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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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