The \(\varkappa \) ring of the moduli of curves of compact type
Recall that the tautological ring \(R^*(\overline{M}_{g,n})\) is defined to be the minimal subalgebra of the Chow ring \(A^*(\overline{M}_{g,n})\) that is closed under push-forwards by boundary morphisms and forgetful morphisms \(\pi: \overline{M}_{g,n}\rightarrow \overline{M}_{g,n-1}\) [\textit{C. Faber} and \textit{R. Pandharipande}, J. Eur. Math. Soc. 7, 96--124 (2005; Zbl 1058.14046)]. Let \(M_{g,n}^{c}\) be the moduli space of ``curves of compact type'' (i.e. its dual graph having no loops). Hence \(M_{g,n} \subset M_{g,n}^{c} \subset \overline{M}_{g,n}\) and \(R^*(M^c_{g,n})\) can be defined from \(R^*(\overline{M}_{g,n})\) by restriction.NEWLINENEWLINEThis self-contained paper gives a detailed study of the structures about the \(\kappa\) ring \(\kappa^*(M^c_{g,n})\subset R^*(M^c_{g,n})\) generated by the \(\kappa\) classes. The investigation is analogous to those of tautological rings conjectured by Faber in the 1990's, which has been a central problem about the moduli space of curves. The author uses the new technique of stable quotients recently introduced in [\textit{A. Marian, D. Oprea} and \textit{R. Pandharipande}, Geom. Topol. 15, No. 3, 1651--1706 (2011; Zbl 1256.14057)]. This fundamental paper also raises many interesting questions about \(\kappa^*(M^c_{g,n})\).
scientific article; zbMATH DE number 6065306
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