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Towards an enumerative geometry of the moduli space of curves. (English) Zbl 0554.14008
Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271-328 (1983).
[For the entire collection see Zbl 0518.00005.]
In this paper the author intends to begin the study of the Chow ring for the moduli space \({\mathcal M}_ g\) of curves of genus g and for its compactification \(\bar {\mathcal M}_ g\), the moduli space of stable curves. The point is that, though \({\mathcal M}_ g\) itself is far more complicated than the Grassmann varieties (the former is not unirational for large g), its Chow ring seems to have a similar simple structure as that of the latter in lower degrees.
In part one, as preliminary, an intersection product is defined in the Chow ring A(\(\bar {\mathcal M}_ g)\) of \(\bar {\mathcal M}_ g\) by using two facts that it has only the quotient singularity and is globally the quotient of a Cohen-Macaulay variety, i.e. the moduli space of curves with level structure. Both the Grothendieck and Baum-Fulton-MacPherson forms of the Riemann-Roch theorem give the necessary tool. - In part II a sequence of what he calls “tautological” classes \(\kappa_ i\in A^ i(\bar {\mathcal M}_ g)\) is defined, and auxiliary classes \(\lambda_ i\) as follows: on \(\bar {\mathcal M}_ g\) there are a canonical family of curves \(\pi:\bar {\mathcal C}_ g\to \bar {\mathcal M}_ g\) whose fibre is C/Aut(C) for [C]\(\in \bar {\mathcal M}_ g\), and a sheaf \(\omega_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g}\) of relative 1-forms which is a Q-sheaf (i.e. quotient of an invertible sheaf by a finite action). With the results in part I one can define the intersection on Chow classes in \(A^.(\bar {\mathcal C}_ g)\) or in \(A^.(\bar {\mathcal M}_ g)\), hence we set \(K_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g}=c_ 1(\omega_{{\mathcal C}_{\bar gg}/\bar {\mathcal M}_ g})\in A^ 1(\bar {\mathcal C}_ g)\) and \(\kappa_ i=(\pi_*K^{i+1}_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g})\in A^ i(\bar {\mathcal M}_ g).\) Moreover for \(E=\pi_*(\omega_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g})\), set \(\lambda_ i=c_ i(E)\). - The author then studies relations between them in \(A^.({\mathcal M}_ g)\) and expresses certain subvarieties such as the hyperelliptic locus with them. One of the important results is that all classes \(\kappa_ i\), \(\lambda_ i\) are polynomials in \(\kappa_ 1,...,\kappa_{g-1}\) in \(A^.({\mathcal M}_ g)\). - In this direction S. Morita has found a number of new interesting relations between them but in \(H^.({\mathcal M}_ g)\) with different, i.e. topological method [see Proc. Japan Acad., Ser. A 60, 373-376 (1984)]. Its geometric counterpart in \(A^.({\mathcal M}_ g)\) is unknown up to now. - In part III, as an example, the author works out \(A^.(\bar {\mathcal M}_ 2)\) completely.
Reviewer: Y.Namikawa

14H10 Families, moduli of curves (algebraic)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
14C40 Riemann-Roch theorems