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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: R. Biggers and M. Fried
Journal: Trans. Amer. Math. Soc. 295 (1986), 59-70
MSC: Primary 14H10; Secondary 12F20, 14H30, 20C32, 32G15
MathSciNet review: 831188
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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 $ (\S1)$, 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}},\S4]$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0831188-3
PII: S 0002-9947(1986)0831188-3
Keywords: Riemann surface, moduli space, Hurwitz monodromy group, permutation representation
Article copyright: © Copyright 1986 American Mathematical Society