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## Автоморфизмы поверхностей марковского типа

Математические заметки. 2021. Т. 110. № 5. С. 744–750.

Affine algebraic surfaces of Markov type of the form

x^2 + y^2 + z^2 − xyz = c

are studied. Their automorphism groups are found.

Perepechko A., Mathematical notes 2021 Vol. 110 No. 5 P. 732–737

Affine algebraic surfaces of Markov type of the form (Formula presented.) are studied. Their automorphism groups are found. ...

Added: October 28, 2022

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

Kuyumzhiyan K., Proceedings of the American Mathematical Society 2020 No. 148 P. 3723–3731

We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_n_i, where the n_i are pairwise distinct, acts m-transitively for each m. ...

Added: August 18, 2020

Perepechko A., Regeta A., Transformation Groups 2023 Vol. 28 P. 401–412

For an affine algebraic variety X, we study the subgroup Autalg(X) of the group of regular automorphisms Aut(X) of X generated by all the connected algebraic subgroups. We prove that Autalg(X) is nested, i.e., is a direct limit of algebraic subgroups of Aut(X), if and only if all the Ga-actions on X commute. Moreover, we ...

Added: October 28, 2022

Trushin A., Математические заметки 2023 Т. 113 № 5 С. 780–784

It is known that the group of automorphisms of the algebra of polynomials in three variables is not generated by elementary automorphisms. In the article, a system of generators is constructed for the group of automorphisms preserving the nontrivial grading by integers. ...

Added: September 18, 2024

Perepechko A., Функциональный анализ и его приложения 2013 Т. 47 № 4 С. 45–52

We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. ...

Added: September 26, 2019

Cerulli Irelli G., Feigin E., Reineke M., / Cornell University. Series math "arxiv.org". 2012. No. 1206.4178.

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study ...

Added: June 29, 2012

Arzhantsev I., Perepechko A., Shakhmatov K., Bulletin des Sciences Mathematiques 2024 Vol. 192 Article 103419

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 ...

Added: April 12, 2024

Chebochko N.G., Kuznetsov M. I., Communications in Algebra 2017 Vol. 45 No. 7 P. 2969–2977

All classes of integrable cocycles in H2(L,L) are obtained for Lie algebra of type G2 over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. ...

Added: October 10, 2017

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

Shafarevich A., Moscow University Mathematics Bulletin 2019 Vol. 74 No. 5 P. 209–211

Let X be an affine toric variety over an algebraically closed field of characteristic zero. Orbits of connected component of identity of automorphism group in terms of dimensions of tangent spaces of the variety X are described. A formula to calculate these dimensions is presented. ...

Added: September 10, 2021

Arzhantsev I., Zaitseva Y., Research in Mathematical Sciences 2024 Vol. 11 No. 2 Article 27

An algebraic variety X is called a homogeneous variety if the automorphism group Aut(X) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we ...

Added: March 23, 2024

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

Kikteva V., Математический сборник 2024 Т. 215 № 10 С. 89–113

We obtain a criterion for the automorphism group of an affine toric variety to be connected in combinatorial terms and in terms of the divisor class group of the variety. The component group of the automorphism group of a non-degenerate affine toric variety is described. In particular, we show that the number of connected components ...

Added: September 30, 2024

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. L., Zarhin Y., / Cornell University. Series math "arxiv.org". 2018. No. 1808.01136.

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are ...

Added: August 8, 2018

Vladimir L. Popov, Transformation Groups 2014 Vol. 19 No. 2 P. 549–568

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: March 17, 2014

Shramov K., Prokhorov Y., / Cornell University. Series arXiv "math". 2019.

We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kaehler manifold of ...

Added: November 19, 2019

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...

Added: July 21, 2013

Zhukova N., В кн.: <i>Международная молодежная школа-семинар "Современная геометрия и ее приложения". Международная конференция "Современная геометрия и ее приложения". Материалы школы-семинара и конференции.</i>. Каз.: Издательство Казанского университета, 2017. С. 48–51.

We introduce a category of rigid geometries on smooth singular spaces of leaves of foliations.
A special category $\mathfrak F_0$ containing orbifolds is allocated. Unlike orbifolds, objects
of $\mathfrak F_0$ can have non-Hausdorff topology and can even not satisfy the separability axiom $T_0$.
It is shown that the rigid geometry $(N,\zeta)$, where $N\in (\mathfrak F_0)$, allows a desingularization. ...

Added: April 1, 2018

Zhukova N., Moscow Mathematical Journal 2018

We introduce a category of rigid geometries on singular spaces which
are leaf spaces of foliations and are considered as leaf manifolds. We
single out a special category F_0 of leaf manifolds containing the orbifold
category as a full subcategory. Objects of F_0 may have non-Hausdorff
topology unlike the orbifolds. The topology of some objects of F_0 does
not satisfy ...

Added: April 2, 2018

Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2018.

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. ...

Added: June 8, 2019

Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2022. No. 2202.07507.

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence ...

Added: September 12, 2022

Nina I. Zhukova, Anna Yu. Dolgonosova .., Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076–2088

The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...

Added: September 28, 2014