Mathematische Annalen

, Volume 368, Issue 1–2, pp 753–809 | Cite as

Lines on quartic surfaces

  • Alex Degtyarev
  • Ilia Itenberg
  • Ali Sinan Sertöz


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.

Mathematics Subject Classification

Primary 14J28 Secondary 14J27 14N25 



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.


  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)Google Scholar
  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 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bogomolov, F.A., Tschinkel, Y.: Density of rational points on elliptic \(K3\) surfaces. Asian J. Math. 4(2), 351–368 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boissière, S., Sarti, A.: Counting lines on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6(1), 39–52 (2007)Google Scholar
  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)Google Scholar
  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)Google Scholar
  9. 9.
    Chen, X.: Rational curves on \(K3\) surfaces. J. Algebraic Geom. 8(2), 245–278 (1999)MathSciNetMATHGoogle 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)Google Scholar
  14. 14.
    Kulikov, V.S.: Surjectivity of the period mapping for \(K3\) surfaces. Uspehi Mat. Nauk 32(4(196)), 257–258 (1977)Google Scholar
  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)]Google Scholar
  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]Google Scholar
  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 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Saint-Donat, B.: Projective models of \(K\)-\(3\) surfaces. Am. J. Math. 96, 602–639 (1974)MathSciNetCrossRefMATHGoogle 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 MathSciNetCrossRefMATHGoogle 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 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Segre, B.: The maximum number of lines lying on a quartic surface. Q. J. Math. Oxford Ser. 14, 86–96 (1943)Google Scholar
  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 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Alex Degtyarev
    • 1
  • Ilia Itenberg
    • 2
    • 3
  • Ali Sinan Sertöz
    • 1
  1. 1.Department of MathematicsBilkent UniversityAnkaraTurkey
  2. 2.Institut de Mathématiques de Jussieu–Paris Rive GaucheUniversité Pierre et Marie CurieParis Cedex 5France
  3. 3.Département de Mathématiques et ApplicationsEcole Normale SupérieureParis Cedex 5France

Personalised recommendations