Irreducibility of moduli spaces of cyclic unramified covers of genus $g$ curves

Abstract:

Let $({C_1}, \ldots ,{C_r}G) = ({\mathbf {C}},G)$ be an $r$-tuple consisting of a transitive subgroup $G$ of ${S_m}$ and $r$ conjugacy classes ${C_1}, \ldots ,{C_r}$ of $G$. We consider the concept of the moduli space $\mathcal {H}({\mathbf {C}},G)$ of compact Riemann surface covers of the Riemann sphere of Nielsen class $({\mathbf {C}},G)$. The irreducibility of $\mathcal {H}({\mathbf {C}},G)$ is equivalent to the transitivity of a specific permutation representation of the Hurwitz monodromy group $(\S 1)$, but there are few general tools to decide questions about this representation. Theorem 2 gives a class of examples of $({\mathbf {C}},G)$ for which $\mathcal {H}({\mathbf {C}},G)$ is irreducible. As an immediate corollary this gives an elementary proof and generalization of the irreduciblity of the moduli space of cyclic unramified covers of genus $g$ curves (for which Deligne and Mumford [${\mathbf {DM}}$, Theorem 5.15] applied Teichmüller theory and Dehn’s theorem). This contrasts with the examples of $({\mathbf {C}},G)$ in $[{\mathbf {BFr}}]$ for which $\mathcal {H}({\mathbf {C}},G)$ is reducible. These kinds of questions combined with the study of the existence of rational subvarieties of $\mathcal {H}({\mathbf {C}},G)$ have application to the realization of a group $G$ as the Galois group of a regular extension of $\mathbb {Q}(t)\;[{\mathbf {Fr3}},\S 4]$.