Morphisms between the moduli spaces of curves with generalized Teichmüller structure (Q1598155)

Let \(C_{g, n}\) be a compact surface of genus \(g\) having \(n\) punctures, let \(T_{g, n}\) be the associated Teichmüller space and let \(\Gamma_{g, n}\) be the corresponding modular group. For any subgroup \(\Gamma \leq \Gamma_{g, n}\) let \(\mathcal{T}_{g, n, \Gamma}\) be the functor from the category of complex spaces to the category of sets, which maps every analytical space \(S\) to the set of isomorphism classes of smooth curves of genus \(g\) having \(n\) punctures and a \(\Gamma\)-marking and which associates with every holomorphic map \(\nu: T \rightarrow S\) between analytical spaces the base change by \(\nu\). Among other results in this work, the author proves that \(T_{g, n, \Gamma}:= T_{g, n}/\Gamma\) is a coarse moduli space for \(\mathcal{T}_{g, n, \Gamma}\), and is a fine moduli space if and only if \(\Gamma\) contains no element of finite order. The main result states that if \(P: C_{g, n} \rightarrow C_{g', n'}\) is a finite covering and \(\Gamma' \leq \Gamma_{g', n'}\) is the subgroup induced by \(\Gamma\), then under suitable conditions there exists a morphism of functors \( \mathcal{T}_{g, n, \Gamma} \rightarrow \mathcal{T}_{g', n', \Gamma'}\) (which he studies) inducing a morphism of coarse moduli spaces \(T_{g, n, \Gamma} \rightarrow T_{g', n', \Gamma'}\).