The \(\varkappa \) ring of the moduli of curves of compact type. (English) Zbl 1273.14057
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 \(\pi: \overline{M}_{g,n}\rightarrow \overline{M}_{g,n-1}\) [C. Faber and 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.
This 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 [A. Marian, D. Oprea and R. Pandharipande, Geom. Topol. 15, No. 3, 1651–1706 (2011; Zbl 1256.14057)]. This fundamental paper also raises many interesting questions about \(\kappa^*(M^c_{g,n})\).
This 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 [A. Marian, D. Oprea and R. Pandharipande, Geom. Topol. 15, No. 3, 1651–1706 (2011; Zbl 1256.14057)]. This fundamental paper also raises many interesting questions about \(\kappa^*(M^c_{g,n})\).
Reviewer: Hao Xu (Pittsburgh)
MSC:
| 14H10 | Families, moduli of curves (algebraic) |
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