Mathematische Annalen

, Volume 373, Issue 1–2, pp 597–623 | Cite as

Construction of Nikulin configurations on some Kummer surfaces and applications

  • Xavier RoulleauEmail author
  • Alessandra Sarti


A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration \(\mathcal {C}\), then X is a Kummer surface \(X=\mathrm{Km}(B)\) where B is an Abelian surface determined by \(\mathcal {C}\). Let B be a generic Abelian surface having a polarization M with \(M^{2}=k(k+1)\) (for \(k>0\) an integer) and let \(X=\mathrm{Km}(B)\) be the associated Kummer surface. To the natural Nikulin configuration \(\mathcal {C}\) on \(X=\mathrm{Km}(B)\), we associate another Nikulin configuration \(\mathcal {C}'\); we denote by \(B'\) the Abelian surface associated to \(\mathcal {C}'\), so that we have also \(X=\mathrm{Km}(B')\). For \(k\ge 2\) we prove that B and \(B'\) are not isomorphic. We then construct an infinite order automorphism of the Kummer surface X that occurs naturally from our situation. Associated to the two Nikulin configurations \(\mathcal {C},\)\(\mathcal {C}'\), there exists a natural bi-double cover \(S\rightarrow X\), which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for \(k=2\) is a Schoen surface.

Mathematics Subject Classification

Primary: 14J28 Secondary: 14J50 14J29 14J10 



The authors thank the anonymous referee for useful remarks improving the exposition of the paper.


  1. 1.
    Alexeev, V., Nikulin, V., Del Pezzo.: and K3 surfaces. MSJ Memoirs, 15. Mathematical Society of Japan, Tokyo, (2006). (ISBN: 4-931469-34-5)Google Scholar
  2. 2.
    Artebani, M., Sarti, A., Taki, S.: K3 surfaces with non-symplectic automorphisms of prime order. Math. Z. 268, 507–533 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barth, W., Nieto, I.: Abelian surfaces of type \((1,3)\) and quartic surfaces with \(16\) skew lines. J. Algebraic Geom. 3(2), 173–222 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barth, W., Hulek, K., Peters, A.M., Van de Ven, A.: Compact complex surfaces, Second edition, Springer, Berlin (2004)Google Scholar
  5. 5.
    Bogomolov, F., Tschinkel, Y.: Lagrangian subvarieties of Abelian fourfolds. Asian J. Math. 4(1), 19–36 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bryan, J., Oberdieck, G., R. Pandharipande, R., Yin, Q.: Curve counting on abelian surfaces and threefolds, to appear in Algebr. GeomGoogle Scholar
  7. 7.
    Ciliberto, C., Mendes-Lopes, M., Roulleau, X.: On Schoen surfaces. Comment. Math. Helv. 90(1), 59–74 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garbagnati, A., Sarti, A.: On symplectic and non-symplectic automorphisms of K3 surfaces. Rev. Mat. Iberoam. 29(1), 135–162 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garbagnati, A., Sarti, A.: Kummer surfaces and K3 surfaces with\((\mathbb{Z}/2\mathbb{Z})^4\) symplectic action. Rocky Mt. J. 46(4), 1141–1205 (2016)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gritsenko, V., Hulek, K.: Minimal Siegel modular threefolds. Math. Proc. Camb. Philos. Soc. 123, 461–485 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Huybrechts, D.: Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, 158 (2016)Google Scholar
  12. 12.
    Hosono, S., Lian, B.H., Oguiso, K., Yau, S.T.: Kummer structures on a K3 surface—an old question of T. Shioda. Duke Math. J. 120(3), 635–647 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Keum, J.H.: Automorphisms of Jacobian Kummer surfaces. Compositio Mathematica 107, 269–288 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kondo, S.: The automorphism group of a generic Jacobian Kummer surface. J. Algebraic Geom. 7, 589–609 (1998)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lange, H.: Principal polarizations on products of elliptic curves, Contemp. Math., 397, Am. Math. Soc., Providence, RI (2006)Google Scholar
  16. 16.
    McMullen, C.: K3 surfaces, entropy and glue. J. Reine Angew. Math. 658, 1–25 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Morrison, D.: On K3 surfaces with large Picard number. Invent. Math. 75(1), 105–121 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Narasimhan, M.S., Nori, M.V.: Polarisations on an abelian variety. Proc. Ind. Acad. Sci. (Math) 90(2), 125–128 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nikulin, V., Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39, : 278–293 (1975) (English translation: Math. USSR. Izv 9(1975), 261–275)Google Scholar
  20. 20.
    Orlov, D.O.: On equivalences of derived categories of coherent sheaves on abelian varieties, Izv. Ross. Akad. Nauk Ser. Mat. 66(3), 131–158 (2002) (translation in Izv. Math. 66 (2002), no. 3, 569–594)Google Scholar
  21. 21.
    Polizzi, F.: Monodromy representations and surfaces with maximal Albanese dimension, Bollettino dell’Unione Matematica Italiana, 1–13 (2017)Google Scholar
  22. 22.
    Penegini, M.: The classification of isotrivially fibred surfaces with \(p_g = q = 2\). Collect. Math., 62(3):239–274 (2011) (With an appendix by Sönke Rollenske)Google Scholar
  23. 23.
    Reid, M.: Chapters on algebraic surfaces, Complex algebraic geometry (Park City, UT, 1993), IAS/Park City Math. Ser., vol. 3, Am. Math. Soc., Providence, RI, pp. 3–159 (1997)Google Scholar
  24. 24.
    Rito, C., Roulleau, X., Sarti, A.: On explicit Schoen surfaces, to appear in Algebr. GeomGoogle Scholar
  25. 25.
    Roulleau, X.: Bounded negativity, Miyaoka-Sakai inequality and elliptic curve configurations. Int Math Res Notices 2017(8), 2480–2496 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Roulleau, X.: Curves with low Harbourne constants on Kummer and Abelian surfaces, to appear in Rend. Circ. Mat. Palermo, II. SerGoogle Scholar
  27. 27.
    Saint-Donat, B.: Projective models of K3 surfaces. Am. J. Math. 96, 602–639 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sarti, A., van Geemen, B.: Nikulin involutions on K3 surfaces. Math. Z. 255(4), 731–753 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shioda, T.: Some remarks on abelian varieties, J. Fac. Sci. Univ. Tokyo, Sect. IA, 24, 11–21 (1977)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aix-Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373MarseilleFrance
  2. 2.Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Université de PoitiersChasseneuilFrance

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