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## Flexibility of normal affine horospherical varieties

Cornell University
,
2018.
No. arXiv:1805.05024.

Gayfullin S., Шафаревич А. А.

Gayfullin S., Shafarevich Anton, Proceedings of the American Mathematical Society 2019 Vol. 147 P. 3317–3330

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only con- stant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety with only constant invertible functions is flexible. ...

Added: October 17, 2019

Gayfullin S., Journal of Algebra 2021 No. 573 P. 364–392

In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov. ...

Added: February 6, 2021

Gayfullin S., / Cornell University. Series arXiv "math". 2018. No. arXiv:1709.09237.

In 2007, Dubouloz introduced Danielewski varieties. Such varieties general- ize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov. ...

Added: September 1, 2018

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

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

Arzhantsev I., Gayfullin S., Mathematische Nachrichten 2017 Vol. 290 No. 5-6 P. 662–671

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of is pointed, the group is a finite extension of . As an application, ...

Added: February 19, 2017

Borovik V., Gayfullin S., Shafarevich A., Mathematische Nachrichten 2024 Vol. 297 No. 9 P. 3174–3183

In this paper, we describe orbits of the automorphism group on an affine horospherical variety in terms of degrees of homogeneous with respect to natural grading locally nilpotent derivations. In case of (possibly nonnormal) toric varieties, a description of orbits of the automorphism group in terms of corresponding weight monoid is obtained. ...

Added: September 18, 2024

Gayfullin S., Shafarevich A., Journal of Pure and Applied Algebra 2024 Vol. 228 No. 6 Article 107616

We investigate modified Makar-Limanov and Derksen invariants of an affine algebraic variety. The modified Makar-Limanov invariant is the intersection of kernels of all locally nilpotent derivations with slices and the modified Derksen invariant is the subalgebra generated by these kernels. We prove that the modified Makar-Limanov invariant coincides with the Makar-Limanov invariant if there exists ...

Added: June 7, 2024

Gayfullin S., / Cornell University. Series arXiv "math". 2018.

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety is flexible. ...

Added: December 6, 2018

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

Boldyrev I., Gayfullin S., Математические заметки 2021 Т. 110 № 6 С. 837–855

Criteria for the flexibility, rigidity, and almost rigidity of nonnormal affine toric varieties are obtained. For rigid and almost rigid toric varieties, automorphism groups are explicitly calculated. ...

Added: February 6, 2022

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

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

Arzhantsev I. V., Gaifullin S.A., Sbornik Mathematics 2010 Vol. 201 No. 1 P. 1–21

We study the Cox realization of an affine variety, that is, a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of ...

Added: December 17, 2014

Shafarevich A., Математический сборник 2017 Т. 208 № 2 С. 285–310

Let k be an algebraically closed field of characteristic zero and Ga = (k, +) the additive group of k. An algebraic variety X is said to be flexible if the tangent space at every regular point of X is generated by the tangent vectors to orbits of various regular actions of Ga. In 1972, ...

Added: December 5, 2018

Perepechko A., Michałek M., Süß H., Mathematische Zeitschrift 2018 Vol. 290 No. 3-4 P. 1457–1478

We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces. ...

Added: September 26, 2019

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., 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., Kuyumzhiyan K., Zaidenberg M., Математический сборник 2012 Т. 203 № 7 С. 3–30

We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on ...

Added: September 12, 2012

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

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

Added: July 10, 2014

Shafarevich A., Proceedings of the American Mathematical Society 2019 Vol. 147 No. 8 P. 3317–3330

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety with only constant invertible functions is flexible. ...

Added: September 10, 2019

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

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