Affine homogeneous varieties and suspensions

TL;DR

This paper establishes criteria for smoothness and homogeneity of algebraic varieties, particularly suspensions and Danielewski surfaces, expanding understanding of their automorphism groups and classification.

Contribution

It provides new criteria for smoothness of suspensions and characterizes when Danielewski surfaces are homogeneous varieties or spaces, including constructions of new examples.

Findings

Criteria for smoothness of suspensions

Conditions for Danielewski surfaces to be homogeneous

Construction of affine suspensions that are homogeneous varieties but not spaces

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.