Birational quadratic transformations of the three dimensional complex projective space. (Transformations birationnelles quadratiques de l’espace projectif complexe à trois dimensions.) (French) Zbl 0987.14009

The paper concerns the classical subject of the classification of all birational morphisms (i.e. Cremona transformations) of \(\mathbb P^n(\mathbb C)\) in particular case \(n=3\) and degree of morphisms 2. The main result is a finite list of birational morphisms of \(\mathbb P^3(\mathbb C)\) of degree 2 (a geometric description of them is also given) such that any other birational morphism \(\phi :\mathbb P^3_x(\mathbb C) \to \mathbb P^3_y(\mathbb C)\) of degree 2 is equal to one in this list up to linear changes of variables in \(\mathbb P^3_x(\mathbb C)\) and \(\mathbb P^3_y(\mathbb C)\). Besides, the authors divide the whole class of birational morphisms of degree 2 in three natural subclasses (non-disjoint) which are locally closed subvarieties in an appropriate Grassmannian.


14E07 Birational automorphisms, Cremona group and generalizations
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