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
Full Text: DOI Numdam EuDML


[1] Algebraische Transformationen und Korrespondenzen, 2.2.B, (1932), Teubner · JFM 59.1291.01
[2] Le superficie razionali, (1939), Zanichelli, Bologna · Zbl 0021.05306
[3] Sulle transformazioni razionali nello spazio, Annali di Mat. ser. II, V, 131-162, (18711873) · JFM 04.0418.02
[4] On varieties of minimal degree, Algebraic Geometry, Bowdoin 1985, 46, 3-13, (1987), Amer. Math. Soc. · Zbl 0646.14036
[5] Classification of degree 2 polynomial automorphisms of \({\Bbb C}^3,\) Publ. Mat., 42, 195-210, (1998) · Zbl 0923.58006
[6] Algebraic Geometry, (1992), Springer Verlag · Zbl 0779.14001
[7] Algebraic Geometry, (1979), Springer Verlag · Zbl 0367.14001
[8] Cremona transformation in Plane and Space, (1927), University Press, Cambridge · JFM 53.0595.01
[9] Sur le multidegré des transformations de Cremona, C.R. Acad. Sci. Paris, Série I, 330, 297-300, (2000) · Zbl 1011.14003
[10] Introduction to Algebraic Geometry, (1949), Claredon Press, Oxford · Zbl 0041.27903
[11] Selected Topics in Algebraic Geometry, (1970), Chelsea Pub. Company, Washington · Zbl 0213.47101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.