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On special quadratic birational transformations of a projective space into a hypersurface

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Abstract

We study transformations as in the title with emphasis on those having smooth connected base locus, called “special”. In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.

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Notes

  1. Note that this is the only place in the proof where we use the smoothness of \(\mathbf{Q }\). Note also that this case does not arise if one first shows that \(\deg (\mathrm{Sec }(Y))=3\). Indeed, let \({\fancyscript{C}}\subset \mathbb{P }^N\) be a cubic hypersurface and \(D\subset {\fancyscript{C}}\) an irreducible divisor contained in \(\mathrm{sing }({\fancyscript{C}})\). Then, for a general plane \(\mathbb{P }^2\subset \mathbb{P }^N\), putting \(C={\fancyscript{C}}\cap \mathbb{P }^2\) and \(\Lambda =D\cap \mathbb{P }^2\), we have \(\Lambda \subseteq \mathrm{sing }(C)\). Hence, by Bézout’s Theorem, being \(C\) a plane cubic curve, it follows \(\deg (D)=\#(\Lambda )=1\).

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Acknowledgments

The author wishes to thank Prof. Francesco Russo for his indispensable suggestions about the topics in this paper.

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  1. Department of Mathematics and Computer Sciences, University of Study of Catania, 95125 , Catania, Italy

    Giovanni Staglianò

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Staglianò, G. On special quadratic birational transformations of a projective space into a hypersurface. Rend. Circ. Mat. Palermo 61, 403–429 (2012). https://doi.org/10.1007/s12215-012-0099-x

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