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Flexibility criterion for affine horospherical varieties

Sergey Gaifullin, Veronika Kikteva

TL;DR

The paper addresses when an affine complexity-zero horospherical variety X is flexible by introducing the regularity cone gamma(X) derived from the horospherical data and linking flexibility to Demazure-root locally nilpotent derivations. The main approach combines LNDs with the combinatorics of cones and Demazure roots to characterize when regular strata can be moved by G_a-actions, yielding an explicit equivalence between flexibility and a geometric condition on gamma(X). The central result states that X is flexible iff gamma(X) is not contained in any hyperplane in N_Q, equivalently iff every non-constant regular function f on X has its zero set meeting the regular locus, which unifies known criteria for toric and horospherical cases and extends to non-normal, mixed-factor scenarios. This criterion provides a practical, constructive framework for analyzing the automorphism group generated by G_a-subgroups and the structure of the regular locus in complexity-zero horospherical varieties.

Abstract

In this paper we obtain a criterion of flexibility for an affine complexity-zero horospherical variety. This result generalizes previously known results on flexibility of normal horospherical varieties, horospherical varieties with an action of a semisimple group, and non-normal toric varieties.

Flexibility criterion for affine horospherical varieties

TL;DR

The paper addresses when an affine complexity-zero horospherical variety X is flexible by introducing the regularity cone gamma(X) derived from the horospherical data and linking flexibility to Demazure-root locally nilpotent derivations. The main approach combines LNDs with the combinatorics of cones and Demazure roots to characterize when regular strata can be moved by G_a-actions, yielding an explicit equivalence between flexibility and a geometric condition on gamma(X). The central result states that X is flexible iff gamma(X) is not contained in any hyperplane in N_Q, equivalently iff every non-constant regular function f on X has its zero set meeting the regular locus, which unifies known criteria for toric and horospherical cases and extends to non-normal, mixed-factor scenarios. This criterion provides a practical, constructive framework for analyzing the automorphism group generated by G_a-subgroups and the structure of the regular locus in complexity-zero horospherical varieties.

Abstract

In this paper we obtain a criterion of flexibility for an affine complexity-zero horospherical variety. This result generalizes previously known results on flexibility of normal horospherical varieties, horospherical varieties with an action of a semisimple group, and non-normal toric varieties.

Paper Structure

This paper contains 7 sections, 11 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally nilpotent derivations
  4. Rational polyhedral cones
  5. Horospherical varieties
  6. LNDs on affine horospherical varieties
  7. Main results

Key Result

Lemma 2.1

Suppose we have a $\mathbb Z^n$-grading on a finitely generated $\mathbb K$-domain $B$. Let $\delta$ be an LND of $B$. Consider the convex hull $P$ of degrees of nonzero homogeneous components of $\delta$. Then the homogeneous components of $\delta$ corresponding to the vertices of $P$ are locally n

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (23)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 13 more