Geodesic
Homogeneous Algebraic Varieties and Transitivity Degree
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 17-30

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Let $X$ be an algebraic variety such that the group $\mathrm {Aut}(X)$ acts on $X$ transitively. We define the transitivity degree of $X$ as the maximum number $m$ such that the action of $\mathrm {Aut}(X)$ on $X$ is $m$-transitive. If the action of $\mathrm {Aut}(X)$ 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. We also discuss a conjecture and open questions related to this invariant.
Keywords: algebraic variety, homogeneous space, quasi-affine variety, transitivity degree, infinite transitivity, toric variety, unirationality.
Mots-clés : automorphism group, algebraic group 
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     author = {Ivan V. Arzhantsev and Yulia I. Zaitseva and Kirill V. Shakhmatov},
     title = {Homogeneous {Algebraic} {Varieties} and {Transitivity} {Degree}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     url = {http://geodesic.mathdoc.fr/item/TM_2022_318_a1/}
}
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Ivan V. Arzhantsev; Yulia I. Zaitseva; Kirill V. Shakhmatov. Homogeneous Algebraic Varieties and Transitivity Degree. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 17-30. http://geodesic.mathdoc.fr/item/TM_2022_318_a1/