Finite subgroups of Cremona group of rank 3 over the field of rational numbers
Alexandr Zaitsev
本文研究了有理数域上三维Cremona群有限子群的阶的显式界。
我们给出了有理数域上三维Cremona群有限子群的阶的显式界。
We give an explicit bound on orders of finite subgroups of Cremona group of rank three over $\mathbb{Q}$.
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This survey deals with the Cremona group via its subgroups.
Bounds for the order for p-elementary subgroups in the plane Cremona group over a perfect field
得到了任意完满域上平面Cremona群中p-初等子群的一个精确界.
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.
p-elementary subgroups of the Cremona group
我们对同构于（Z/p）^r的Cremona群的子群进行共轭分类，其中p是素数，r是极大子群。
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.
The conjugacy classes of finite nonsolvable subgroups in the plane Cremona group
给出了平面Cremona群中非可解子群共轭类的代数簇的几何刻画。
On stable conjugacy of finite subgroups of the plane Cremona group, II
证明了除少数情况外，平面Cremona群的有限子群的稳定线性化蕴涵线性化。
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。
On stable conjugacy of finite subgroups of the plane Cremona group, I
讨论了Cremona群的有限子群的稳定共轭性问题。我们证明了群$H^1(G,Pic(X))$是一个稳定的双有理不变量，并在某些情况下计算了这个群。
2-elementary subgroups of the space Cremona group
本文给出了有理连通三重双有理自同构的初等交换2-群的阶的一个精确界。
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.
p-elementary subgroups of the Cremona group of rank 3
对于Cremona群$\mathrm{Cr}_3(\mathbb C)$的子群具有$(\boldsymbolμ_p)^s$的形式，其中$p$是素数，我们得到了$s$的一个上界.如果$p\ge 17$，则我们的界是尖锐的.
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。
A Minkowski-style bound for the order of the finite subgroups of the Cremona group of rank 2 over an arbitrary field
设Cr（k）是域k上秩为2的Cremona群，即k（X，Y）的所有k-自同构的群。我们确定l.c.m. Cr（k）的阶与k的特征数互素的有限子群的阶。
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。
A Cremona transformation is a birational self-map of the projective space $ \mathbb{P}^{n} $. Cremona transformations of $ \mathbb{P}^{n} $ form a group and this group has a rational action on subvarieties of $ \mathbb{P}^{n} $ and hence on its Hilbert scheme. We study this action on the family of rational curves of $ \mathbb{P}^{3} $ and we prove the rectifiability of any one dimensional family. This shows that any uniruled surface is Cremona equivalent to a scroll and it answers a question of Bogomolov-Böhning related to the study of uniformly rational varieties. We provide examples of infin…
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。
New rational cubic fourfolds arising from Cremona transformations
本文证明判别式20的非常一般三次四维流形中Fourier-Mukai等价蕴含双有理等价，并借助Veronese曲面定义的Cremona变换发现新的有理三次四维流形。
Simple finite subgroups of the Cremona group of rank 3
我们对秩为3的Cremona群中的所有有限单群进行了分类
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 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.
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 give some examples of non-nilpotent locally nilpotent, and hence nonlinear subgroups of the planar Cremona group.
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 $r \in…
Consider an algebraically closed field k and the Cremona group of all birational transformations of the projective plane over k. We characterize infinite order elements of this group having a non-zero power generating a proper normal subgroup of the Cremona group.
Linearization problem for finite subgroups of the plane Cremona group
该论文解决了特征零代数闭域上平面Cremona群有限子群的线性化问题，提供了完整解法。
Jordan property for Cremona groups
在Borisov-Alexeev-Borisov猜想的假设下，证明了存在一个常数$J=J(n)$，使得对任意维数为$n$的有理连通簇$X$和任意有限子群$G\subset Bir(X)$，存在一个指数至多为$J$的正规交换子群$A\subset G$.特别地，我们得到了Cremona群$Cr_3=Bir(P^3)$具有Jordan性质.
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。
Elements and cyclic subgroups of finite order of the Cremona group
我们给分类的元素-分别循环子群-有穷级的克雷莫纳群，共轭。给出了与正亏格不动曲线相关的共轭类的自然参数化。
Linearisation of finite abelian subgroups of the Cremona group of the plane
本文给出了[J. Blanc，Finite Abel subgroups of the Cremona group of the plane，C.R. Acad. Sci.巴黎先生I 344（2007），21-26.]以及有理曲面的自同构的一些描述。给出了平面Cremona群的有限Abel子群与极小曲面的自同构群群是否双有理共轭的一种判定方法。特别地，我们证明了平面的双有理变换的有限循环群是线性化的当且仅当它的非平凡元素都不固定正亏格的曲线。对于有限阿贝尔群，只有一个令人惊讶的例外，即同构于Z/2 ZxZ/4 Z的群，其非平凡元素不固定正亏格的曲线，但不共轭于极小有理曲面的自同构自同构群。我们也给出了del Pezzo曲面和锥丛的自同构（不一定是有穷级的）的一些描述。
$p$-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。
Two birational subvarieties of P^n are called Cremona equivalent if there is a Cremona modification of P^n mapping one to the other. If the codimension of the varieties is at least 2 then they are always Cremona Equivalent. For divisors the question is much more subtle and a general answer is unknown. In this paper I study the case of rational quartic surfaces and prove that they are all Cremona equivalent to a plane.
