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Limits of Pluri–Tangent Planes to Quartic Surfaces

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Algebraic and Complex Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 71))

Abstract

We describe, for various degenerations S → Δ of quartic K3 surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as t ∈ Δ ∗ tends to 0 of the Severi varieties V δ (S t ), parametrizing irreducible δ-nodal plane sections of S t . We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic K3 surfaces.

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Acknowledgements

We thank Erwan Brugallé (who, together with G. Mikhalkin, was able to provide the enumerative numbers of Theorems 1 and 3 using tropical methods), and Concettina Galati for numerous discussions, both enlightening and motivating.

This project profited of various visits of the second author to the first, which have been made possible by the research group GRIFGA, in collaboration between CNRS and INdAM.

The first author is a member of GNSAGA of INdAM and was partly supported by the project “Geometry of Algebraic varieties” funded by the Italian MIUR. The second author is a member of projects CLASS and MACK of the French ANR.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica, 00133, Roma, Italy

    Ciro Ciliberto

  2. Institut de Mathématiques de Toulouse (CNRS UMR 5219), Université Paul Sabatier, 31062, Toulouse Cedex 9, France

    Thomas Dedieu

Authors
  1. Ciro Ciliberto
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  2. Thomas Dedieu
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Correspondence to Ciro Ciliberto .

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Editors and Affiliations

  1. Leibniz Universität Hannover, Hannover, Germany

    Anne Frühbis-Krüger

  2. Humboldt-University Berlin, Berlin, Germany

    Remke Nanne Kloosterman

  3. Leibniz Universität Hannover, Hannover, Germany

    Matthias Schütt

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Ciliberto, C., Dedieu, T. (2014). Limits of Pluri–Tangent Planes to Quartic Surfaces. In: Frühbis-Krüger, A., Kloosterman, R., Schütt, M. (eds) Algebraic and Complex Geometry. Springer Proceedings in Mathematics & Statistics, vol 71. Springer, Cham. https://doi.org/10.1007/978-3-319-05404-9_6

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