Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus

Given a smooth, projective curve $Y$ of genus $\geq 1$ and a finite group $G$, let $H^G_n (Y)$ be the Hurwitz space which parameterizes the $G$-equivalence classes of $G$-coverings of $Y$ branched in n points. This space is a finite étale covering of $Y^{(n)} \setminus \triangle$, where $\triangle$ is the big diagonal. In this paper we calculate explicitly the monodromy of this covering. This is an extension to curves of positive genus of a well known result in the case of $Y \approxeq \mathbb{P}^1$, and may be used for determining the irreducible components of $H^G_n (Y)$ in particular cases.

Keywords

Hurwitz space, Galois covering, braid group

2010 Mathematics Subject Classification

Primary 14H10. Secondary 14H30, 20F36.

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Published 7 October 2014