Mathematische Zeitschrift

, Volume 258, Issue 2, pp 319–331 | Cite as

Picard group of moduli of hyperelliptic curves

  • Sergey Gorchinskiy
  • Filippo Viviani


The main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves. We compute their Picard groups, giving explicit description of the generators. We get an application to the non-existence of a tautological family over the coarse moduli space.


Hyperelliptic curves Coarse moduli scheme Stack Picard group 

Mathematics Subject Classification (2000)

14D22 14H10 14C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arbarello E. and Cornalba M. (1987). The Picard groups of the moduli space of curves. Topology 26: 153–171 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arsie A. and Vistoli A. (2004). Stacks of cyclic covers of projective spaces. Compos. Math. 140: 647–666 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cornalba, M.: The Picard group of the moduli stack of stable hyperelliptic curves. Preprint available at math.AG/0605531Google Scholar
  4. 4.
    Edidin D. and Graham W. (1998). Equivariant intersection theory. Invent. Math. 131: 595–634 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gorchinskiy, S., Viviani, F.: Families of hyperelliptic curves. Preprint available at math.AG/0511627Google Scholar
  6. 6.
    Harer J. (1983). The second homology group of the mapping class group of an orientable surface. Invent. Math. 72: 221–239 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Harris J. and Morrison D. (1998). Moduli of Curves GTM 187. Springer, New York Google Scholar
  8. 8.
    Igusa J. (1960). Arithmetic variety of moduli for genus two. Ann. of Math. 72: 612–649 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Lonsted K. (1984). The singular points on the moduli spaces for smooth curves. Math. Ann. 266: 397–402 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lonsted K. and Kleiman S. (1979). Basics on families of hyperelliptic curves. Comp. Math. 38: 83–111 zbMATHMathSciNetGoogle Scholar
  11. 11.
    Mestrano N. and Ramanan S. (1985). Poincaré bundles for families of curves. J. Reine Angew. Math. 362: 169–178 zbMATHMathSciNetGoogle Scholar
  12. 12.
    Mumford, D.: Picard groups of moduli problems. In: Proceedings and Conference on Arithmetical Algebraic Geometry, Purdue University, Harper and Row, New York 33–81 (1963)Google Scholar
  13. 13.
    Oort F. (1975). Singularities of the moduli scheme for curves of genus three. Indag. Math. 37: 170–174 MathSciNetGoogle Scholar
  14. 14.
    Popp H. (1969). The singularities of the moduli schemes of curves. J. Number Theory 1: 90–107 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rauch H.E. (1962). The singularities of the modulus space. Bull. Am. Math. Soc. 68: 390–394 zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Vistoli A. (1998). The Chow ring of \({\mathcal{M}}_{2}\) Invent. Math. 131: 635–644 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Dip. MatematicaUniversità di Roma Tor VergataRomeItaly

Personalised recommendations