Fedor Bogomolov, Yuri Prokhorov
We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant and compute this group in some cases.
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The aim of this paper is to give a finer geometric description of the algebraic varieties parametrizing conjugacy classes of nonsolvable subgroups in the plane Cremona group.
Conjugacy classes of linear actions in the plane Cremona group
本文在Cremona群共轭意义下，对有限群在射影平面上的正则一般自由作用进行了分类。
We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.
This note presents the study of the conjugacy classes of elements of some given finite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if n is even, n=3 or n=5, and that it is equal to 3 (respectively 9) if n=9 (respectively 15), and is exactly 1 for all remaining odd orders. Some precise representative elements of the classes are given.
We give the classification of the maximal infinite algebraic subgroups of the real Cremona group of the plane up to conjugacy and present a parametrisation space of each conjugacy class. Moreover, we show that the real plane Cremona group is not generated by a countable union of its infinite algebraic subgroups.
On conjugacy of additive actions in the affine Cremona group
本文研究仿射Cremona群中加法作用的共轭性，证明射影空间上的任意两个加法作用可通过仿射Cremona群内由局部代数基本多项式给出的元素实现共轭，推进了代数群作用与双有理几何的相关研究。
We describe the conjugacy classes of affine automorphisms in the group $Aut(n,\K)$ (respectively $Bir(\K^n)$) of automorphisms (respectively of birational maps) of $\K^n$. From this we deduce also the classification of conjugacy classes of automorphisms of $\Pn$ in the Cremona group $Bir(\K^n)$.
We consider countably many three dimensional $\mathtt{PSL}_2(\mathbb{F}_7)$-del Pezzo surface fibrations over $\mathbb{P}^1$. Conjecturally they are all irrational except two families, one of which is the product of a del Pezzo surface with $\mathbb{P}^1$. We show that the other model is $\mathtt{PSL}_2(\mathbb{F}_7)$-equivariantly birational to $\mathbb{P}^2\times\mathbb{P}^1$. Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group $\mathtt{Cr}_3(\mathbb{C})$.
This survey deals with the Cremona group via its subgroups.
We classify, up to conjugacy, the subgroups of the Cremona group isomorphic to (Z/p)^r, where p is prime and r is maximal.
We obtain a sharp bound for p-elementary subgroups in the plane Cremona group over an arbitrary perfect field.
We prove that a finite $3$-group in the Cremona group $\mathrm{Cr}_3(\mathbb{C})$ 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.
Finite subgroups of Cremona group of rank 3 over the field of rational numbers
本文研究了有理数域上三维Cremona群有限子群的阶的显式界。
Algebraic subgroups of the plane Cremona group over a perfect field
该研究针对完美域上平面Cremona群的代数子群问题，证明了其任意无限代数子群都含于某个极大代数子群，并通过双有理映射共轭关系完成了极大群及有理点子群的分类。
We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds.
We classify all finite subgroups of the plane Cremona group which have a fixed point. In other words, we determine all rational surfaces X with an action of a finite group G such that X is equivariantly birational to a surface which has a G-fixed point.
For the subgroups of the Cremona group $\mathrm{Cr}_3(\mathbb C)$ having the form $(\boldsymbolμ_p)^s$, where $p$ is prime, we obtain an upper bound for $s$. Our bound is sharp if $p\ge 17$.
Maximal subgroups in the Cremona group
研究证明，在n≥5时，Cremona群中存在不被任何极大连通代数子群包含的连通代数子群，方法基于稳定有理但非有理三重面的存在。
Let Cr(k) be the Cremona group of rank 2 over a field k, i.e. the group of all k-automorphisms of k(X,Y). We determine the l.c.m. of the orders of the finite subgroups of Cr(k) of order prime to the characteristic of k.
We consider 3-subgroups in groups of birational automorphisms of rationally connected threefolds and show that any 3-subgroup can be generated by at most five elements. Moreover, we study groups of regular automorphisms of terminal Fano threefolds and prove that in all cases which are not among several explicitly described exceptions any 3-subgroup of such group can be generated by at most four elements.
Simple subgroups of the real space Cremona group
本文研究实三维射影空间双有理自映射群的有限非交换单子群，证明其中仅存在交错群$\mathfrak{A}_5$与$\mathfrak{A}_6$两类。
The derived length of subgroups of the Cremona group
本文证明了平面Cremona群可解子群的最大导出长度为5，有限可解子群为4，并通过中心化子定理和Blanc分类等工具给出证明。
An equivariant stable birational invariant of an action of a finite group on a smooth projective variety is the first cohomology group of the Picard module. Bogomolov-Prokhorov and Shinder computed this for actions of cyclic groups on rational surfaces, with maximal stabilizers, in terms of the geometry of the fixed point locus. Using the Brauer group of the quotient stack, we extend the computation to more general actions and relate it to the equivariant Burnside group formalism.
