Lines on quartic surfaces


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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  1. 1.

    Akyol, A., Degtyarev, A.: Geography of irreducible plane sextics. Proc. Lond. Math. Soc. (3) 111(6), 1307–1337 (2015). doi:10.1112/plms/pdv053

  2. 2.

    Barth, W.: Lectures on \(K3\)- and Enriques surfaces. In: Algebraic geometry, Sitges (Barcelona), 1983, Lecture Notes in Math., vol. 1124, pp. 21–57. Springer, Berlin (1985). doi:10.1007/BFb0074994

  3. 3.

    Beauville, A.: Application aux espaces de modules. Geometry of \(K3\) surfaces: moduli and periods (Palaiseau, 1981/1982). Astérisque 126, 141–152 (1985)

  4. 4.

    Bogomolov, F., Hassett, B., Tschinkel, Y.: Constructing rational curves on K3 surfaces. Duke Math. J. 157(3), 535–550 (2011). doi:10.1215/00127094-1272930

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bogomolov, F.A., Tschinkel, Y.: Density of rational points on elliptic \(K3\) surfaces. Asian J. Math. 4(2), 351–368 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Boissière, S., Sarti, A.: Counting lines on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6(1), 39–52 (2007)

  7. 7.

    Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris (1968)

  8. 8.

    Caporaso, L., Harris, J., Mazur, B.: How many rational points can a curve have? In: The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, pp. 13–31. Birkhäuser, Boston (1995)

  9. 9.

    Chen, X.: Rational curves on \(K3\) surfaces. J. Algebraic Geom. 8(2), 245–278 (1999)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Degtyarev, A.: Smooth models of singular \(K3\)-surfaces (2016, To appear). arXiv:1608.06746

  11. 11.

    Degtyarev, A., Ekedahl, T., Itenberg, I., Shapiro, B., Shapiro, M.: On total reality of meromorphic functions. Ann. Inst. Fourier (Grenoble) 57(6), 2015–2030 (2007).

  12. 12.

    GAP—Groups, Algorithms, and Programming, Version 4.7.7. (2015)

  13. 13.

    Gauss, C.F.: Disquisitiones arithmeticae. Translated and with a preface by Arthur A. Clarke, Revised by William C. Waterhouse, Cornelius Greither and A. W. Grootendorst and with a preface by Waterhouse. Springer-Verlag, New York (1986)

  14. 14.

    Kulikov, V.S.: Surjectivity of the period mapping for \(K3\) surfaces. Uspehi Mat. Nauk 32(4(196)), 257–258 (1977)

  15. 15.

    Meyer, W.F.: Flächen vierter und höchere ordnung. Encykl. Math. Wiss. 3, 1533–1779 (1908)

    Google Scholar 

  16. 16.

    Miranda, R., Morrison, D.R.: Embeddings of integral quadratic forms. (2009, electronic)

  17. 17.

    Mori, S.: On degrees and genera of curves on smooth quartic surfaces in \({\bf P}^3\). Nagoya Math. J. 96, 127–132 (1984).

  18. 18.

    Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus \(11\). In: Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, pp. 334–353. Springer, Berlin (1983). doi:10.1007/BFb0099970

  19. 19.

    Nikulin, V.V.: Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43(1), 111–177, 238 (1979) [English translation: Math USSR-Izv. 14 (1979), no. 1, 103–167 (1980)]

  20. 20.

    Pjateckiĭ-Šapiro, I.I., Šafarevič, I.R.: Torelli’s theorem for algebraic surfaces of type \({\text{K}}3\). Izv. Akad. Nauk SSSR Ser. Mat. 35, 530–572 (1971) [English translation: Math. USSR-Izv. 5, 547–588]

  21. 21.

    Rams, S., Schütt, M.: At most 64 lines on smooth quartic surfaces (characteristic 2) (2012, To appear). arXiv:1512.01358

  22. 22.

    Rams, S., Schütt, M.: 64 lines on smooth quartic surfaces. Math. Ann. 362(1–2), 679–698 (2015). doi:10.1007/s00208-014-1139-y

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Saint-Donat, B.: Projective models of \(K\)-\(3\) surfaces. Am. J. Math. 96, 602–639 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Salmon, G.: A treatise on the analytic geometry of three dimensions. Hodges, Smith and Co., Cambridge (1862)

    Google Scholar 

  25. 25.

    Schur, F.: Ueber eine besondre Classe von Flächen vierter Ordnung. Math. Ann. 20(2), 254–296 (1882). doi:10.1007/BF01446525

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Schütt, M.: \(K3\) surfaces with Picard rank 20. Algebra Number Theory 4(3), 335–356 (2010). doi:10.2140/ant.2010.4.335

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Segre, B.: The maximum number of lines lying on a quartic surface. Q. J. Math. Oxford Ser. 14, 86–96 (1943)

  28. 28.

    Yau, S.T., Zaslow, E.: BPS states, string duality, and nodal curves on \(K3\). Nucl. Phys. B 471(3), 503–512 (1996). doi:10.1016/0550-3213(96)00176-9

    MathSciNet  Article  MATH  Google Scholar 

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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.

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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).

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Mathematics Subject Classification

  • Primary 14J28
  • Secondary 14J27
  • 14N25