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Chow rings of moduli spaces of curves. I: The Chow ring of \(\bar {\mathcal M}_ 3\). (English) Zbl 0721.14013
Let \(\bar {\mathcal M}_ g\) denote the Deligne-Mumford compactification of the moduli space \(\bar {\mathcal M}_ g\) of curves of genus g. Mumford has determined the Chow ring of \(\bar {\mathcal M}_ 2\) in terms of generators and relations. Let \(\lambda_ 2, \kappa_ 2\) denote the tautological classes in \(\Lambda^ 1(\bar {\mathcal M}_ 3)\) and \(\delta_ 0, \delta_ 1\) denote the \({\mathbb{Q}}\)-class of the boundary components \(\Delta_ 0\) and \(\Delta_ 1\) of \(\bar {\mathcal M}_ 3\). Then, the Chow ring of \(\bar {\mathcal M}_ 3\) (as the author shows) is generated by \(\lambda_ 2, \delta_ 0, \delta_ 1\) and \(\kappa_ 2\) modulo an ideal I generated by three relations in codimension 3 and six relations in codimension 4. The dimensions of the Chow groups are 1, 3, 7, 10, 7, 3, 1 and the pairing of the Chow groups in complementary dimensions is perfect. The proof is by an exhaustive case by case analysis. The ample divisor classes of \(\bar {\mathcal M}_ 3\) are also found.
[See also part II of this paper, ibid. 132, No.3, 421-449 (1990).]

MSC:
14H10 Families, moduli of curves (algebraic)
14C05 Parametrization (Chow and Hilbert schemes)
14D20 Algebraic moduli problems, moduli of vector bundles
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