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Grassmannians, flag varieties, and Gelfand-Zetlin polytopes
P. 179–226.
The aim of these notes is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V. Kiritchenko, V. Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics of Gelfand-Zetlin polytopes.
Shafarevich A., Research in the Mathematical Sciences 2025 Vol. 12 No. 1 Article 6
We describe complete simplicial toric varieties on which a unipotent group acts with a finite number of orbits. We also provide a complete list of such varieties in the cases when the dimension is equal to 2 or the divisor class group is Z. ...
Added: March 10, 2025
Kikteva V., Sbornik Mathematics 2024 Vol. 215 No. 10 P. 1351–1373
We obtain a criterion for the automorphism group of an affine toric variety to be connected, stated in combinatorial terms and in terms of the divisor class group of the variety. We describe the component group of the automorphism group of a nondegenerate affine toric variety. In particular, we show that the number of connected components of ...
Added: January 27, 2025
Roman Avdeev, Vladimir Zhgoon, / Series arXiv "math". 2024. No. 2312.03377.
Given a connected reductive algebraic group G and a spherical G-variety X, a B-root subgroup on X is a one-parameter additive group of automorphisms of X normalized by a Borel subgroup B⊂G. We obtain a complete description of all B-root subgroups on a certain open subset of X. When X is horospherical, we extend the construction of standard B-root subgroups introduced earlier by Arzhantsev and Avdeev for affine X and obtain a ...
Added: December 17, 2024
Kikteva V., Математический сборник 2024 Т. 215 № 10 С. 89–113
We obtain a criterion for the automorphism group of an affine toric variety to be connected in combinatorial terms and in terms of the divisor class group of the variety. The component group of the automorphism group of a non-degenerate affine toric variety is described. In particular, we show that the number of connected components ...
Added: September 30, 2024
Zaitseva Y., Results in Mathematics 2024 Vol. 79 Article 249
We give a classification of noncommutative algebraic monoid structures on normal affine varieties such that the group of invertible elements of the monoid is connected, solvable, and has a one-dimensional unipotent radical. We describe the set of idempotents and the center of such a monoid and give a criterion for existence of the zero element. ...
Added: September 13, 2024
Arzhantsev I., Perepechko A., Shakhmatov K., Bulletin des Sciences Mathematiques 2024 Vol. 192 Article 103419
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 ...
Added: April 12, 2024
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
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
Ignatyev Mikhail V., Shevchenko A., Bochkarev M., Journal of Algebra 2016 Vol. 465 P. 259–286
We study tangent cones to Schubert subvarieties of the flag variety of a complex reductive group G. Let T be a maximal torus of G, B be a Borel subgroup of G containing T, Φ be the root system of G with respect to T, W be the Weyl group of Φ, and F = G/B be the flag variety. We prove that if every irreducible component of Φ is of type Bn or ...
Added: October 11, 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
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
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
Shafarevich A., Indagationes Mathematicae 2023 Vol. 34 No. 1 P. 42–53
Let G_a be the additive group of the field of complex numbers ℂ. We say that an irreducible algebraic variety X of dimension n admits an additive action if there is a regular action of the group G_a^n =G_a×…×G_a (n times) on X with an open orbit. In 2017 Baohua Fu and Jun-Muk Hwang introduced a class of Euler-symmetric varieties. They gave a classification ...
Added: February 6, 2023
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
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
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
С. М. Гусейн-Заде, Труды Математического института им. В.А. Стеклова РАН 2022 Т. 318 С. 66–72
Хорошо известная формула для эйлеровой характеристики полного пересечения в комплексном торе в терминах носителей многочленов Лорана — левых частей уравнений, задающих полное пересечение (фактически в терминах выпуклых оболочек носителей — многогранников Ньютона), была анонсирована в краткой заметке Д.Н. Бернштейна, А.Г. Кушниренко и А.Г. Хованского (1976). Её доказательство было дано А.Г. Хованским (1978), но было несамозамкнутым (существенно опиралось на результаты работы другого ...
Added: November 3, 2022
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
Ivan Arzhantsev, Roman Avdeev, Selecta Mathematica, New Series 2022 Vol. 28 No. 3 Article 60
Added: April 28, 2022
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