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Log terminal singularities, platonic tuples and iteration of Cox rings
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 coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions. As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.
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
Popov V., / Bielefeld University. Series LAGRS "Linear Algebraic Groups and Related Structures". 2012. No. 485.
We construct counterexamples to the rationality conjecture regar-ding the new version of the Makar-Limanov invariant introduced in A. Liendo, Ga-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653–3665. ...
Added: January 9, 2013
Gayfullin S., Chunaev D., Фундаментальная и прикладная математика 2023 Т. 25
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
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
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
Vladimir L. Popov, Journal of the Ramanujan Mathematical Society 2013 Vol. 28A No. Special Issue-2013 dedicated to C.S.Seshadri's 80th birthday P. 409-415
We construct counterexamples to the rationality conjecture regarding the new version of the Makar-Limanov invariant formulated in A. Liendo, G_a-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653--3665. ...
Added: June 20, 2013
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
Vladimir L. Popov, European Journal of Mathematics 2016 Vol. 2 No. 1 P. 283-290
According to the classical theorem, every 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 projective space of a positive dimension. We show that the classical proof of this theorem
actually works only in characteristic 0 and we give a characteristic free
proof ...
Added: February 2, 2016
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
Ayzenberg A., Cherepanov V., / 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
Popov V., Transformation Groups 2023 Vol. 28 No. 3 P. 1277-1297
Considering a certain construction of algebraic varieties X endowed with an
algebraic action of the group Aut(F_n), n < \infty, we obtain a criterion for the faithfulness
of this action. It gives an in nite family F of Xs such that Aut(F_n) embeds into Aut(X).
For n > 2, this implies nonlinearity, and for n > 1, the ...
Added: January 7, 2024
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
Попов В. Л., Известия РАН. Серия математическая 2019 Т. 84 № 4 С. 194-225
The rst 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: July 31, 2019
Ayzenberg A., / 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
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
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
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
Ayzenberg A., Masuda M., / 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
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
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
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
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
Zürich : European Mathematical Society Publishing house, 2010
Fascinating and surprising developments are taking place in the classification of algebraic varieties. Work of Hacon and McKernan and many others is causing a wave of breakthroughs in the Minimal Model Program: we now know that for a smooth projective variety the canonical ring is finitely generated. These new results and methods are reshaping the ...
Added: October 11, 2013