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Chow rings of moduli spaces of curves. II: Some results on the Chow ring of \(\overline{{\mathcal M}_ 4}\). (English) Zbl 0735.14021

As summarized by the author: “In the second part of this paper we gather our results concerning the Chow ring of the moduli space of stable curves of genus 4. These results are not complete. We find generators for the Chow ring of \({\mathcal M}_ 4\) and for the Chow groups in codimension 1 and 2 of \(\overline {\mathcal M}_ 4\). For \(A^ 2(\overline {\mathcal M}_ 4)\) we find fourteen generators. Using test surfaces we prove that the dimension of \(A^ 2(\overline{\mathcal M}_ 4)\) is at least 13 and explicitly determine the single relation between the fourteen generators which still can exist. Finally, we have two proofs that this relation does indeed hold, so that the dimension of \(A^ 2(\overline{\mathcal M}_ 4)\) equals 13. This enables us to determine the Chow ring of \({\mathcal M}_ 4.\)”
In addition, the author shows that the Chow ring of \({\mathcal M}_ 4\) is generated by the tautological classes. The paper depends strongly on its part I [ibid., No. 2, 331-419 (1990; Zbl 0721.14013)].

MSC:

14H10 Families, moduli of curves (algebraic)
14C05 Parametrization (Chow and Hilbert schemes)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D20 Algebraic moduli problems, moduli of vector bundles

Citations:

Zbl 0721.14013
Full Text: DOI