The Cremona group and its subgroups

TL;DR

This survey explores the structure and properties of the Cremona group by examining its various subgroups, providing an overview of current knowledge and open problems in the field.

Contribution

It offers a comprehensive overview of the Cremona group's subgroups, highlighting recent developments and open questions in the study of this algebraic group.

Findings

Detailed classification of subgroups of the Cremona group

Identification of key properties and structures of these subgroups

Summary of open problems and future research directions

Abstract

This survey deals with the Cremona group via its subgroups.

Figures

Equations

$$\phi\colon(z_{0}:z_{1}:z_{2})\dashrightarrow\big{(}\phi_{0}(z_{0},z_{1},z_{2}):\phi_{1}(z_{0},z_{1},z_{2}):\phi_{2}(z_{0},z_{1},z_{2})\big{)}$$

$$\delta q_0={\varphi }_0(p_0,p_1,p_2),\delta q_1={\varphi }_1(p_0,p_1,p_2),\delta q_2={\varphi }_2(p_0,p_1,p_2)$$

$$\alpha {\varphi }_0+\beta {\varphi }_1+\gamma {\varphi }_2=0$$

$$\alpha p_0={\psi }_0(q_0,q_1,q_2)$$

$$\alpha {\varphi }_0+\beta {\varphi }_1+\gamma {\varphi }_2=0$$

$${\alpha }^{\prime }\overline{{\varphi }_0}+{\beta }^{\prime }\overline{{\varphi }_1}+{\gamma }^{\prime }\overline{{\varphi }_2}=0$$

$$(z_{0}:z_{1}:z_{2})\dashrightarrow\big{(}\overline{\phi_{0}}(z_{0},z_{1},z_{2}):\overline{\phi_{1}}(z_{0},z_{1},z_{2}):\overline{\phi_{2}}(z_{0},z_{1},z_{2})\big{)}$$

$$n=L\cdot {\Lambda }_{\varphi }=\varphi (L)\cdot \varphi ({\Lambda }_{\varphi })={\Lambda }_{{\varphi }^{-1}}\cdot L^{\prime }=n^{\prime }.$$

$${\mathbb{P}}_{\mathbb{C}}^3⇢{\mathbb{P}}_{\mathbb{C}}^3$$

Full text

After a short introduction which explains in an informal style some selected results and techniques Chapter LABEL:chap:hyperbolicspace gives a description of the hyperbolic space on which the Cremona group in two variables acts, and which has turned out to provide some of the key techniques to understand the plane Cremona group. In Chapter LABEL:Chapter:algebraicsubgroup the Zariski topology is described. Chapter LABEL:chapter:gen gives an overview of various presentations of the plane Cremona group. Chapter LABEL:chapter:alg treats some group theoretical properties of the plane Cremona group. Chapter LABEL:chapter:finite surveys some results about finite (mostly abelian) subgroups of the plane Cremona group. Chapter LABEL:chapter:uncountable surveys results about various subgroups using techniques that rely on the base-field being uncountable. Chapter LABEL:chap:hyper gives a big variety of important results that can be deduce from the action of the plane Cremona group on the hyperbolic space, such as the Tits alternative or the non-simplicity of the group. Chapter LABEL:chapter:dyn gives an introduction to some notions from dynamics and their relationship to the plane Cremona group.

À Benoît

Preface

The main purpose of the present treatise is to draw a portrait of the $n$-dimensional Cremona group $\mathrm{Bir}(\mathbb{P}^{n}_{\mathbb{C}})$. The study of this group started in the XIXth century; the subject has known a lot of developments since the beginning of the XXIth century. Old and new results are discussed; unfortunately we will not be exhaustive. The Cremona group is approached through the study of its subgroups: algebraic, finite, normal, nilpotent, simple, torsion subgroups are evoked but also centralizers of elements, representation of lattices, subgroups of automorphisms of positive entropy etc

Let us introduce birational self maps of the plane and the plane Cremona group from a geometrical point of view.

A plane collineation is a one-to-one map from $\mathbb{P}^{2}_{\mathbb{C}}$ to itself such that the images of collinear points are themselves collinear. Such maps leave the projective properties of curves unaltered. In advancing beyond such properties let us introduce other maps of the plane to itself that establish relations between curves of differents orders and possessing different sets of singularities. The most general rational map of the plane is defined by equations of the form

[TABLE]

where $\phi_{0}$, $\phi_{1}$ and $\phi_{2}$ are homogeneous polynomials of degree $n$ without common factor of positive degree. Such a map makes correspond to a point $p$ with coordinates $(p_{0}:p_{1}:p_{2})$ a point $\phi(p)=q$ with coordinates $(q_{0}:q_{1}:q_{2})$ where

[TABLE]

with $\delta$ in $\mathbb{C}^{*}$.

