Related papers: Cremona maps of de Jonqui\`eres type
We study determinantal Cremona maps, i.e. birational maps whose base ideal is the maximal minors ideal of a given matrix $\Phi$, via the resolution of the polynomials systems defined by $\Phi$. Using convex geometry, this approach leads in…
Motivated by the study of the Kahan--Hirota--Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques…
We give a complete classification of maximal algebraic subgroups of the Cremona group of the plane and provide algebraic varieties that parametrize the conjugacy classes. ----- Nous donnons une classification compl\`ete des sous-groupes…
We present a complete set of criteria for determining A-types of plane-to-plane map-germs of corank one with A-codimension <7, which provides a new insight into the A-classification theory from the viewpoint of recognition problem. As an…
Veneroni maps are a class of birational transformations of projective spaces. This class contains the classical Cremona transformation of the plane, the cubo-cubic transformation of the space and the quatro-quartic transformation of…
We construct a map from fundamental groups of complements to some plane configurations to the groups $G_{n}^{k}$ for large $k$. We discuss connection between the groups $G_{n}^{k}$ for different $G_{n}^{k}$ and their geometric realization.
This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space $\sM_{\dd}^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system,…
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…
We obtain a sharp bound for p-elementary subgroups in the plane Cremona group over an arbitrary perfect field.
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.
The Cremona group is the group of birational transformations of the plane. A birational transformation for which there exists a pencil of lines which is sent onto another pencil of lines is called a Jonqui\`eres transformation. By the…
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…
Two infinite families of Cremona maps depending on one real parameter are given. For all integers $n \ge 1$ the first family of Cremona maps consists of group elements in $Bir \left( \mathbb{P}^{n} \right)$ with bidegree $(n, n)$, the…
It is well known that degree two Deligne cohomology groups can be identified with groups of isomorphism classes of holomorphic line bundles with connections. There is also a geometric description of degree three Deligne cohomology, due to…
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…
In this paper we describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.
A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a…
We show that every automorphism of the group $\mathcal{G}_n:= \textrm{Aut}(\mathbb{A}^n)$ of polynomial automorphisms of complex affine $n$-space $\mathbb{A}^n=\mathbb{C}^n$ is inner up to field automorphisms when restricted to the subgroup…
The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to enumerate such…
We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds.