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On the Cremona transformations of bidegree (3,3). (Sur les transformations de Cremona de bidegré (3,3).) (French) Zbl 0921.14005

Let \(\mathbb{P}^3\) be three-dimensional projective space over an algebraically closed field \(k\) of characteristic zero. In the paper all birational transformations \(T\) of \(\mathbb{P}^3\) of bidegree \((3,3)\) (this means that equally \(T\) and \(T^{-1}\) have degree \(3\)) are classified into three (not disjoint) classes:
(1) the class of determinantal transformations (i.e. components of \(T\) are minors of a \(4\times 3\) matrix the entries of which are linear forms on \(k^4\)),
(2) the Jonquières class of transformations (i.e. the strict transformation \(\overline{T^{-1}(L)}\) of a generic line \(L\) is a plane cubic),
(3) the ruled class of transformations (i.e. the strict transformation \(\overline{T^{-1}(H)}\) of a generic plane \(H\) is a ruled cubic surface).

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
14E05 Rational and birational maps
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