Affine T-varieties of complexity one and locally nilpotent derivations

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.

31 pages. Minor changes in the structure. Fixed some typos

References in corpus (5)

• Locally nilpotent derivations on affine surfaces with a $\C^*$-action
• Ga-actions of fiber type on affine T-varieties
• Canonical divisors on T-varieties
• Toroidal embeddings and polyhedral divisors
• Rational singularities of normal T-varieties

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