Relations on \(\overline{\mathcal{M}}_{g,n}\) via orbifold stable maps (Q2832803)

Let \(\overline{\mathcal M}_{g,n}\) be the moduli space of stable genus \(g\) curves with \(n\) marked points. The tautological ring of \(\overline{\mathcal M}_{g,n}\) is the minimal subring inside the Chow ring (or cohomology ring) that is stable under pullbacks and pushforwards via forgetful and attaching maps between these moduli spaces. The natural classes \(\kappa\), \(\psi\), \(\lambda\), and boundary classes are all contained in the tautological ring. The study of relations between tautological classes dates back to Mumford, and has been investigated intensively in the last three decades.NEWLINENEWLINEIn this paper the author studies the equivariant cycle of the moduli space of stable maps to \([\mathbb C/\mathbb Z_r]\), or equivalently, the vanishing of high-degree Chern classes of a certain vector bundle over the moduli space of stable maps to \(B\mathbb Z_r\), where \(B\mathbb Z_r\) denotes the stack consisting of a single point with \(\mathbb Z_r\) isotropy. As a result, the author obtains relations in the Chow ring of \(\overline{\mathcal M}_{g,n}(B\mathbb Z_r, 0)\), and they push forward to yield relations on \(\overline{\mathcal M}_{g,n}\).