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## Limit points and additive group actions

Ricerche di Matematica. 2021.

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*.

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

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., Automorphisms of Danielewski varieties / 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

Ayzenberg A., Труды Математического института им. В.А. Стеклова РАН 2018 Т. 302 С. 23-40

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated xed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a ltration of the orbit manifold by orbit dimensions. The subset ...

Added: October 15, 2018

Gayfullin S., Zaitseva Y., Journal of Algebra and its Applications 2019 Vol. 18 No. 10 P. 1950196-1-1950196-14

We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras. ...

Added: September 10, 2019

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

Added: April 28, 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

Ayzenberg A., Torus action on quaternionic projective plane and related spaces / Cornell University. Series arXiv "math". 2019. No. 1903.03460.

For an action of a compact torus T on a smooth compact manifold~X with isolated fixed points the number 12dimX−dimT is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that HP2/T3≅S5 and S6/T2≅S4, for the homogeneous spaces HP2=Sp(3)/(Sp(2)×Sp(1)) and S6=G2/SU(3). Here the maximal tori of the corresponding Lie ...

Added: October 23, 2019

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

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

Gayfullin S., Шафаревич А. А., Flexibility of normal affine horospherical varieties / Cornell University. Series arXiv "math". 2018. No. arXiv:1805.05024.

Added: September 1, 2018

Arzhantsev I., Liendo A., Stasyuk T., Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499

Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...

Added: July 29, 2020

Arzhantsev I., Hausen J., Mathematical Research Letters 2007 Vol. 14 No. 1 P. 129-136

Given a multigraded algebra A, it is a natural question whether or not for
two homogeneous components A_u and A_v, the product A_nuA_nv is the whole component
A_nu+nv for n big enough. We give combinatorial and geometric answers to this question. ...

Added: July 10, 2014

Ayzenberg A., Cherepanov V., Torus actions of complexity one in non-general position / Cornell University. Series arXiv "math". 2019. No. 1905.04761.

Let the compact torus Tn−1 act on a smooth compact manifold X2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X2n/Tn−1 if the action is cohomologically equivariantly formal (which essentially means that Hodd(X2n;Z)=0). It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite ...

Added: October 23, 2019

Ayzenberg A., Бухштабер В. М., Математический сборник 2021

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space
$M_{\St_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove this space to be a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe ...

Added: November 6, 2020

Ayzenberg A., Masuda M., Orbit spaces of equivariantly formal torus actions / Cornell University. Series arXiv "math". 2019.

Let a compact torus T=T^{n−1} act on a smooth compact manifold X=X^{2n} effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If H^{odd}(X)=0 and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space Q=X/T is a homology (n+1)-sphere. If, in addition, π_1(X)=0, then Q is homeomorphic to S^{n+1}. ...

Added: January 14, 2020

Bilich B., Classification of noncommutative monoid structures on normal affine surfaces / 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

Ayzenberg A., Cherepanov V., Osaka Journal of Mathematics 2021 Vol. 58 No. 4 P. 839-853

Let the compact torus Tn1 act on a smooth compact manifold X2n eectively with nonempty nite set of xed points. We pose the question: what can be said
about the orbit space X2n{Tn1 if the action is cohomologically equivariantly formal
(which essentially means that HoddpX2n;Zq 0)? It happens that homology of the orbit
space can be arbitrary ...

Added: October 31, 2019

Ayzenberg A., Algebraic and Geometric Topology 2020 Vol. 20 No. 6 P. 2957-2994

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized S\edt{c}hr\"{o}dinger operator we describe all spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The space $X_{n,\lambda}$ carries a natural effective action of a compact $(n-1)$-torus. ...

Added: January 14, 2020

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

Arzhantsev I., Braun L., Hausen J. et al., European Journal of Mathematics 2018 Vol. 4 No. 1 P. 242-312

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension ...

Added: March 4, 2018

Ayzenberg A., Arnold Mathematical Journal 2020 P. 1-24

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number 12dimX−dimT12dimX−dimT is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that HP2/T3≅S5HP2/T3≅S5 and S6/T2≅S4S6/T2≅S4, for the homogeneous spaces HP2=Sp(3)/(Sp(2)×Sp(1))HP2=Sp(3)/(Sp(2)×Sp(1)) and S6=G2/SU(3)S6=G2/SU(3). Here, the maximal tori of ...

Added: November 19, 2020

Cherepanov V., Математический сборник 2019

We consider effective actions of a compact torus Tn−1 on an even-dimensional smooth manifold M2n with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to Sn+1∖(U1⊔…⊔Ul) where Sn+1 is the (n+1)--sphere and U1,…,Ul are open domains. We apply the results to ...

Added: October 28, 2020

Arzhantsev I., Kuyumzhiyan K., Zaidenberg M., Advances in Mathematics 2019 Vol. 351 P. 1-32

An affine algebraic variety X of dimension ≥2 is called flexible if the subgroup SAut(X)⊂Aut(X) generated by the one-parameter unipotent subgroups acts m-transitively on reg(X) for any m≥1. In the previous paper we proved that any nondegenerate toric affine variety X is flexible. In the present paper we show that one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same ...

Added: May 15, 2019