Since the end of the 19th century, we have known 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 has motivated the authors' work, which essentially deals with these quadratic maps.

The authors 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. The authors prove that a finite number of generic quadratic birational maps generates a free group. They 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.

The authors study some dynamical properties: following an idea of Guillot, they translate some invariants for foliations; in particular, they obtain that if two generic quadratic birational maps are birationally conjugate, then they are conjugate by an automorphism of the projective plane. The authors are also interested in invariant objects: curves, foliations, fibrations. They study birational maps of degree \(3\) and, by considering the different possible configurations of the exceptional curves, they give the “classification” of these maps and can deduce from it that the set of the birational maps of degree \(3\) exactly is irreducible, and is, in fact, rationally connected.