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Some loci of rational cubic fourfolds

  • Michele Bolognesi
  • Francesco Russo
  • Giovanni Staglianò
Article
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Abstract

In this paper we investigate the divisor \({\mathcal {C}}_{14}\) inside the moduli space of smooth cubic hypersurfaces in \({\mathbb {P}}^5\), whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in \({\mathbb {P}}^5\) contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in \({\mathcal {C}}_{14}\) is rational. Combining our proof with the Hodge theoretic definition of \({\mathcal {C}}_{14}\), we deduce that on a smooth cubic fourfold every class \(T\in {\text {H}}^{2,2}(X,\mathbb {Z})\) with \(T^2=10\) and \(T\cdot h^2=4\) is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in \({\mathcal {C}}_{14}\).

Notes

Acknowledgements

We have received support from the Research Network Program GDRE-GRIFGA, by PRIN Geometria delle Varietà Algebriche and by the Labex LEBESGUE. We would like to thank: A. Auel (a comment of whom inspired Theorem 4.7), M. Bernardara, C. Ciliberto, B. Hassett, A. Kuznetsov, D. Markushevich and H. Nuer for stimulating conversations and exchange of ideas. We heartfully thank the referees for their careful reading and for many suggestions, which lead to a significant improvement of the exposition.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Michele Bolognesi
    • 1
  • Francesco Russo
    • 2
  • Giovanni Staglianò
    • 3
  1. 1.Institut Montpellierain Alexander GrothendieckUniversité de Montpellier, CNRSMontpellier Cedex 5France
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di CataniaCataniaItaly
  3. 3.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

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