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.

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

##### MSC:

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 |