## Abstract

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.

### Keywords

- 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.*

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## References

Al Arakelov, S.,

*Families of algebraic curves with fixed degeneracies*, Izv. Akad. Nauk,*35*(1971).Arbarello, E.,

*Weierstrass points and moduli of*curves, Comp. Math.,*29*(1974), pp. 325–342.Atiyah, M., and Bott, R.,

*The Yang-Mills equations over Riemann surfaces*,to appear.Baum, P., Fulton, W., and MacPherson, R.,

*Riemann-Roch for**singular varieties*, Publ. I.H.E.S.,.*(5*(1975), pp. 101–145.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.Fulton, W.,

*Rational equivalence on singular varieties*,Publ. I.H.E.S.*45*(1975), pp. 147–167.Fulton, W.,

*Intersection Theory*, Springer-Verlag, 1983.Fulton, W. and MacPherson, R.,

*Categorical framework for the study of singular spaces*, Memoirs A.M.S. 243, (1981).Harer, J., The second homology group of the mapping class group of an orientable surface, to appear.

Harris, J., and Mumford, D., On

*the Kodaira dimension of the moduli space of curves*, Inv. Math.,*67*(1982), pp. 23–86.Hironaka, H.,

*Bimeromorphic smoothing of a complex-analytic space*, Acta. Math. Vietnamica,*2*(1977), pp. 103–168.Igusa, J.I.,

*Arithmetic theory of moduli for genus two*, Annals of Math.,*72*(1960), pp. 612–649.Knudsen, F.,

*The projectivity of the moduli space of stable curves*, Math. Scand. to appear.Kleiman, S.,

*Towards a numerical theory of ampleness*, Annals of Math.,*84*(1966), pp. 293–344.Matsusaka, T.,

*Theory of Q-varieties*, Publ. Math. Soc. Japan,*8*(1964)Miller, E.,

*The homology of the mapping class group of surfaces*,in preparation.Mumford, D.,

*Stability of projective varieties*, L’Ens, Math.,*24*(1977), pp. 39–110.Mumford, D.,

*Hirzebruch’s proportionality principle in the non-compact case*, Inv. math.,*42*(1977), pp. 239–272.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.

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*Dedicated to Igor Shafarevitch on his 60th birthday*

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© 1983 Springer Science+Business Media New York

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Mumford, D. (1983). Towards an Enumerative Geometry of the Moduli Space of Curves. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol 36. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-9286-7_12

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DOI: https://doi.org/10.1007/978-1-4757-9286-7_12

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