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 Cours Spécialisés--Collection SMF 2013; 223 pp; softcover Number: 19 ISBN-13: 978-2-85629-770-4 List Price: US$108 Member Price: US$86.40 Order Code: COSP/19 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. Table of Contents Introduction Transformations rationnelles et birationnelles quadratiques Germes de flots birationnels quadratiques Transformations rationnelles, feuilletages, conjugaison dans les $$\sum{^i}$$ Quelques propriétés dynamiques des transformations birationnelles quadratiques Propriétés algébriques du groupe de Cremona Transformations birationnelles de degré 3 Index Index des notations Bibliographie