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## Three embeddings of the Klein simple group into the Cremona group of rank three

Transformation Groups. 2012. Vol. 17. No. 2. P. 303-350.

Cheltsov I., Shramov K.

We study the action of the Klein simple group PSL2(*F*7 ) 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(*F*7 ). As a by-product, we prove that *X* admits a Kähler–Einstein metric, and we construct a smooth polarized *K*3 surface of degree 22 with an action of the group PSL2( *F*7 ).

Unless explicitly stated otherwise, varieties are assumed to be projective, normal and complex.

Avilov A., / Cornell University. Series math "arxiv.org". 2022.

In this paper we classify nodal rational non-Q-factorial del Pezzo threefolds of degree 2 which can be G-birationally rigid for some subgroup G ⊂ Aut(X). ...

Added: December 8, 2022

Prokhorov Y., 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: April 9, 2014

Trepalin A., Central European Journal of Mathematics 2014

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: October 14, 2013

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

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

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

Prokhorov Y., Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 215-229

We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds. ...

Added: January 24, 2014

Popov V., / Cornell University. Series math "arxiv.org". 2012. No. arXiv:1207.5205v3.

We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the 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 generalizing to disconnected groups Bialynicki-Birula's results of 1966-67. We prove ``fusion ...

Added: January 9, 2013

Yuri Prokhorov, / 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

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

Andrey S. Trepalin, Central European Journal of Mathematics 2014 Vol. 12 No. 2 P. 229-239

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: December 3, 2013

Bogataya S., Bogatyi S., Кудрявцева Е. А., Математический сборник 2012 Т. 203 № 4 С. 103-118

We prove that the bound from the theorem on ‘economic’ maps is best possible. Namely, for m > n + d we construct a map from an n-dimensional simplex to an m-dimensional Euclidean space for which (and for any close map) there exists a d-dimensional plane whose preimage has cardinality not less than the upper ...

Added: October 30, 2012

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

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Loginov K., / Cornell University. Series math "arxiv.org". 2021.

We prove that a finite 3-group in the Cremona group Cr_3(ℂ) can be generated by at most 4 elements. This provides the last missing piece in bounding the ranks of finite p-subgroups in the space Cremona group. ...

Added: February 10, 2021

Avilov A., European Journal of Mathematics 2018 Vol. 4 No. 3 P. 761-777

We classify three-dimensional singular cubic hypersurfaces with an action of a finite group G, which are not G-rational and have no birational structure of G-Mori fiber space with the base of positive dimension. Also we prove the 𝔄5A5-birational superrigidity of the Segre cubic. ...

Added: September 16, 2018

Vladimir L. Popov, / 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

Cheltsov I., Shramov K., CRC Press, 2015

Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A5 in them. The book surveys known facts about surfaces with an action of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and gives a proof of its A5-birational rigidity.
The authors ...

Added: October 6, 2015

Prokhorov Y., Shramov K., American Journal of Mathematics 2016 Vol. 138 No. 2 P. 403-418

Assuming the Borisov-Alexeev-Borisov conjecture, we prove that there is a constant $J=J(n)$ such that for any rationally connected variety $X$ of dimension $n$ and any finite subgroup $G\subset{\rm Bir}(X)$ there exists a normal abelian subgroup $A\subset G$ of index at most $J$. In particular, we obtain that the Cremona group ${\rm Cr}_3={\rm Bir}({\Bbb P}^3)$ enjoys ...

Added: August 31, 2016

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

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

Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2016.

We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks 2 and 3. ...

Added: September 26, 2016

Avilov A., Математический сборник 2023 Т. 214 № 6 С. 3-40

In this paper we classify nodal rational non-Q-factorial del Pezzo threefolds of degree 2 which can be G-birationally rigid for some subgroup G ⊂ Aut(X). ...

Added: December 7, 2022

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