A Rationality Problem of Some Cremona Transformation
我们将证明在一定的克雷莫纳变换下二元函数域的合理性。
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。
Tori in the Cremona groups
我们分类共轭的子群的某些类型的充分，在仿射，并在特殊的仿射克雷莫纳群体。我们证明了这些子群的正规化子是代数的。作为应用，我们在推广到不连通群的线性化问题中得到了新的结果，Bia \lynicki -Birula在1966-67年的结果。我们证明了“融合定理”的$n$维环面的仿射和特殊的仿射克雷莫纳集团的秩$n$。在最后一节中，我们介绍并讨论了Cremona群的若尔当分解和挠素数的概念。
Finite subgroups of the plane Cremona group
我们完成了古典和现代的工作分类的共轭类的有限子群的双有理变换组的复射影平面。
Cremona maps of de Jonquières type
本文利用$n\geq 3$将平面de Jonquières映射推广到高维空间$\mathbb{P}^n$.对于$\mathbb{P}^n$的每一个给定的点，都有一个由这样的映射组成的维度为$n$的整个克雷莫纳群的子群。一个研究这一概念的几何和群论性质。在这种情况下，其中$n=3$一个描述了一个明确的一组发电机的组，并给出了一个同调表征的基本子群。
In this paper we describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.
Normal subgroups in the Cremona group (long version)
设k是一个代数闭域。证明了k上射影平面P^2的所有双有理变换的Cremona群不是单群。该策略利用双曲几何，几何群论，代数几何产生的克雷莫纳群，产生非平凡的正规子群的元素。
The Cremona group is compactly presentable
这篇文章表明，克雷莫纳集团是compatible presentable。为了证明这一点，我们表明，它是一个广义的合并产品的三个代数子群（自同构的平面和Hirzebruch表面）除以一个关系。
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。
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。
On Characteristic Classes Of Determinantal Cremona Transformations
我们计算的multidegrees和一般行列式克雷莫纳变换的Segre数，一般减少基计划，通过专门的标准克雷莫纳变换和计算其Segre类通过混合卷的合理多面体。
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…
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。
Sous-groupes algébriques du groupe de Cremona
给出了平面Cremona群的极大代数子群的完全分类，并给出了共轭类的参数化代数簇。--我们将克雷莫纳地区最大的亚组进行了完整的分类，并明确了以共轭类为参数的各种类型。
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.
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。
Bass' triangulability problem
研究了仿射Cremona群的幂幺代数子群的Bass可三角化问题，证明了一个可三角化准则，证明了Cremona群的不可三角化连通可解仿射代数子群的存在性，以及这类子群的稳定可三角化性，特别地，在稳定范围内，我们回答了Bass可三角化问题是肯定的.为此，我们证明了一个定理的不变子域的$1$ -扩张。我们还得到了一般的建设，所有合理的三角化子群的Cremona集团，并作为应用，分类合理的三角化连接一维幂幺仿射代数子群的Cremona集团共轭。
Birational classification of curves on rational surfaces
本文研究了（S，L）对的双有理分类，其中S是有理曲面，L是S上的线性系。我们给出了一个分类定理，这样的对，我们确定，对于每个不可约的平面曲线B，它的“克雷莫纳最小”模型，即那些平面曲线是等价的B通过克雷莫纳变换，并在此条件下具有最小的程度。
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 $\operatorname{char}(\mathbf{K})=0$.
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。
Three embeddings of the Klein simple group into the Cremona group of rank three
研究了由168个元素组成的Klein单群G在两个有理三重空间上的作用：三维射影空间和反正则度为22且指数为1的光滑Fano三重空间X.证明了秩为3的Cremona群至少有3个非共轭子群同构于G.作为副产品，我们证明了X允许Kahler-Einstein度规，我们构造了一个光滑的极化K3曲面的度22与群G的作用。
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study …
Some remarks about the Zariski topology of the Cremona group
对于一个代数簇$X$，我们研究了从一个代数簇到$X$的双有理映射群$\bir(X)$的代数态射的行为，并作为应用，得到了关于$\bir(X)$的所谓的Zerki拓扑与这个群的代数结构之间关系的一些见解，其中$X$是一个有理簇.
Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras
对于任何$n\geq 3$，我们证明，这些显然无关的对象之间存在等价：不可约$n$维非退化的投影品种$X\subset \mathbb P^{2n+1}$不同的合理正常涡卷和3-覆盖扭曲的三次曲线，射影等价;四次二次克雷莫纳变换的$ \mathbb P^{n-1}$，线性等价; $n$维复杂的约旦代数秩三，高达合痕。我们还提供了一些应用程序的分类的特定类别的品种在上述定义的类和quadro-quadric克雷莫纳变换，证明也是一个结构定理，这些双有理映射和品种3-覆盖扭曲的三次通过重新解释这些对象的可解性的根的约旦代数。
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