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Cremona transformations and special double structures. (English) Zbl 1093.14018

The objects of this very interesting paper are the Cremona transformations on \({\mathbb P}^N({\mathbb C})\), whose base locus is a locally complete intersection scheme \(X\), such that \(X_{\text{red}}\) is irreducible, smooth and \(\deg(X)=2\deg(X_{\text{red}})\), (i.e. a so-called “Fossum-Ferrand double structure” on a smooth variety as support). The main results are:
1. if \(\dim X\leq \frac{2}{3}N\) or \(\text{codim}(X)=2\), then \(N=3\) and one deals with a cubo-cubic Cremona transformation, not defined along a skew cubic;
2. the same conditions holds if the Hartshorne conjecture about complete intersection is true;
3. the same conditions holds for Cremona transformations defined by degree \(3\) forms.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
14M10 Complete intersections
14N05 Projective techniques in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M06 Linkage
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