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On the tautological ring of $$\overline{\mathcal M}_{g,n}$$. (English) Zbl 1040.14007
Let $${\mathcal M}_{g,n}$$ denote the moduli space of smooth $$n$$-pointed curves of genus $$g$$ and $$\overline{\mathcal M}_{g,n}$$ its Deligne-Mumford compactification, the moduli space of stable $$n$$-pointed curves. Let $$A^*(\overline{\mathcal M}_{g,n})$$ denote the Chow ring and $$R^*(\overline{\mathcal M}_{g,n})$$ its subring, the tautological ring of $$\overline{\mathcal M}_{g,n}$$. C. Faber and R. Pandharipande [Mich. Math. J. 48, Spec. Vol., 215–252 (2000; Zbl 1090.14005)], in analogy with a previous conjecture of Faber on $${\mathcal M}_{g}$$, stated a conjecture (called by them a speculation) on $$R^*(\overline{\mathcal M}_{g,n})$$, saying that it is a Gorenstein ring with socle in codimension $$3g-3$$.
The first step to prove this conjecture is to check that the tautological ring has rank 1 in maximal codimension $$3g-3+n$$. This is proved in the paper under review. The essential ingredient of the proof is a formula of T. Ekedahl, S. Lando, M. Shapiro and A. Vainshtein [Invent. Math. 146, No. 2, 297–327 (2001; Zbl 1073.14041)], for the Hurwitz number $$H^g_{\alpha_1\ldots\alpha_n}$$ of genus $$g$$ irreducible branched covers of $${\mathbb P}^1$$ of degree $$\sum\alpha_i$$, with simple branching above $$r$$ fixed points, branching with monodromy type $$(\alpha_1, \ldots, \alpha_n)$$ above $$\infty$$ and no other branching.

##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14H10 Families, moduli of curves (algebraic) 14C15 (Equivariant) Chow groups and rings; motives 14E20 Coverings in algebraic geometry