain A, so that D generates a k_+-action on X that is normalized by the T-action. We provide a complete classification of pairs (X,D) 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 non-rational varieties with trivial Makar-Limanov invariant.