Irreducibility of moduli spaces of cyclic unramified covers of genus $g$ curves
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- by R. Biggers and M. Fried PDF
- Trans. Amer. Math. Soc. 295 (1986), 59-70 Request permission
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 $\mathbb {Q}(t)\;[{\mathbf {Fr3}},\S 4]$.References
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Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 295 (1986), 59-70
- MSC: Primary 14H10; Secondary 12F20, 14H30, 20C32, 32G15
- DOI: https://doi.org/10.1090/S0002-9947-1986-0831188-3
- MathSciNet review: 831188