Transformations birationnelles de petit degré

Abstract

Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals with these quadratic maps. We establish algebraic properties such as the classification of one parameter groups of quadratic birational maps or the smoothness of the set of quadratic birational maps in the set of rational maps. We prove that a finite number of generic quadratic birational maps generates a free group. We show that if f is a quadratic birational map or an automorphism of the projective plane, the normal subgroup generated by f is the full group of birational maps of the projective plane, which implies that this group is perfect. We study some dynamical properties: following an idea of Guillot, we translate some invariants for foliations in our context, in particular we obtain that if two generic quadratic birational maps are birationally conjugated, then they are conjugated by an automorphism of the projective plane. We are also interested in the presence of "invariant objects": curves, foliations, fibrations. Then follows a more experimental part: we draw orbits of quadratic birational maps with real coefficients and sets analogous to Julia sets for polynomials of one variable. We study birational maps of degree 3 and, by considering the different possible configurations of the exceptional curves, we give the "classification" of these maps. We can deduce from this that the set of the birational maps of degree 3 exactly is irreducible, in fact rationally connected.