Homogeneous algebraic varieties and transitivity degree

TL;DR

This paper introduces the concept of transitivity degree for algebraic varieties, computes it for specific classes like toric varieties and homogeneous spaces, and discusses related conjectures and open questions.

Contribution

It defines the transitivity degree for algebraic varieties and computes it for quasi-affine toric varieties and many homogeneous spaces, advancing understanding of automorphism group actions.

Findings

Transitivity degree computed for all quasi-affine toric varieties.

Transitivity degree determined for many homogeneous spaces.

Discussion of conjectures and open questions related to the invariant.

Abstract

Let $X$ be an algebraic variety such that the group $\text{Aut}(X)$ acts on $X$ transitively. We define the transitivity degree of $X$ as a maximal number $m$ such that the action of $\text{Aut}(X)$ on $X$ is $m$-transitive. If the action of $\text{Aut}(X)$ is $m$-transitive for all $m$, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. Also we discuss a conjecture and open questions related to this invariant.