We prove that if $X$ is a rationally connected threefold and $G$ is a $p$-subgroup in the group of birational selfmaps of $X$, then $G$ is an abelian group generated by at most $3$ elements provided that $p\ge 17$. We also prove a similar result for $p\ge 11$ under an assumption that $G$ acts on a (Gorenstein) $G$-Fano threefold, and show that the same holds for $p\ge 5$ under an assumption that $G$ acts on a $G$-Mori fiber space.
The Cremona group $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is the group of birational self-maps of $\mathbb{P}^2_\mathbb{C}$. Using the action of $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ on the Picard-Manin space of $\mathbb{P}^2_\mathbb{C}$ we characterize its solvable subgroups. If $\mathrm{G}\subset\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is solvable, non abelian, and infinite, then up to finite index: either any element of $\mathrm{G}$ is of finite order or conjugate to an automorphism of $\mathbb{P}^2_\mathbb{C}$, or $\mathrm{G}$ preserves a unique fibration that is rational or elliptic, or $…
We classify all finite simple subgroups in the Cremona group of rank 3
It is well known that all Borel subgroups of a linear algebraic group are conjugate. This result also holds for the automorphism group ${\mathrm{Aut}} (\mathbb A^2)$ of the affine plane \cite{BerestEshmatovEshmatov2016} (see also \cite{FurterPoloni2018}). In this paper, we describe all Borel subgroups of the complex Cremona group ${{\rm Bir}({\mathbb P}^2)}$ up to conjugation, proving in particular that they are not necessarily conjugate. More precisely, we prove that ${{\rm Bir}({\mathbb P}^2)}$ admits Borel subgroups of any rank $r \in \{ 0,1,2 \}$ and that all Borel subgroups of rank $r \in…
The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an appliction, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.
We give some examples of non-nilpotent locally nilpotent, and hence nonlinear subgroups of the planar Cremona group.
Assuming 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 $G\subset Bir(X)$ there exists a normal abelian subgroup $A\subset G$ of index at most $J$. In particular, we obtain that the Cremona group $Cr_3=Bir(P^3)$ enjoys the Jordan property.
On abelian extensions of finite abelian subgroups of Cremona groups
本文研究Cremona群有限阿贝尔子群的扩张性质，证明在维数至多4时扩张与直积一致（除一例外），从而对有限阿贝尔群情形下忠实作用提升至终端Fano簇的问题给出肯定回答。
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 Bia\lynicki-Birula's results of 1966-67. We prove "fusion theorems" for $n$-dimensional tori in the affine and in the special affine Cremona groups of rank $n$. In the final section we introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.
Consider an algebraically closed field k and the Cremona group of all birational transformations of the projective plane over k. We characterize infinite order elements of this group having a non-zero power generating a proper normal subgroup of the Cremona group.
Linearization problem for finite subgroups of the plane Cremona group
该论文解决了特征零代数闭域上平面Cremona群有限子群的线性化问题，提供了完整解法。
This article gives the proof of results announced in [J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris, Sér. I 344 (2007), 21-26.] and some description of automorphisms of rational surfaces. Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive gen…
$p$-elementary non-cyclic subgroups of the Cremona group of the plane
研究平面克雷莫纳群中同构于\((\mathbb{Z}/p\mathbb{Z})^r\)（\(p\)素数，\(r \geq 2\)）的子群，在特定代数闭域上进行共轭分类，给出代表列表和共轭性研究，还给出特征等于\(p\)时的相关结果。
We give the classification of elements - respectively cyclic subgroups - of finite order of the Cremona group, up to conjugation. Natural parametrisations of conjugacy classes, related to fixed curves of positive genus, are provided.
On linearization problems in the plane Cremona group
研究平面 Cremona 群中可能稳定可线性化的有限非线性化子群。
This paper is concerned with suitable generalizations of a plane de Jonquières map to higher dimensional space $\mathbb{P}^n$ with $n\geq 3$. For each given point of $\mathbb{P}^n$ there is a subgroup of the entire Cremona group of dimension $n$ consisting of such maps. One studies both geometric and group-theoretical properties of this notion. In the case where $n=3$ one describes an explicit set of generators of the group and gives a homological characterization of a basic subgroup thereof.
This article shows that the Cremona group is compactly presentable. To prove this we show that it is a generalised amalgamated product of three of its algebraic subgroups (automorphisms of the plane and Hirzebruch surfaces) divided by one relation.
