Ga-actions of fiber type on affine T-varieties

TL;DR

This paper classifies certain algebraic group actions on affine T-varieties, explores their implications for the structure of varieties with trivial ML invariants, and introduces a new invariant related to rationality.

Contribution

It provides a classification of Ga-actions of fiber type on affine T-varieties and introduces the FML invariant, linking triviality to rationality in low dimensions.

Findings

Varieties with trivial ML invariant are birationally decomposable as Y×P^2.

Existence of affine varieties with trivial ML invariant birational to Y×P^2 for any Y.

The FML invariant's triviality implies rationality in dimensions up to 3.

Abstract

Let X be a normal affine T-variety, where T stands for the algebraic torus. We classify Ga-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\times P^2, for some Y. Conversely, given a variety Y, there exists an affine variety X with trivial ML invariant birational to Y\times 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. This conjecture holds in dimension at most 3.