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## Affine monoids of corank one

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.

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

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

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

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

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

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., 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

Arzhantsev I., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87–98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2021 Т. 500 № 1 С. 52–54

It is explored to which extent the group variety of an algebraic group determines its group structure. ...

Added: November 18, 2021

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1502.02167.

According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and the s-dimensional projectice space with positive s. We show that the classical proof of this theorem actually works only in characteristic 0 and ...

Added: February 10, 2015

Ivan Arzhantsev, Roman Avdeev, Selecta Mathematica, New Series 2022 Vol. 28 No. 3 Article 60

Added: April 28, 2022

Beldiev I., Results in Mathematics 2023 Vol. 78 No. 5 Article 192

We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group G_a^m with an open orbit that can be extended to a regular action on the ambient projective space. We prove that if a projective hypersurface admits an induced additive action, then it is unique if and only if the hypersurface is ...

Added: August 7, 2023

Popov V., Известия РАН. Серия математическая 2022 Т. 86 № 5 С. 73–96

We explore to what extent the group variety of a connected algebraic group or the group manifold of a real Lie group determines its group structure. ...

Added: June 9, 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

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.

Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...

Added: May 28, 2021

Amosov G. G., Zhdanovskiy I., Journal of Mathematical Sciences 2016 Vol. 215 No. 6 P. 659–676

We study the noncommutative operator graph ℒ θ depending on a complex parameter θ recently introduced by M. E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We define a noncommutative group G and an algebra A θ which is the quotient of ℂG by a special algebraic relation depending on θ such that ...

Added: July 8, 2016

Shirokov D., Marchuk N., Advances in Applied Clifford Algebras 2008 Vol. 18 No. 2 P. 237–254

For the complex Clifford algebra <img /> (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These ...

Added: June 16, 2015

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2017. No. 1707.06914 [math.AG].

We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) ...

Added: July 24, 2017

Amerik E., Campana F., / Cornell University library. Series arxiv.org "algebraic geometry". 2013.

This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different, very short solution in the non-algebraic case and make some ...

Added: April 9, 2013

V. L. Popov, Izvestiya: Mathematics, England 2019 Vol. 83 No. 4 P. 830–859

The first 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: September 29, 2019

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

Р.С. Авдеев, Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3–48

For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...

Added: October 22, 2014