Towards an Enumerative Geometry of the Moduli Space of Curves

  • David Mumford
Part of the Progress in Mathematics book series (PM, volume 36)


The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli space M g of curves of genus g and its compactification M g, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in M g in terns of these classes. We take as a model for this the enumerative geometry of the Grassmannians. Here the basic classes are the Chern classes of the tautological or universal bundle that lives over the Grassmannian, and the most basic cycles are the loci of linear spaces satisfying various Schubert conditions: the so-called Schubert cycles. However, since Harris and I have shown that for g large, M g is not unir.ational [H-M] it is not possible to expect that M g has a decomposition into elementary cells or that the Chow ring of M g is as simple as that of the Grassmannian. But in the other direction, J. Harer [Ha] and P. Miller [Mi] have strong results indicating that at least the low dimensional homology groups of M g behave nicely. Moreover, it appers that many geometrically natural cycles are all expressible in terms of a small number of basic classes.


Modulus Space Abelian Variety Chern Class Double Point Mapping Class Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    Al Arakelov, S., Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk, 35 (1971).Google Scholar
  2. [Arb]
    Arbarello, E., Weierstrass points and moduli of curves, Comp. Math., 29 (1974), pp. 325–342.MathSciNetzbMATHGoogle Scholar
  3. [A_B]
    Atiyah, M., and Bott, R., The Yang-Mills equations over Riemann surfaces,to appear.Google Scholar
  4. [B-F-M]
    Baum, P., Fulton, W., and MacPherson, R., Riemann-Roch for singular varieties, Publ. I.H.E.S.,. (5 (1975), pp. 101–145.Google Scholar
  5. [D-M]
    Deligne, P., and Mumford, D., The irreducibility of the space of curves of given genus, Publ. I.H.E.S. 36 (1969) pp. 75–109.MathSciNetzbMATHGoogle Scholar
  6. [Fi]
    Fulton, W., Rational equivalence on singular varieties,Publ. I.H.E.S. 45 (1975), pp. 147–167.Google Scholar
  7. [F2]
    Fulton, W., Intersection Theory, Springer-Verlag, 1983.Google Scholar
  8. [F-M]
    Fulton, W. and MacPherson, R., Categorical framework for the study of singular spaces, Memoirs A.M.S. 243, (1981).Google Scholar
  9. [Ha]
    Harer, J., The second homology group of the mapping class group of an orientable surface, to appear.Google Scholar
  10. [H-M]
    Harris, J., and Mumford, D., On the Kodaira dimension of the moduli space of curves, Inv. Math., 67 (1982), pp. 23–86.MathSciNetzbMATHGoogle Scholar
  11. [H]
    Hironaka, H., Bimeromorphic smoothing of a complex-analytic space, Acta. Math. Vietnamica, 2 (1977), pp. 103–168.MathSciNetzbMATHGoogle Scholar
  12. [I]
    Igusa, J.I., Arithmetic theory of moduli for genus two, Annals of Math., 72 (1960), pp. 612–649.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [K]
    Knudsen, F., The projectivity of the moduli space of stable curves, Math. Scand. to appear.Google Scholar
  14. [Kl]
    Kleiman, S., Towards a numerical theory of ampleness, Annals of Math., 84 (1966), pp. 293–344.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Ma]
    Matsusaka, T., Theory of Q-varieties, Publ. Math. Soc. Japan, 8 (1964)Google Scholar
  16. [Mi]
    Miller, E., The homology of the mapping class group of surfaces,in preparation.Google Scholar
  17. [M]
    Mumford, D., Stability of projective varieties, L’Ens, Math., 24 (1977), pp. 39–110.Google Scholar
  18. [M2]
    Mumford, D., Hirzebruch’s proportionality principle in the non-compact case, Inv. math., 42 (1977), pp. 239–272.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [N]
    Namikawa, Y., A new compactification of the Siegel space and degeneration of abelian varieties I and II, Math. Annalen, 221 (1976), pp. 97–141, 201–241.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • David Mumford
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations