Affine $ \mathbb{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 ∂ be a homogeneous locally nilpotent derivation on the normal affine Z n -graded domain A, so that ∂ generates a k + -action on X that is normalized by the T-action.We provide a complete classification of pairs (X, ∂) 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.

On rigidity of factorial trinomial hypersurfaces

“…Assume that a finitely generated domain A is graded by a lattice M. If A admits a nonzero locally nilpotent derivation, then it admits a homogeneous one [21,Lemma 1.10]. So in order to prove that A is rigid it suffices to check that A admits no nonzero homogeneous locally nilpotent derivation.…”

“…A description of homogeneous locally nilpotent derivations of horizontal type for an affine T -variety X of complexity one in terms of the combinatorial data (Y, D) is given in [21], see also [5,Section 1.4]. To represent such a derivation, as a first step one needs to fix a vertex v z for every coefficient ∆ z of the polyhedral divisor D in such a way that all but two of these vertices are in N, see [5, Definition 1.8(iii)].…”

“…Further we assume that n i l i1 > 1 for i = 0, 1, 2 or, equivalently, f does not contain a linear term; otherwise the hypersurface X is isomorphic to the affine space K n−1 . Under this assumption, the trinomial hypersurface X is factorial if and only if any two of d 0 , If X is a toric (not necessary affine) variety with the acting torus T , then the actions G a × X → X normalized by T can be described combinatorially in terms of the so-called Demazure roots; see [8], [22,Section 3.4] for the original approach and [21,5,3] for generalizations. It is easy to deduce from this description that every affine toric variety different from a torus admits a non-trivial G a -action.…”

“…If K[X] admits a nonzero locally nilpotent derivation, then it admits a homogeneous one. Our idea is to represent X by a proper polyhedral divisor on a curve following Altmann and Hausen [1] and to use a description of homogeneous locally nilpotent derivations due to Liendo [21]. This allows to prove the "if" part of Theorem 1, while the "only if" part is elementary.…”

Многообразия Флагов, Торические Многообразия И Надстройки: Три Примера Бесконечной Транзитивности

“…В частности, это выполняется для многообразий всех трех видов из теоремы 0.2 (для первых двух из них см. также [20], [21; 3.16] и [22]). С другой стороны, если X гибко, то инвариант ML(X) тривиален.…”