On flexibility of trinomial varieties

TL;DR

This paper investigates the conditions under which trinomial varieties, a class of affine varieties defined by three-term polynomials, exhibit flexibility, meaning their automorphism groups act transitively on their smooth points.

Contribution

It generalizes Gaifullin's sufficient condition for flexibility from trinomial hypersurfaces to all trinomial varieties, broadening understanding of their automorphism groups.

Findings

Provides a new sufficient condition for flexibility of trinomial varieties.

Extends previous results from hypersurfaces to general trinomial varieties.

Enhances understanding of automorphism groups acting on these varieties.

Abstract

Trinomial varieties are affine varieties given by a system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal varieties with torus action of complexity one. For an affine variety $X$ we consider the subgroup $\mathrm{SAut}(X)$ of the automorphism group generated by all algebraic subgroups isomorphic to the additive group of the ground field. By definition, an affine variety is flexible if $\mathrm{SAut}(X)$ acts transitively on its regular locus. Gaifullin proved a sufficient condition for a trinomial hypersurface to be flexible. We give a generalization of his results, proving a sufficient condition to be flexible for an arbitrary trinomial variety.