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## Three plots about the Cremona groups

Izvestiya: Mathematics, England. 2019. Vol. 83. No. 4. P. 830-859.

The first 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.

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

Kleptsyn V., Alvarez S., Malicet D. et al., Groups with infinitely many ends acting analytically on the circle / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

Vladimir L. Popov, Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 185-213

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.
The appendix is an expanded version ...

Added: April 28, 2014

Ivan Cheltsov, Constantin Shramov, Transactions of the American Mathematical Society 2014 Vol. 366 No. 3 P. 1289-1331

We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we obtain that the Cremona group of rank 3 has at least five non-conjugate subgroups isomorphic to ...

Added: October 10, 2013

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

Tokyo : American Mathematical Society, World Scientific, 2017

Preface
The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.
Recently, there have been remarkable developments in algebraic geometry and related fields, ...

Added: July 12, 2017

Filimonov D., Клепцын В. А., Nonlinearity 2014 Vol. 27 No. 6 P. 1205-1223

We study possible one-end finitely presented subgroups of <img />, acting without finite orbits. Our main result, theorem 1, establishes that any such action possesses the so-called property (<img />), that allows one to make distortion-controlled expansion and is thus sufficient to conclude that the action is Lebesgue-ergodic. We also propose a path towards full ...

Added: October 23, 2014

Gusein-Zade S., Раух А. Я., Функциональный анализ и его приложения 2021 Т. 55 № 1 С. 56-64

V.I.Arnold classified simple (i.e. having no moduli for the classification) singularities (function germs) and also simple boundary singularities: function germs invariant with respect to the action
σ(x1;y1,…,yn)=(−x1;y1,…,yn) of the group Z2. In particular, it was shown that a function germ (a germ of a boundary singularity) is simple if and only if the intersection form (respectively, ...

Added: February 3, 2021

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

Trepalin A., International Journal of Mathematics 2019 Vol. 30 No. 11

Let $\ka$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\Aut(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\ka$. Obviously, if there are no smooth $\ka$-points on $X / G$ then it is not $\ka$-rational. ...

Added: October 19, 2019

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

Yuri Prokhorov, On stable conjugacy of finite subgroups of the plane Cremona group, II / Cornell University. Series math "arxiv.org". 2013.

We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability. ...

Added: October 10, 2013

Vladimir L. Popov, Three plots about the Cremona groups / Cornell University. Series math "arxiv.org". 2018. No. 1810.00824.

The first 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: October 2, 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., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

Cheltsov I., Известия РАН. Серия математическая 2014 Т. 78 № 2 С. 167-224

We prove two new local inequalities for divisors on smooth surfaces and consider several applications of these inequalities. ...

Added: December 6, 2013

Amerik E., Campana F., On families of lagrangian tori on hyperkaehler manifolds / . 2013.

This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different, very short solution in the non-algebraic case and make some ...

Added: April 9, 2013

Gusein-Zade S., Математические заметки 2020 Т. 107 № 6 С. 855-864

V.I.Arnold has classified simple (i.e., having no moduli for the classification) singularities (function germs), and also simple boundary singularities: function germs invariant with respect to the action σ (x1; y1, …, yn) = (−x1; y1, …, yn) of the group ℤ2. In particular, it was shown that a function germ (a boundary singularity germ) is ...

Added: October 27, 2020

Popov V., Izvestiya. Mathematics 2013 Vol. 77 No. 4 P. 742-771

We classify up to conjugacy the subgroups of certain types in the full, affine, and special affine Cremona groups.
We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the linearization problem by generalizing Bia{\l}ynicki-Birula's results of 1966--67 to disconnected groups.
We prove fusion theorems for n-dimensional tori in ...

Added: August 23, 2013

Ebeling W., Gusein-Zade S., International Mathematics Research Notices 2021 Vol. 2021 No. 16 P. 12305-12329

A.Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement about the equivariant Saito duality ...

Added: August 26, 2021

Gusein-Zade S., Функциональный анализ и его приложения 2018 Т. 52 № 2 С. 78-81

Let G be a finite Abelian group acting (linearly) on space ℝn and, therefore, on its complexification ℂn, and let W be the real part of the quotient ℂn/G (in the general case, W ≠ ℝn/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the ...

Added: October 27, 2020

Ю. Г. Прохоров, Известия РАН. Серия математическая 2013 Т. 77 № 3 С. 199-222

We study elements $\tau$ of order two in the birational automorphism groups of rationally connected three-dimensional algebraic varieties such that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program, we give a rough classification of such elements. ...

Added: July 1, 2013

Vladimir L. Popov, Jordan groups and automorphism groups of algebraic varieties / 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

Cheltsov I., Shramov K., Transformation Groups 2012 Vol. 17 No. 2 P. 303-350

We study the action of the Klein simple group PSL2(F7 ) consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to PSL2(F7 ). As a ...

Added: August 30, 2012