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Classification of noncommutative monoid structures on normal affine surfaces
Cornell University
,
2021.
No. 2106.04884.
Bilich B.
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 a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.
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., 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
Ivan Arzhantsev, Roman Avdeev, Selecta Mathematica, New Series 2022 Vol. 28 No. 3 Article 60
Added: April 28, 2022
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
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
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
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
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
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
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
Kotenkova P., Beitrage zur Algebra und Geometrie 2014 Vol. 55 No. 2 P. 621-634
Let X be a normal affine algebraic variety with regular action of a torus T and T ⊂ T be a subtorus. We prove that each root of X with respect to T can be obtained by restriction of some root of X with respect to T. This allows to get an elementary proof of ...
Added: September 17, 2015
Vladimir L. Popov, Documenta Mathematica 2015 Vol. Extra Volume: Merkurjev's Sixtieth Birthday P. 513-528
A “rational” version of the strengthened form of the Commuting Derivation Conjecture, in which the assumption of commutativity
is dropped, is proved. A systematic method of constructing in any dimension greater than 3 the examples answering in the negative a question by M. El Kahoui is developed. ...
Added: September 25, 2015
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
Arzhantsev I., Ricerche di Matematica 2024 Vol. 73 No. 2 P. 715-724
We show that an effective action of the one-dimensional torus G_m on a normal affine algebraic variety X can be extended to an effective action of a semi-direct product G_m⋌G_a with the same general orbit closures if and only if there is a divisor D on X that consists of G_m-fixed points. This result is applied to the study of orbits of the automorphism group Aut(X) on X. ...
Added: August 16, 2021
Белев С. А., Tyurin N. A., Теоретическая и математическая физика 2013 Т. 175 № 2 С. 147-158
We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds. ...
Added: February 18, 2013
Zaitseva Y., Математические заметки 2019 Т. 105 № 6 С. 824-838
В работе получено описание однородных локально нильпотентных дифференцирований алгебры регулярных функций некоторого класса триномиальных гиперповерхностей. Данный класс включает в себя все нефакториальные триномиальные гиперповерхности. ...
Added: September 19, 2019
Ayzenberg A., Cherepanov V., / Cornell University. Series arXiv "math". 2019. No. 1905.04761.
Let the compact torus Tn−1 act on a smooth compact manifold X2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X2n/Tn−1 if the action is cohomologically equivariantly formal (which essentially means that Hodd(X2n;Z)=0). It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite ...
Added: October 23, 2019
Бухштабер В.М., Бунькова Е. Ю., Математические заметки 2020 Т. 108 № 1 С. 17-32
Построены алгебры Ли систем из 2g градуированных операторов теплопроводности Q0,Q2,…,Q4g−2, определяющих сигма-функции σ(z,λ) гиперэллитических кривых рода g=1,2 и 3. В качестве следствия получено, что системы из трех операторов Q0, Q2 и Q4 уже достаточно, чтобы определить сигма-функции. Оператор Q0 является оператором Эйлера, а каждый из операторов Q2k, k>0, задает g-мерное уравнение Шрёдингера с квадратичным потенциалом по z в неголономном репере векторных полей в C2g с координатами λ. Для любого решения φ(z,λ) системы уравнений теплопроводности мы вводим градуированное кольцо Rφ, порожденное логарифмическими производными от функции φ(z,λ) порядка не менее 2 и в явном ...
Added: June 16, 2021
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
Ilten N. O., Lewis J., Victor Przyjalkowski, Journal of Algebra 2013 Vol. 374 P. 104-121
We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of ...
Added: July 2, 2013
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
Matveev D., Математический сборник 2019 Т. 210 № 11 С. 103-128
Let X be an affine algebraic variety endowed with an action of complexity one of an algebraic torus T. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions K[X] can be described in terms of proper polyhedral divisors corresponding to T-variety X. We prove that homogeneous locally nilpotent derivations commute if ...
Added: November 21, 2019
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