# Transformations Birationnelles de Petit Degré

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*Dominique Cerveau; Julie Déserti*

A publication of the Société Mathématique de France

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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in birational maps.