The number of conjugacy classes of elements of the Cremona group of some given finite order

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.Comment: 14 page

On the inertia group of elliptic curves in the Cremona group of the plane

We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that if the group contains a non-trivial torsion, the fixed curve is the image of a smooth cubic by a birational transformation of the plane. We show that for a smooth cubic, the group is generated by its elements of degree 3, and prove that it contains a free product of Z/2Z, indexed by the points of the curve.Comment: 14 pages, no figur

Conjugacy classes of affine automorphisms of K^n and linear automorphisms of P^n in the Cremona groups

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).Comment: 17 pages, no figure

Simple relations in the Cremona group

We give a simple set of generators and relations for the Cremona group of the plane. Namely, we show that the Cremona group is the amalgamated product of the de Jonqui\`eres group with the group of automorphisms of the plane, divided by one relation which is $\sigma\tau=\tau\sigma$, where $\tau=(x:y:z)\mapsto (y:x:z)$ and \sigma=(x:y:z)\dasharrow (yz:xz:xy)

On degenerations of plane Cremona transformations

This article studies the possible degenerations of plane Cremona transformations of some degree into maps of smaller degree.Comment: 21 pages, corrected typos and Proposition 3.21, Mathematische Zeitschrift, published online on 2015/9/2

Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

We characterise smooth curves in P^3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) $4d-30 \le g\le 14$ or $(g,d) = (19,12)$, (ii) there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C. This generalises the classical similar situation for the blow-up of points in P^2. We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.Comment: 36 pages, to appear in Proc. London Math. So

Dynamical degrees of birational transformations of projective surfaces

The dynamical degree $\lambda(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of $\lambda(f)$ and the structure of the conjugacy class of $f$. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.Comment: 65 page

The group of Cremona transformations generated by linear maps and the standard involution

This article studies the group generated by automorphisms of the projective space of dimension $n$ and by the standard birational involution of degree $n$. Every element of this group only contracts rational hypersurfaces, but in odd dimension, there are simple elements having this property which do not belong to the group. Geometric properties of the elements of the group are given, as well as a description of its intersection with monomial transformations

Automorphisms of cluster algebras of rank 2

We compute the automorphism group of the affine surfaces with the coordinate ring isomorphic to a cluster algebra of rank 2.Comment: To appear in Transform. Group