Advertisement

The Moduli Space of 5 Points on ℙ1 and K3 Surfaces

Chapter
Part of the Progress in Mathematics book series (PM, volume 260)

Abstract

We show that the moduli space of 5 ordered points on ℙ1 is isomorphic to an arithmetic quotient of a complex ball by using the theory of periods of K3 surfaces. We also discuss a relation between our uniformization and the one given by Shimura [S], Terada [Te], Deligne-Mostow [DM].

Keywords

Moduli K3 surfaces quartic del Pezzo surfaces complex ball uniformization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bol]
    O. Bolza, On binary sextics with linear transformations into themselves, Amer. J. Math., 10 (1888), 47–70.CrossRefMathSciNetGoogle Scholar
  2. [Borel]
    A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Diff. Geometry, 6 (1972), 543–560.zbMATHMathSciNetGoogle Scholar
  3. [DM]
    P. Deligne, G. W. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IHES, 63 (1986), 5–89.zbMATHMathSciNetGoogle Scholar
  4. [D]
    I. Dolgachev, Lectures on Invariant Theory, London Math. Soc. Lecture Note Ser., 296, Cambridge 2003.Google Scholar
  5. [DGK]
    I. Dolgachev, B. van Geemen, S. Kondō, A complex ball uniformaization of the moduli space of cubic surfaces via periods of K3 surfaces, math.AG/0310342, J. reine angew. Math. (to appear).Google Scholar
  6. [K1]
    S. Kondō, A complex hyperbolic structure of the moduli space of curves of genus three, J. reine angew. Math., 525 (2000), 219–232.MathSciNetzbMATHGoogle Scholar
  7. [K2]
    S. Kondō, The moduli space of curves of genus 4 and Deligne-Mostow’s complex reflection groups, Adv. Studies Pure Math., 36 (2002), Algebraic Geometry 2000, Azumino, 383–400.Google Scholar
  8. [K3]
    S. Kondō, The moduli space of 8 points on1 and automorphic forms, to appear in the Proceedings of the Conference “Algebraic Geometry in the honor of Igor Dolgachev”.Google Scholar
  9. [Mo]
    G. W. Mostow, Generalized Picard lattices arising from half-integral conditions, Publ. Math. IHES, 63 (1986), 91–106.zbMATHMathSciNetGoogle Scholar
  10. [Mu]
    D. Mumford, K. Suominen, Introduction to the theory of moduli, Algebraic Geometry, Oslo 1970, F. Oort, ed., Wolters-Noordholff 1971.Google Scholar
  11. [Na]
    Y. Namikawa, Periods of Enriques surfaces, Math. Ann., 270 (1985), 201–222.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [NU]
    Y. Namikawa, K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta math., 9 (1973), 143–186.zbMATHMathSciNetGoogle Scholar
  13. [N1]
    V. V. Nikulin, Integral symmetric bilinear forms and its applications, Math. USSR Izv., 14 (1980), 103–167.zbMATHCrossRefGoogle Scholar
  14. [N2]
    V. V. Nikulin, Finite automorphism groups of Kähler K3 surfaces, Trans. Moscow Math. Soc., 38 (1980), 71–135.Google Scholar
  15. [N3]
    V. V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections, J. Soviet Math., 22 (1983), 1401–1475.zbMATHCrossRefGoogle Scholar
  16. [PS]
    I. Piatetski-Shapiro, I. R. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv., 5 (1971), 547–587.CrossRefGoogle Scholar
  17. [S]
    G. Shimura, On purely transcendental fields of automorphic functions of several variables, Osaka J. Math., 1 (1964), 1–14.zbMATHMathSciNetGoogle Scholar
  18. [Te]
    T. Terada, Fonction hypergéométriques F 1 et fonctions automorphes, J. Math. Soc. Japan 35 (1983), 451–475; II, ibid., 37 (1985), 173–185.zbMATHMathSciNetGoogle Scholar
  19. [Th]
    W. P. Thurston, Shape of polyhedra and triangulations of the sphere, Geometry & Topology Monograph, 1 (1998), 511–549.MathSciNetCrossRefGoogle Scholar
  20. [V]
    E. B. Vinberg, Some arithmetic discrete groups in Lobachevskii spaces, in “Discrete subgroups of Lie groups and applications to moduli”, Tata-Oxford (1975), 323–348.Google Scholar
  21. [Vo]
    S. P. Vorontsov, Automorphisms of even lattices that arise in connection with automorphisms of algebraic K3 surfaces, Vestnik Mosk. Univ. Mathematika, 38 (1983), 19–21.MathSciNetGoogle Scholar
  22. [YY]
    T. Yamazaki, M. Yoshida, On Hirzebruch’s examples of surfaces with c 12 = 3c 2, Math. Ann., 266 (1984), 421–431.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan

Personalised recommendations