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## Around the Abhyankar–-Sathaye conjecture

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.

Gayfullin S., Chunaev D., Фундаментальная и прикладная математика 2023 Т. 24 № 4 С. 47–59

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

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

Popov V. L., Mathematical notes 2019 Vol. 105 No. 3-4 P. 580–581

It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf–Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018), is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems. ...

Added: May 27, 2019

Kochetkov Y., Mathematical notes 2011 Vol. 89 No. 1-2 P. 150–155

The action of the group of totally positive units on the convex hull of the semigroup of totally positive integers of a real cubic Galois field is studied. The fundamental domain of this action has a simple description in the case of so called regular field. ...

Added: March 10, 2013

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

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

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2014. No. 1409.6330.

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 24, 2014

Vladimir L. Popov, Pacific Journal of Mathematics 2015 Vol. 279 No. 1--2 (Special issue In memoriam: Robert Steinberg) P. 423–446

For the coordinate algebras of connected affine algebraic groups, we explore
the problem of finding a presentation by generators and relations canonically
determined by the group structure. ...

Added: December 27, 2015

Kochetkov Y., Математические заметки 2011 Т. 89 № 1 С. 139–144

The action is considered of a group of totally positive units of a real cubic Galois field on the border of convex hull of its totally positive integers. In case of so called regular fields the fundamental domain of this action has a simple description. ...

Added: May 15, 2012

Popov V., Труды Математического института им. В.А. Стеклова РАН 2023 Т. 320 С. 287–297

Построена новая бесконечная серия рациональных аффинных алгебраических многообразий, группа автоморфизмов которых содержит группу Aut}(F_n) автоморфизмов свободной группы ранга n. Группы автоморфизмов таких многообразий нелинейны и содержат группу кос B_n c n нитями при n > 2, а при n > 1 неаменабельны. В качестве приложения доказано, что при n > 2 каждая группа Кремоны ранга ...

Added: June 9, 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

Popov V., Известия РАН. Серия математическая 2013 Т. 77 № 4 С. 103–134

We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups.
We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bialynicki-Birula's results of 1966--67. We prove ``fusion theorems'' ...

Added: June 3, 2013

V. L. Popov, Doklady Mathematics 2016 Vol. 93 No. 3 P. 307–309

A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rational triangulability. The ...

Added: June 29, 2016

Vladimir L. Popov, Proceedings of the Steklov Institute of Mathematics 2016 Vol. 292 P. 209–223

For every algebraically closed field k of characteristic different from 2, we prove
the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type
of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple
of algebraically independent (over k) rational functions of the structure constants. (2) There
exists an “algebraic normal form” to ...

Added: March 29, 2016

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

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

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

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2014. No. 1401.0278.

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: January 3, 2014

Popov V., Успехи математических наук 2023 Т. 78 № 4(472) С. 209–210

The Picard group of the group variety of any connected affine algebraic group defined over an arbitrary algebraically closed field is calculated. ...

Added: January 7, 2024

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

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1508.02860.

For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure. ...

Added: August 13, 2015

Gayfullin S., Шафаревич А. А., / Cornell University. Series arXiv "math". 2018. No. arXiv:1805.05024.

Added: September 1, 2018

V. L. Popov, Proceedings of the Steklov Institute of Mathematics, Springer 2019 Vol. 307 P. 193–197

We prove that, for any prime integer $p\geqslant 2$, there exists an algebraic action of the two-dimensional Witt group $W_2(p)$ on an al\-geb\-raic variety $X$ and a point $x\in X$ such that the closure of the $W_2(p)$-orbit of $x$
in $X$ contains infinitely many $W_2(p)$-orbits.\;This is related to the problem of extending from characteristic zero to ...

Added: March 30, 2020

Arzhantsev I., Communications in Algebra 2008 Vol. 36 No. 12 P. 4368–4374

Added: July 10, 2014