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Affine homogeneous varieties and suspensions

Ivan Arzhantsev, Yulia Zaitseva

Abstract

An algebraic variety $X$ is called a homogeneous variety if the automorphism group $\mathrm{Aut}(X)$ acts on $X$ transitively, and a homogeneous space if there exists a transitive action of an algebraic group on $X$. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces.

Affine homogeneous varieties and suspensions

Abstract

An algebraic variety is called a homogeneous variety if the automorphism group acts on transitively, and a homogeneous space if there exists a transitive action of an algebraic group on . We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces.
Paper Structure (4 sections, 11 theorems, 29 equations)

This paper contains 4 sections, 11 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Regular suspensions
  3. The case of surfaces
  4. Homogeneous varieties of higher dimension

Key Result

Theorem 1

Suppose that an irreducible affine variety $X$ of positive dimension is flexible. Then any suspension over $X$ is flexible as well.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Theorem 1: AKZ2012
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • Remark 1
  • Remark 2
  • ...and 23 more