Lines on quartic surfaces

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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Correspondence to Ali Sinan Sertöz.

Additional information

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.

Communicated by Jean-Yves Welschinger.

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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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Mathematics Subject Classification

  • Primary 14J28
  • Secondary 14J27
  • 14N25