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Infinite transitivity on affine varieties
P. 1–13.
In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.
Keywords: Affine algebraic variety
Arzhantsev I., Zaidenberg M., Kuyumzhiyan K., Sbornik Mathematics 2012 Vol. 203 No. 7 P. 923–949
We say that a group G acts infinitely transitively on a set X if for every m ∈ N the induced diagonal action of G is transitive on the cartesian m-th power X^m\Δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their ...
Added: June 12, 2025
Kikteva V., Siberian Mathematical Journal 2024 Vol. 64 No. 5 P. 1167–1178
We obtain a generalization of the ABC Theorem on locally nilpotent derivations to the case of the polynomials with m monomials such that each variable is included just in a sole monomial. As applications of this result we provide some construction of rigid and semirigid algebras and describe the Makar-Limanov invariant of algebras of a ...
Added: September 21, 2024
Arzhantsev I., Zaitseva Y., Research in Mathematical Sciences 2024 Vol. 11 No. 2 Article 27
An algebraic variety X is called a homogeneous variety if the automorphism group 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 ...
Added: March 23, 2024
Kikteva V., Сибирский математический журнал 2023 Т. 64 № 5 С. 1009–1022
We obtain a generalization of the ABC Theorem on locally nilpotent derivations to the case of the polynomials with m monomials such that each variable is included just in a sole monomial. As applications of this result we provide some construction of rigid and semirigid algebras and describe the Makar-Limanov invariant of algebras of a ...
Added: December 3, 2023
Popov V., Труды Математического института им. В.А. Стеклова РАН 2023 Т. 320 С. 287–297
Построена новая бесконечная серия рациональных аффинных алгебраических многообразий, группа автоморфизмов которых содержит группу Aut}(F_n) автоморфизмов свободной группы ранга n. Группы автоморфизмов таких многообразий нелинейны и содержат группу кос B_n c n нитями при n > 2, а при n > 1 неаменабельны. В качестве приложения доказано, что при n > 2 каждая группа Кремоны ранга ...
Added: June 9, 2022
Vladimir L. Popov, / Series math "arxiv.org". 2021. No. 2102.08032.
Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained. ...
Added: February 17, 2021
Arzhantsev I., Arkiv for Matematik 2018 Vol. 56 No. 1 P. 1–14
It is known that if the special automorphism group SAut(X) of a quasiaffine variety X of dimension at least 2 acts transitively on X, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group Aut(X) to act infinitely transitively on X. We show that this is the case, provided X admits a nontrivial G_a- or G_m-action. Moreover, 2-transitivity of ...
Added: May 2, 2018
Vladimir L. Popov, , in: Affine Algebraic Geometry, Proceedings of the Conference, Osaka, Japan, 3-6 March 2011.: L., Singapore, New Jersey: World Scientific Publishing Co. Pte. Ltd, 2013. Ch. 10 P. 213–242.
We explore algebraic subgroups of the Cremona group Cn over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of Cn that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on An is intimately related to rationality of the invariant fields ...
Added: August 2, 2013
L., Singapore, New Jersey: World Scientific Publishing Co. Pte. Ltd, 2013.
Ths is Proceedings of a Conference on Affine Algebraic Geometry that was held at Osaka Umeda Campus of Kwansei Gakuin University during the period 3--6 March, 2011 on the occasion of the seventieth birthday of Professor Masayaoshi Miyanishi. ...
Added: August 2, 2013
Singapore: World Scientific Publishing Co. Pte. Ltd, 2013.
This volume is dedicated to Professor M. Miyanishi on the occasion of his 70th birthday. ...
Added: March 27, 2013
Kuyumzhiyan K., Arzhantsev I. V., Zaidenberg M. G., Sbornik Mathematics 2012 Vol. 203 No. 7 P. 923–949
We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on ...
Added: February 4, 2013