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## On the Existence of B -Root Subgroups on Affine Spherical Varieties

Doklady Mathematics . 2022. Vol. 105. No. 2. P. 51-55.

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 connected with an open *G*-orbit by means of a suitable *B*-normalized one-parameter additive action.

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

Aleksei Golota, / Cornell University. Series arXiv "math". 2019.

For a polarized variety (X,L) and a closed connected subgroup G⊂Aut(X,L) we define a G-invariant version of the δ-threshold. We prove that for a Fano variety (X,−KX) and a connected subgroup G⊂Aut(X) this invariant characterizes G-equivariant uniform K-stability. We also use this invariant to investigate G-equivariant K-stability of some Fano varieties with large groups of ...

Added: October 7, 2019

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

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., 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., 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

Zhgoon V., Journal of Lie Theory 2013 Vol. 23 P. 607-638

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of Local Structure Theorems obtained by Knop and Timashev, which describe the action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of Vinberg and Timashev ...

Added: February 6, 2013

Kiritchenko Valentina, Krishna A., Transformation Groups 2013 Vol. 18 No. 2 P. 391-413

We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank. ...

Added: February 18, 2013

Kiritchenko V., Annales de l'Institut Fourier 2006 Vol. 56 No. 4 P. 1225-1256

In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. When a complete intersection is a curve, this formula gives an explicit answer for the ...

Added: October 7, 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

Smirnov E., Journal of Mathematical Sciences 2020 Vol. 248 No. 3 P. 338-373

This paper is a review of results on multiple flag varieties, i.e., varieties of the form G/P1×· · ·×G/Pr. We provide a classification of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of B-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the ...

Added: July 6, 2020

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

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

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

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

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016

Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216

Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...

Added: December 4, 2017

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016

Buchstaber V., Limonchenko I., / Cornell University. Series math "arxiv.org". 2018. No. 1808.08851.

We introduce the notions of algebraic and geometric direct families of polytopes and develop a theory of such families. The theory is then applied to the problem of existence of nontrivial higher Massey products in cohomology of moment-angle-complexes. ...

Added: September 29, 2019

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Shiryaev A., Zhitlukhin M., Ziemba W., / SSRN. Series Social Science Research Network "Social Science Research Network". 2013.

We study the land and stock markets in Japan circa 1990. While the Nikkei stock average in the late 1980s and its -48% crash in 1990 is generally recognized as a financial market bubble, a bigger bubble and crash was in the golf course membership index market. The crash in the Nikkei which started on ...

Added: March 9, 2014

Maslov V., Теоретическая и математическая физика 2019 Т. 201 № 1 С. 65-83

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint
using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes
a boson at the instant of separating from a fermionic nucleus. We study the further transformations of
boson and fermion states of separation in a ...

Added: November 1, 2019

Litvin Y. V., Абрамов И. В., Технологии техносферной безопасности 2016 № 66

Advanced approach to the assessment of a random time of arrival fire fighting calculation on the object of protection, the time of their employment and the free combustion. There is some quantitative assessments with the review of analytical methods and simulation ...

Added: August 27, 2016