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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.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
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References:

[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
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