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Lines on quartic surfaces

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Abstract

We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic.

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Acknowledgements

A large part of the work on this project was accomplished during our visits to a number of institutions worldwide. We are grateful to these organizations for their hospitality and support: École Normale Supérieure (first author), Hiroshima University, supported by the Japan Society for the Promotion of Science (first author), Institut des Hautes Études Scientifiques (first and second authors), International Centre for Theoretical Physics (first and second authors), Max-Planck-Institut für Mathematik (first and second authors), Université Pierre et Marie Curie - Paris 6 (first author). We extend our gratitude to Dmitrii Pasechnik, Sławomir Rams, Matthias Schütt, Ichiro Shimada, Tetsuji Shioda, and Davide Veniani for the motivation and fruitful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Bilkent University, 06800, Ankara, Turkey

    Alex Degtyarev & Ali Sinan Sertöz

  2. Institut de Mathématiques de Jussieu–Paris Rive Gauche, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 5, France

    Ilia Itenberg

  3. Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 5, France

    Ilia Itenberg

Authors
  1. Alex Degtyarev
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  2. Ilia Itenberg
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  3. Ali Sinan Sertöz
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Correspondence to Ali Sinan Sertöz.

Additional information

Communicated by Jean-Yves Welschinger.

A. Degtyarev was supported by the JSPS grant L15517 and TÜBİTAK grant 114F325. I. Itenberg was supported in part by the FRG Collaborative Research grant DMS-1265228 of the U.S. National Science Foundation. A. S. Sertöz was supported by the TÜBİTAK grant 114F325.

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Degtyarev, A., Itenberg, I. & Sertöz, A.S. Lines on quartic surfaces. Math. Ann. 368, 753–809 (2017). https://doi.org/10.1007/s00208-016-1484-0

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