Related papers: On solvable subgroups of the Cremona group

We describe the nilpotent subgroups of the group Bir(P^2(C)) of birational transformations of the complex projective plane. Let N be a nilpotent subgroup of class k>1; then either each element of N has finite order, or N is virtually…

We initiate the study of the ''algebraic growth'' of groups of automorphisms and birational transformations of algebraic varieties. Our main result concerns $\text{Bir}(\mathbb{P}^2)$, the Cremona group in $2$ variables. This group is the…

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…

We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of…

We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the well-known subgroup of the Cremona group made up of the polynomial automorphisms…

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space, in which case Bir(X) is the Cremona group…

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.

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…

We explore algebraic subgroups of of the Cremona group $\mathcal C_n$ over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of $\mathcal C_n$ that we call flattenable. It contains…

We give an explicit set of generators for various natural subgroups of the real Cremona group Bir_R(P^2). This completes and unifies former results by several authors.

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…

We show that for any $n\geq5$ there exist connected algebraic subgroups in the Cremona group $\mathrm{Bir}(\mathbb{P}^n)$ that are not contained in any maximal connected algebraic subgroup. Our approach exploits the existence of stably…

The Cremona group of birational transformations of $\mathbb{P}^2_\mathbb{C}$ acts on the space $\mathbb{F}(2)$ of holomorphic foliations on the complex projective plane. Since this action is not compatible with the natural graduation of…

We give some properties of the subgroup $G_n(\mathbb{C})$ of the group of birational self-maps of $\mathbb{P}^n_\mathbb{C}$ generated by the standard involution and the group of automorphisms of $\mathbb{P}^n_\mathbb{C}$. We prove that…

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…

In this paper, we show that Cremona groups are sofic. We actually introduce a quantitative notion of soficity, called sofic profile, and show that the group of birational transformations of a d-dimensional variety has sofic profile at most…

We study the automorphism groups of Mori Del Pezzo fibrations over a smooth projective curve $C$ of positive genus. From that, we obtain a classification of maximal connected algebraic subgroups of $\mathrm{Bir}(C\times \mathbb{P}^2)$. Our…

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g.…

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, S\'er. I 344 (2007), 21-26.] and some description of automorphisms of rational surfaces.…

We prove that any group of cardinality at most the one of $\mathbb{C}$ is a quotient of any Cremona group of rank at least $4$. This provides a definitive answer to the question of what the quotients of Cremona groups can be. As a…