Homogeneous algebraic varieties and transitivity degree

Ivan Arzhantsev; Kirill Shakhmatov; Yulia Zaitseva

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arXiv · 2204.08056

Homogeneous algebraic varieties and transitivity degree

Ivan Arzhantsev·Kirill Shakhmatov·Yulia Zaitseva

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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.

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Ivan Arzhantsev, Kirill Shakhmatov, Yulia Zaitseva. 2022-04-17. Homogeneous algebraic varieties and transitivity degree. https://arxiv.org/abs/2204.08056

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