### ?

## The automorphism group of a rigid affine variety

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, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.

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. 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

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

Arzhantsev I., 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. ...

Added: August 16, 2021

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

Added: September 1, 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., 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

Piontkovski D., Advances in Mathematics 2019 Vol. 343 P. 141-156

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to ...

Added: February 26, 2019

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

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

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

Rumynin D., Taylor J., Linear Algebra and its Applications 2021 Vol. 610 P. 135-168

We use the structure of finite-dimensional graded algebras to develop the theory of antilinear representations of finite C_2-graded groups. A finite C_2-graded group is a finite group with a subgroup of index 2. In this theory the subgroup acts linearly, while the other coset acts antilinearly. We introduce antilinear blocks, whose structure is a crucial ...

Added: September 7, 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

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

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

Buchstaber V.M., Terzić S., Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 397-463

The family of the complex Grassmann manifolds G(n,k) with the canonical action of the torus T-n = T-n and the analogue of the moment map mu: G(n,k) ->Delta(n,)(k) for the hypersimplex Delta(n,) (k), is well known. In this paper we study the structure of the orbit space G(n,k)/T-n by developing the methods of toric geometry ...

Added: June 18, 2021

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., Acta Arithmetica 2018 Vol. 186 No. 1 P. 87-99

We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwarz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwarz-Halphen's Theorem ...

Added: October 20, 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

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., Бухштабер В. М., / Arxiv Cornell University Library. Series 1803.10449 "1803.10449". 2018. No. 10449.

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space MStn,λ of Hermitian arrow (n+1)×(n+1)-matrices with fixed simple spectrum λ. We prove that this space is 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: October 15, 2018

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

Added: July 10, 2014