Consider the net of curves $\Lambda_{\phi}$ defined by the equation

[TABLE]

where $\alpha$, $\beta$ and $\gamma$ are arbitrary parameters. As $p$ describes a line in $\mathbb{P}^{2}_{\mathbb{C}}$, then $q=\phi(p)$ describes a curve $\mathcal{C}$ of $\Lambda_{\phi}$. The curves of the net $\Lambda_{\phi}$ are thus correlated by $\phi$ with the lines of the plane. Conversely given any net $\Lambda$ of curves such as $\Lambda_{\phi}$ a linear representation of the curves of $\Lambda$ on the lines of the plane is equivalent to a rational map of the plane.

The curves of $\Lambda_{\phi}$ may have base-points $p_{i}$ common to them all. Each such point is a common zero of $\phi_{0}$, $\phi_{1}$ and $\phi_{2}$, so the equations (0.1) to determine its corresponding point are illusory. Conversely each point, termed a base-point of $\phi$, which renders equation (0.1) illusory is a base-point of $\Lambda_{\phi}$. In other words

Theorem**.**

The base-points of any rational map are the base-points of the associated net of curves.

Any two general curves $\mathcal{C}$ and $\mathcal{C}^{\prime}$ of $\Lambda_{\phi}$ define a pencil of curves $\mathcal{C}+\alpha\mathcal{C}^{\prime}$ of the net. Denote by $n$ the number of free intersections of $\mathcal{C}$ and $\mathcal{C}^{\prime}$ not occuring at the base-points $p_{i}$ of $\Lambda_{\phi}$; denote by $r_{1}$, $r_{2}$, $\ldots$, $r_{n}$ these points. The integer $n$ is called the grade of $\Lambda_{\phi}$.

To curves of the arbitrary pencil $\mathcal{C}+\alpha\mathcal{C}^{\prime}$ there correspond by the map $\phi$ lines of a pencil $L+\alpha L^{\prime}$. Furthermore if the base-point of the latter pencil is $q$, then clearly every point $r_{i}$ corresponds to $q$. Conversely if any two points of the plane have the same preimage $q$, then they belong to the same free intersection set of some pencil in $\Lambda_{\phi}$.

Theorem**.**

Let $\phi$ be a rational self map of the plane. Let $\Lambda_{\phi}$ be its associated net and let $n$ be the grade of $\Lambda_{\phi}$. An arbitrary point $q$ is the transform of $n$ points $r_{1}$, $r_{2}$, $\ldots$, $r_{n}$ which together form the free intersection set of a pencil of curves of $\Lambda_{\phi}$.

In other words the general rational map of the plane is a $(n,1)$ correspondence between the points $p$ and $q$. And this means that, when the ratios of $q_{0}$, $q_{1}$, $q_{2}$ are given the equations (0.1) have in general $n$ distinct solutions for the ratios of $p_{0}$, $p_{1}$ and $p_{2}$. If $n=1$, i.e. if these equations have only one solution, $(p_{0}:p_{1}:p_{2})$ are rational functions of $(q_{0}:q_{1}:q_{2})$. In this case the equations of the reverse map will be of the form

[TABLE]

where $\psi_{0}$, $\psi_{1}$ and $\psi_{2}$ are homogeneous polynomials of degree $n^{\prime}$. A Cremona map is a rational map whose reverse is also rational, we also speak about birational self map of the plane. The plane Cremona group is the group of birational self maps of the plane.

A homaloidal net of curves in the plane is one whose grade is $1$.

Equations (0.1) define a birational map $\phi$ if and only if the associated net $\Lambda_{\phi}$ is homaloidal. Conversely from any given homaloidal net we can derive many birational self maps of the plane; if $\overline{\phi_{0}}$, $\overline{\phi_{1}}$ and $\overline{\phi_{2}}$ are three independent linear combinations of $\phi_{0}$, $\phi_{1}$ and $\phi_{2}$, the net

[TABLE]

can also be expressed in the form

[TABLE]

and the map defined by

[TABLE]

is based on the same net. Moreover

Theorem**.**

To any birational self map of the plane there is associated a homaloidal net of curves.

Conversely any homaloidal net of curves generates an infinity of birational self maps of the plane, any of which is the product of any other by a plane collineation.

A collineation is the simplest kind of birational self map of the plane whose homaloidal net is composed of the lines of the plane.

