Jérémy Blanc

Dynamical degrees of birational transformations of projective surfaces

The dynamical degree lambda( f )xa0 of a birational transformation f measures the exponential growth rate of the degree of the formulae that define the n -th iterate of fxa0 . 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 fxa0 . 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.

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

We characterise smooth curves in P3 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

Topologies and structures of the Cremona groups

We study the algebraic structure of the n-dimensional Cremona group and show that it is not an algebraic group of innite dimension (ind-group) if n 2. We describe the obstruction to this, which is of a topological nature. By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.

Elements and cyclic subgroups of finite order of the Cremona group

We give the classification of elements – respectively cyclic subgroups – of finite order of the Cremona group, up to conjugation. Natural parametrisations of conjugacy classes, related to fixed curves of positive genus, are provided.

Degree growth of birational maps of the plane

This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth. The classification elements of infinite order with a bounded sequence of degrees is achieved, the case of elements of finite order being already known. The coefficients of the linear and quadratic growth are then described, and related to geometrical properties of the map. The dynamical number of base-points is also studied. Applications of our results are the description of embeddings of the Baumslag-Solitar groups and GL(2,Q) into the Cremona group.

Geometrically rational real conic bundles and very transitive actions

In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

Simple relations in the Cremona group

We give a simple set of generators and relations for the Cre- mona group of the plane. Namely, we show that the Cremona group is the amalgamated product of the de Jonquieres group with the group of auto- morphisms of the plane, divided by one relation .

Automorphisms of the plane preserving a curve

We study the group of automorphisms of the affine plane preserv- ing some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect.

The correspondence between a plane curve and its complement

Abstract Given two irreducible curves of the plane which have isomorphic complements, it is natural to ask whether there exists an automorphism of the plane that sends one curve on the other. This question has a positive answer for a large family of curves and H. Yoshihara conjectured that it is true in general. We exhibit counterexamples to this conjecture, over any ground field. In some of the cases, the curves are isomorphic and in others not; this provides counterexamples of two different kinds. Finally, we use our construction to find the existence of surprising non-linear automorphisms of affine surfaces.

MODULI SPACES OF QUADRATIC RATIONAL MAPS WITH A MARKED PERIODIC POINT OF SMALL ORDER

The surface corresponding to the moduli space of quadratic endomorphisms of P1 with a marked periodic point of order n is studied. It is shown that the surface is rational over Q when n 5 and is of general type for n = 6. An explicit description of the n = 6 surface lets us find several infinite families of quadratic endomorphisms f : P1-> P1 defined over Q with a rational periodic point of order 6. In one of these families, f also has a rational fixed point, for a total of at least 7 periodic and 7 preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over Q admits rational periodic points of order n > 3.