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Автоморфизмы алгебраических многообразий и бесконечная транзитивность
Алгебра и анализ. 2022. Т. 34. № 2. С. 1-55.
Arzhantsev I., St Petersburg Mathematical Journal 2023 Vol. 34 No. 2 P. 143-178
We survey recent results on multiple transitivity for automorphism groups of affine algebraic varieties. We consider the property of infinite transitivity of the special automorphism group, which is equivalent to flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of ...
Added: March 30, 2023
Avdeev R., Zhgoon V., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2022 Т. 503 № 1 С. 5-10
Let X be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group G. In this paper we provide sufficient conditions, formulated in terms of weight combinatorics, for the existence of one-parameter additive actions on X normalized by a Borel subgroup B⊂G. As an application, we prove that every G-stable prime divisor in X can be ...
Added: June 1, 2022
Gayfullin S., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 318 С. 43-50
In 2013 Bazhov proved a criterion for two points on a complete toric variety to lie in the same orbit of the neutral component of the automorphism group. This criterion is formulated in terms of the divisor class group. The same year Arzhantsev and Bazhov obtained a similar criterion for affine toric varieties. We prove ...
Added: December 12, 2022
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
Dzhunusov S., Zaitseva Y., Forum Mathematicum 2021 Vol. 33 No. 1 P. 177-191
We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n-1 or n on a normal affine variety of dimension n. ...
Added: January 15, 2021
Arzhantsev I., Perepechko A., / Bulletin des sciences mathématiques. Series 22-00305 "BULSCI-D". 2023.
We consider complete toric varieties X such that a maximal unipotent subgroup U of the automorphism group Aut(X) acts on X with an open orbit. It turns out that such varieties can be characterized by several remarkable properties. We study the set of Demazure roots of the corresponding complete fan, describe the structure of a maximal unipotent subgroup U in Aut(X), and find all regular subgroups ...
Added: October 6, 2023
I. Arzhantsev, Kaliman S., M. Zaidenberg, Advances in Mathematics 2024 Vol. 437 Article 109449
It was shown by Kaliman and Zaidenberg (2023) that the affine cones over flag manifolds and rational smooth projective surfaces are elliptic in the sense of Gromov. The latter remains true after successive blowups of points on these varieties. In the present article we extend this to smooth projective spherical varieties (in particular, toric varieties) successively ...
Added: December 17, 2023
Ivan Arzhantsev, Roman Avdeev, Selecta Mathematica, New Series 2022 Vol. 28 No. 3 Article 60
Added: April 28, 2022
Попов В. Л., Известия РАН. Серия математическая 2019 Т. 84 № 4 С. 194-225
The rst group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...
Added: July 31, 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
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
Bilich B., / Cornell University. Series math "arxiv.org". 2021. No. 2106.04884.
In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by ...
Added: June 13, 2021
Arzhantsev I., Zaidenberg M., International Mathematics Research Notices 2022 Vol. 2022 No. 11 P. 8162-8195
Given a toric affine algebraic variety X and a collection of one-parameter unipotent subgroups U_1,…,U_s of Aut(X), which are normalized by the torus acting on X, we show that the group G generated by U_1,…,U_s verifies the following alternative of Tits type: either G is a unipotent algebraic group or it contains a non-abelian free subgroup. We deduce that if G is 2-transitive on a G-orbit in X, then G contains a non-abelian ...
Added: January 31, 2021
Popov V., / Bielefeld University. Series LAGRS "Linear Algebraic Groups and Related Structures". 2012. No. 485.
We construct counterexamples to the rationality conjecture regar-ding the new version of the Makar-Limanov invariant introduced in A. Liendo, Ga-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653–3665. ...
Added: January 9, 2013
Arzhantsev I., Bragin S., Zaitseva Y., Communications in Contemporary Mathematics 2020 Vol. 22 No. 8 P. 1950064: 1
We study commutative associative polynomial operations A^n×A^n→A^n with unit on the affine space A^n over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric ...
Added: September 19, 2019
Arzhantsev I., Zaitseva Y., Russian Mathematical Surveys 2022 Vol. 77 No. 4 P. 571-650
We survey recent results on open embeddings of the affine space C^n into a complete algebraic variety X such that the action of the vector group G_a^n on C^n by translations extends to an action of G_a^n on X. We begin with the Hassett–Tschinkel correspondence describing equivariant embeddings of \mathbb{C}^n into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag ...
Added: February 26, 2023
Arzhantsev I., Kuyumzhiyan K., Zaidenberg M., Advances in Mathematics 2019 Vol. 351 P. 1-32
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 the previous paper we proved that any nondegenerate toric affine variety X is flexible. In the present paper we show that one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same ...
Added: May 15, 2019
Kishimoto T., Yuri Prokhorov, Zaidenberg M., Algebraic Geometry 2014 Vol. 1 No. 1 P. 46-56
In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface ...
Added: October 10, 2013
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
Shakhmatov K., Математические заметки 2021 Т. 109 № 6 С. 929-937
An open translation-equivariant embedding of the affine space A^n into a complete nonprojective algebraic variety X is constructed for any n >= 3. The main tool is the theory of toric varieties. In the case n = 3, the orbit structure of the obtained action on the variety X is described. ...
Added: June 6, 2021
Galkin S., / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.
Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...
Added: May 4, 2014
И. В. Аржанцев, Ю. И. Зайцева, Успехи математических наук 2022 Т. 77 № 4(466) С. 3-90
Работа содержит обзор недавних результатов об открытых вложениях аффинного пространства C^n в полные алгебраические многообразия X, для которых действие векторной группы G_a^n на C^n параллельными переносами продолжается до действия G_a^n на X. В первой части работы мы подробно изучаем соответствие Хассетта–Чинкеля, описывающее эквивариантные вложения C^n в проективные пространства, и приводим его обобщение на случай вложений в проективные гиперповерхности. Последующие разделы посвящены изучению вложений в многообразия флагов и в их вырождения, в полные торические ...
Added: August 4, 2022
Leonid Monin, Smirnov E., Seminaire Lotharingien de Combinatoire 2023 Vol. 89B Article 76
In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties.
In this paper we extend Pukhlikov--Khovanskii type presentation to the case of K-theory ...
Added: October 26, 2023
Arzhantsev I., Indagationes Mathematicae 2023 Vol. 34 No. 4 P. 812-819
We prove that every non-degenerate toric variety, every homogeneous space of a connected linear algebraic group without non-constant invertible regular functions, and every variety covered by affine spaces admit a surjective morphism from an affine space. ...
Added: May 24, 2023