Skip to main content
Log in

The mapping class group and the Meyer function for plane curves

  • Published:
Mathematische Annalen Aims and scope Submit manuscript


For each d ≥ 2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d = 4, using our Meyer function, we can define the local signature for four-dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Ashikaga, T., Endo, H.: Various aspects of degenerate families of Riemann surfaces, SUGAKU EXPOSITIONS, vol. 19, Number 2 (2006)

  2. Ashikaga T., Konno K.: Global and local properties of pencils of algebraic curves, algebraic geometry 2000 azumino. Adv. Stud. Pure Math. 36, 1–49 (2000)

    MathSciNet  Google Scholar 

  3. Birman, J., Hilden, H.: On mapping class groups of closed surfaces as covering spaces. Advances in the Theory of Riemann Surfaces. Ann. Math. Stud., vol. 66, pp. 81–115. Princeton University Press, New Jersey (1971)

  4. Barth, W. Hulek, K., Peters, C., Van de Ven, A.: Compact Complex Surfaces, 2nd enlarged edn. Springer, New York (2003)

  5. Cohen F.R.: Homology of mapping class groups for surfaces of low genus. Contemp. Math. 58, 21–30 (1987)

    Google Scholar 

  6. Dimca A.: Singularities and Topology of Hypersurfaces. Springer, New York (1992)

    MATH  Google Scholar 

  7. Endo H.: Meyer’s signature cocycle and hyperelliptic fibrations. Math. Ann. 316, 237–257 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gelfand I.M., Kapranov M.M., Zelevinsky A.V.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Basel (1994)

    MATH  Google Scholar 

  9. Kawazumi N.: On the homotopy type of the moduli space of n-point sets of \({\mathbb{P}^1}\). J. Fac. Sci. Univ. Tokyo Ser. IA 37, 263–287 (1990)

    MATH  MathSciNet  Google Scholar 

  10. Lamotke K.: The topology of complex projective varieties after S. Lefschetz. Topology 20, 15–51 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  11. Matsumoto, Y.: Lefschetz fibrations of genus two—a topological approach. In: Proceedings of the 37th Taniguchi Symposium on “Topology and Teichmüller Spaces”, World Scientific, Singapore, pp. 123–148 (1996)

  12. Meyer W.: Die Signatur von Flächenbündeln. Math. Ann. 201, 239–264 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  13. Morifuji T.: On Meyer’s function of hyperelliptic mapping class groups. J. Math. Soc. Jpn. 55, 117–129 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nag S.: The Complex Analytic Theory of Teichmüller Spaces. Wiley-Interscience, New York (1988)

    MATH  Google Scholar 

  15. Yoshikawa, K.: A local signature for generic 1-parameter deformation germs of a complex curve. In: Algebraic Geometry and Topology of Degenerations, Coverings and Singularities 2000, pp. 188–200 (in Japanese) (2000)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yusuke Kuno.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuno, Y. The mapping class group and the Meyer function for plane curves. Math. Ann. 342, 923–949 (2008).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification (2000)