The degree of a birational self map of the plane is the degree of the curves of its generating homaloidal net.

Let $\phi$ be a birational self map of the plane of degree $n$. Denote by $n^{\prime}$ the degree of its inverse $\phi^{-1}$. If the number of intersections of two curves $\mathcal{C}$ and $\mathcal{C}^{\prime}$ is denoted by $\mathcal{C}\cdot\mathcal{C}^{\prime}$ and if $L$ and $L^{\prime}$ are lines, then

[TABLE]

Hence

Theorem**.**

A birational self map of the plane and its inverse have the same degree.

Let us finish this introduction by pointing out that this statement is not true in higher dimension:

[TABLE]

is a birational self map of $\mathbb{P}^{3}_{\mathbb{C}}$ of degree $2$ whose inverse

[TABLE]

has degree $3$. As we will see there are many other differences between the $2$-dimensional Cremona group and the $n$-dimensional Cremona group, $n\geq 3$.

Note that the study of $\mathrm{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is central: if $S$ is a complex rational surface, then its group of birational self maps is isomorphic to $\mathrm{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$.

We now deal with the content of the manuscript. Chapter LABEL:chapter:intro contains introductory examples and the very basic techniques used to study birational maps of the projective plane. This chapter explains in particular the importance of divisors and linear systems in the study of the plane Cremona groups.

Chapter LABEL:chap:hyperbolicspace builds up on Chapter LABEL:chapter:intro by explaining how to blow-up all points in $\mathbb{P}^{2}_{\mathbb{C}}$ and subsequent blown-up surfaces. It gives rise to an infinite hyperbolic space on which the Cremona group acts. This space plays a fundamental role in the study of Cremona groups, as it allows to apply tools from geometric group theory to study subgroups of the Cremona group, as well as degree growth and dynamical behaviours of birational maps.

Chapter LABEL:Chapter:algebraicsubgroup presents two natural topologies on the Cremona group and their properties, and the notion of algebraic subgroups of the Cremona groups. The construction of one of the topologies - the Zariski topology - is defined via the concept of morphisms. It links to the concept of an algebraic group acting on a variety, which is discussed in this chapter as well.

Chapter LABEL:chapter:gen adresses a very basic and classical interest while dealing with a group: finding a ”nice” and generating set and ”nice” structures of the group, such as an amalgamated structure. This is quite an important topic in research on Cremona groups because for the plane Cremona group there are ”nice” generating sets, and many statements are proven by using them. In higher dimensions no nice generating sets are known: this is one of the many reasons why working with Cremona groups in higher dimensions is very hard.

Chapter LABEL:chapter:alg discusses other group geometric properties of plane Cremona groups. While Chapter LABEL:chap:hyperbolicspace presents a representation of the Cremona group in terms of isometries of an infinite hyperbolic space this chapter deals with linear representations (there are none) and representations of subgroups of $\mathrm{SL}(n,\mathbb{Z})$, $n\geq 3$, inside the plane Cremona group.

Chapter LABEL:chapter:finite deals with results on finite subgroups of the plane Cremona groups. They have been of much interest for a very long time, and a short overview of the progress made in the last $80$ years is given. The chapter focuses on the classification results of finite abelian and finite cyclic subgroups by Blanc and Dolgachev and Iskovskikh.

Chapter LABEL:chapter:uncountable is an extension of Chapter LABEL:chapter:finite; it deals with infinite abelian subgroups of the plane Cremona group. It then moves on the related topic of endomorphisms of Cremona groups, subject already mentioned in Chapter LABEL:chapter:alg.

Chapter LABEL:chap:hyper picks up the topic of Chapter LABEL:chap:hyperbolicspace which is the action of the plane Cremona group on an infinite hyperbolic space by isometries. The action and its properties have been very fruitful and has played a vital role in many recent results on the plane Cremona group.

Chapter LABEL:chapter:dyn has a more dynamical flavour. We first give three answers to the question ”when is a birational self map of $\mathbb{P}^{2}_{\mathbb{C}}$ birationally conjugate to an automorphism ?” We then recall some constructions of automorphisms of rational surfaces with positive entropy. And then we realize $\mathrm{SL}(2,\mathbb{Z})$ as a subgroup of automorphisms of a rational surface with the property that every element of infinite order has positive entropy.

Acknowledgments

I would like to thank all people working on the Cremona groups and around. I have a very special thought for Serge Cantat and Dominique Cerveau who made me discover this wonderful group.

I am grateful to the referees for very constructive recommendations.

Enfin merci à mon petit univers…