?
Infinite transitivity and special automorphisms
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 the automorphism group implies infinite transitivity.
Arzhantsev I., Kuyumzhiyan K., Zaidenberg M., Математический сборник 2012 Т. 203 № 7 С. 3-30
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: September 12, 2012
Ivan V. Arzhantsev, Yulia I. Zaitseva, Kirill V. Shakhmatov, Proceedings of the Steklov Institute of Mathematics 2022 Vol. 318 No. 1 P. 13-25
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as the maximum number m such that the action of Aut(X) on X is m-transitive. If the action of 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 ...
Added: November 5, 2022
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
Arzhantsev I., Zaitseva Y., Shakhmatov K., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 318 С. 17-30
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as a maximal number m such that the action of Aut(X) on X is m-transitive. If the action of 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 ...
Added: November 4, 2022
Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al., Compositio Mathematica 2011 Vol. 147 No. 2 P. 428-466
Let k be a field of characteristic zero, let G be a connected reductive algebraic group
over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces
the adjoint action of G on g. We investigate the ...
Added: March 17, 2013
Vladimir L. Popov, / Cornell University. 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
Perepechko A., Функциональный анализ и его приложения 2013 Т. 47 № 4 С. 45-52
We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. ...
Added: September 26, 2019
Perepechko A., Forum Mathematicum 2021 Vol. 33 No. 2 P. 339-348
Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on ...
Added: January 15, 2021
Singapore : World Scientific, 2013
This volume is dedicated to Professor M. Miyanishi on the occasion of his 70th birthday. ...
Added: March 27, 2013
V. V. Kikteva, Siberian Mathematical Journal 2023 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: December 3, 2023
Gayfullin S., Journal of Algebra 2021 No. 573 P. 364-392
In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov. ...
Added: February 6, 2021
Arzhantsev I., Gayfullin S., Mathematische Nachrichten 2017 Vol. 290 No. 5-6 P. 662-671
An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of is pointed, the group is a finite extension of . As an application, ...
Added: February 19, 2017
Gayfullin S., Shafarevich Anton, Proceedings of the American Mathematical Society 2019 Vol. 147 P. 3317-3330
We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only con- stant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety with only constant invertible functions is flexible. ...
Added: October 17, 2019
Gayfullin S., Chunaev D., Фундаментальная и прикладная математика 2023 Т. 25
In this work we obtain sufficient conditions for a variety with a torus action of complexity one to have finite number of automorphism group orbits. ...
Added: December 2, 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
Arzhantsev I., Bazhov I., Central European Journal of Mathematics 2013 Vol. 11 No. 10 P. 1713-1724
Let X be an affine toric variety. The total coordinates on X provide a canonical presentation !X -> X of X as a quotient of a vector space !X by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the ...
Added: November 13, 2013
Kuyumzhiyan K., Arzhantsev I., Zaidenberg M., / Cornell University. Series arXiv "math". 2018.
An affine algebraic variety X of dimension ≥ 2 is called flexible if the subgroup SAut(X) ⊂ Aut(X) generated by the one-parameter unipotent subgroups acts m-transitively on reg (X) for any m ≥ 1. In a preceding paper ([4]) we proved that any nondegenerate toric affine variety X is flexible. Here we show that if such a toric variety X is ...
Added: December 6, 2018
Arzhantsev I., Алгебра и анализ 2022 Т. 34 № 2 С. 1-55
В работе дан обзор результатов последних лет о кратной транзитивности действий групп автоморфизмов аффинных алгебраических многообразий. Рассматривается свойство бесконечной транзитивности действия группы специальных автоморфизмов и эквивалентное ему свойство гибкости многообразия. Данные свойства имеют важные алгебраические и геометрические следствия, и при этом они выполнены для широких классов многообразий. Отдельно изучаются случаи, когда бесконечная транзитивность имеет место ...
Added: March 14, 2022
Lerman L., Trifonov K., Математические заметки 2020 Т. 108 № 3 С. 474-476
This paper studies the topological properties of the automorphisms of the 4-torus R4/Z4 that are
generated by integer symplectic transformations of R4. Such transformations are customarily called the
symplectic automorphisms of the torus. The purpose is a classification of the possible types of behavior
of the trajectories of symplectic automorphisms of T4. ...
Added: August 28, 2020
Gayfullin S., Шафаревич А. А., / Cornell University. Series arXiv "math". 2018. No. arXiv:1805.05024.
Added: September 1, 2018
Arzhantsev I., Communications in Algebra 2018 Vol. 46 No. 8 P. 3539-3552
A non-degenerate toric variety X is called S-homogeneous if the subgroup of the automorphism group Aut(X) generated by root subgroups acts on X transitively. We prove that maximal S-homogeneous toric varieties are in bijection with pairs (P,A), where P is an abelian group and A is a finite collection of elements in P such that A generates the group P and for every a∈A the element a is contained in the semigroup generated by A∖{a}. We show that any ...
Added: April 20, 2018
L., Singapore, New Jersey : World Scientific, 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
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
Perepechko A., Michałek M., Süß H., Mathematische Zeitschrift 2018 Vol. 290 No. 3-4 P. 1457-1478
We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces. ...
Added: September 26, 2019