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Les transformations de Cremona stellaires. (The stellar Cremona transformations). (French) Zbl 0966.14009
The author studies birational transformations \(F\) of \({\mathbb P}^n\) with the following property: There are points \(P,Q\in {\mathbb P}^n\) such that \(F\) induces a birational transformation between the set of lines through \(P\) and the set of lines through \(Q\). The group St of these stellar transformations with fixed \(P=Q\) fits in an exact sequence: \[ 1 \to PGL(2,K(y_1,\dots,y_{n-1}))\to\text{St}\to \text{Cr}({\mathbb P}^{n-1})\to 1 \] The author describes the elements of St in terms of their image in the group \(\text{Cr}({\mathbb P}^{n-1})\) of Cremona transformations of \({\mathbb P}^{n-1}\). With this description, the author computes the dimension of subgroups of St whose image is a fixed subgroup of \(\text{Cr}({\mathbb P}^{n-1})\).

MSC:
14E07 Birational automorphisms, Cremona group and generalizations
14N99 Projective and enumerative algebraic geometry
14E05 Rational and birational maps
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