On the tautological ring of \(\overline{\mathcal M}_{g,n}\). (Q2719828)

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 \(\overline{\mathcal M}_{g,n}\). \textit{C. Faber} and \textit{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 \(3g-3\).NEWLINENEWLINEThe 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 \textit{T. Ekedahl, S. Lando, M. Shapiro} and \textit{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.