We complete the classical and modern work on the classification of conjugacy classes of finite subgroups of the group of birational transformations of the complex projective plane.
On p-Jordan constant of Cremona group of rank 2 in odd characteristic
研究2阶Cremona群在奇特征域上正常阿贝尔子群的索引界限，揭示其结构特性。
In this paper we describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.
This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on …
Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.
Let $G$ be a simple algebraic group over an algebraically closed field $K$ and let $N = N_G(T)$ be the normalizer of a fixed maximal torus $T\leq G$. Further, let $U$ be the unipotent radical of a fixed Borel subgroup $B$ that contains $T$ and let $U^-$ be the unipotent radical of the opposite Borel subgroup $B^-$. The Bruhat decomposition implies the decomposition $G = NU^-UN$. The Zariski closed subset $U^-U\subset G$ is isomorphic to the affine space $A_K^m$ where $m = \dim G -\dim T$ is the number of roots in the corresponding root system. Here we construct a subgroup $\mathcal{N}\leq \ope…
Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded
该研究证明平面Cremona群中仅含代数元素的有限生成子群次数有界，基于无穷直和上“良好”作用的更一般结论，还刻画了这类子群的次数增长特性。
For perfect fields $k$ satisfying $[\bar k:k]>2$, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank $n$ over (subfields of) the complex numbers is not simple for $n\geq3$.
Cremona groups of real surfaces
本文为实Cremona群Bir_R(P^2)的自然子群提供显式生成元，完善并统一了以往结果。
I study embeddings of alternating group of rank five into the Cremona group of rank three. I find all embeddings induced by A_5-del Pezzo fibrations and I study their conjugacy. As an application, I show that there is a series of continuous families of pairwise non-conjugate embeddings of alternating group of rank five into Cr$_3(\mathbb{C})$.
Linearization of finite subgroups of Cremona groups over non-closed fields
研究全局域上Cremona群有限子群的线性化性质，构造了在整体域上非k-线性化但在所有位点k_v-线性化的双有理对合，主要工具是推广Manin-Voskresenskiı和Bogomolov-Prokhorov的不变量。
We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of order of power of p larger than 2 and it does not contain elements of order p^2 unless p =2. Also we describe conjugacy classes of elements of order 4.
This paper contains a new proof of the classification of elements of prime order in the Cremona group Bir(P^2), up to conjugation. In addition, we give explicit geometric constructions of these Cremona transformations, and provide a parameterization of their conjugacy classes. Analogous constructions in higher dimensions are also discussed.
Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem is the affirmative. To this end we prove a theorem on invariant subfields of $1$-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationall…
Using a filtration on the Grothendieck ring of triangulated categories, we define the categorical dimension of a birational map between smooth projective varieties. We show that birational automorphisms of bounded categorical dimension form subgroups, which provide a nontrivial filtration of the Cremona group. We discuss some geometrical aspect and some explicit example. In the case of threefolds, we can moreover recover the genus of a birational automorphism, and the filtration defined by Frumkin.
Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the group of birational transformations of the projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out an overall study of centralizers of elements of infinite order in $\operatorname{Cr}_2(\mathbf{K})$ which leads to a classification of embeddings of $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well as a classification of maximal non-torsion abelian subgroups of $\operatorname{Cr}_2(\mathbf{K})$ when $\operatorname{char}(\mathbf{K})=0$.
We study the action of the Klein simple group G 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 G. As a by-product, we prove that X admits a Kahler-Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group G.
We give a way of estimating quickly the length of the hyperbolic isometry associated to a plane Cremona transformation that is presented as the product of two involutions. We also show, using the ideas of Cantat and Lamy, that, provided that the ground field is either algebraic of characteristic not 2 or of characteristic zero, the normal closure of a high power of such a transformation is a proper subgroup of the Cremona group; this gives an effective instantiation of some of their results. Moreover, we show that any hyperbolic Cremona transformation is rigid and give a simple criterion for i…
Representations of finite subgroups of Cremona groups
本文研究 Cremona 群有限子群忠实表示的最小维数，给出秩 1、2 的精确值，并证明正特征高秩情形为无穷大、特征 0 情形有限且下界随秩指数增长。
We present some (unfortunately not all) known properties on the Cremona group; when it's possible we mentioned links with the most known group of polynomial automorphisms of the affine plane. The mentioned properties are essentially algebraic properties: generators, relations, finite subgroups, subgroups of finite type, automorphisms of the Cremona group, Tits alternative... but also dynamical properties: classification of birational maps, centralizer, dynamic of an Heisenberg subgroup... We deal with the construction of entropic automorphisms on rational surfaces.
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