Mathematische Annalen

, Volume 354, Issue 2, pp 465–496 | Cite as

Moduli of theta-characteristics via Nikulin surfaces



We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R g of Prym curves of genus g. We observe a striking analogy with Mukai’s well-known work on ordinary K3 surfaces. Many of Mukai’s results have a very precise Prym-Nikulin analogue, for instance a general Prym curve from R g is a section of a Nikulin surface if and only if g ≤ 7 and g ≠ 6. Furthermore, R 7 has the structure of a fibre space over the corresponding moduli space of polarized Nikulin surfaces. We then use these results to study the geometry of the moduli space \({S_g^+}\) of even spin curves, with special emphasis on the transition case of \({S_8^+}\) which is a 21-dimensional Calabi-Yau variety.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aprodu, M., Farkas, G.: Green’s Conjecture for general covers. arXiv:1105.3933 (2011)Google Scholar
  2. 2.
    Beauville A., Merindol Y.: Sections hyperplanes des surfaces K3. Duke Math. J. 4, 873–878 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Catanese F.: On the rationality of certain moduli spaces related to curves of genus 4. Springer Lect. Notes Math. 1008, 30–50 (1983)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cornalba, M.: Moduli of curves and theta-characteristics. In: Lectures on Riemann surfaces, Trieste, pp. 560–589 (1987)Google Scholar
  5. 5.
    Cossec, F., Dolgachev, I.: Enriques surfaces, Progress in Mathematics vol. 76, Birkhäuser (1989)Google Scholar
  6. 6.
    Dolgachev I.: Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81, 2599–2630 (1996)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dolgachev I.: Rationality of \({\mathcal{R}_2}\) and \({\mathcal{R}_3}\) . Pure Appl. Math. Q. 4, 501–508 (2008)MathSciNetMATHGoogle Scholar
  8. 8.
    Donagi R.: The unirationality of \({\mathcal{A}_5}\) . Ann. Math. 119, 269–307 (1984)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Eisenbud D., Harris J.: The Kodaira dimension of the moduli space of curves of genus ≥ 23. Invent. Math. 90, 359–387 (1987)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Farkas G.: The birational type of the moduli space of even spin curves. Adv. Math. 223, 433–443 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Farkas G., Ludwig K.: The Kodaira dimension of the moduli space of Prym varieties. J. Eur. Math. Soc. 12, 755–795 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Farkas G., Popa M.: Effective divisors on \({\overline{\mathcal{M}_g}}\) , curves on K3 surfaces and the Slope Conjecture. J. Algebraic Geom. 14, 151–174 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Farkas, G., Verra, A.: The geometry of the moduli space of odd spin curves. arXiv:1004.0278 (2010)Google Scholar
  14. 14.
    Garbagnati A., Sarti A.: Projective models of K3 surfaces with an even set. Adv. Geom. 8, 413–440 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    van Geemen B., Sarti A.: Nikulin involutions on K3 surfaces. Mathematische Zeitschrift 255, 731–753 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Green M., Lazarsfeld R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1986)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Green M., Lazarsfeld R.: Special divisors on curves on a K3 surface. Invent. Math. 89, 357–370 (1987)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Harris J., Mumford D.: On the Kodaira dimension of \({\overline{\mathcal{M}_g}}\) . Invent. Math. 67, 23–88 (1982)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Harris J., Morrison I.: Slopes of effective divisors on the moduli space of stable curves. Invent. Math. 99, 321–355 (1990)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Izadi E., Lo Giudice M., Sankaran G.: The moduli space of étale double covers of genus 5 is unirational. Pac. J. Math. 239, 39–52 (2009)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Lazarsfeld R.: Brill-Noether-Petri without degenerations. J. Differ. Geom. 23, 299–307 (1986)MathSciNetMATHGoogle Scholar
  22. 22.
    Morrison D.: On K3 surfaces with large Picard number. Invent. Math. 75, 105–121 (1984)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Mukai, S.: Curves, K3 surfaces and Fano 3-folds of genus ≤ 10. In: Algebraic geometry and commutative algebra, pp. 357–377. Kinokuniya, Tokyo (1988)Google Scholar
  24. 24.
    Mukai S.: Curves and Grassmannians. In: Yang, J-H., Namikawa, Y., Ueno, K. (eds) Algebraic Geometry and Related Topics, pp. 19–40. International Press, Boston (1992)Google Scholar
  25. 25.
    Mukai S.: Curves and K3 surfaces of genus eleven. In: Moduli of vector bundles, Lecture Notes in Pure and Applied Mathematics, vol. 179, pp. 189–197. Dekker, New York (1996)Google Scholar
  26. 26.
    Nikulin V.V.: Kummer surfaces. Izvestia Akad. Nauk SSSR 39, 278–293 (1975)MathSciNetMATHGoogle Scholar
  27. 27.
    Saint-Donat B.: Projective models of K3 surfaces. Am. J. Math. 96, 602–639 (1974)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Verra A.: A short proof of the unirationality of \({\mathcal{A}_5}\) . Indagationes Math. 46, 339–355 (1984)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Verra A.: On the universal principally polarized abelian variety of dimension 4. In: Curves and abelian varieties, Proceedings of Internation Conference at Athens, Georgia, 2007, Contemporary Mathematics, vol. 345, pp. 253–274 (2008)Google Scholar
  30. 30.
    Voisin C.: Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface. J. Eur. Math. Soc. 4, 363–404 (2002)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Welters G.: A theorem of Gieseker-Petri type for Prym varieties. Annales Scientifique École Normale Supérieure 18, 671–683 (1985)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut Für Mathematik, Humboldt-Universität zu BerlinBerlinGermany
  2. 2.Dipartimento di MatematicaUniversitá Roma TreRomeItaly

Personalised recommendations