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# Pure and Applied Mathematics Quarterly

## Volume 10 (2014)

### Number 2

### Special Issue: In Memory of Andrey Todorov, Part 3 of 3

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

Pages: 193 – 222

DOI: http://dx.doi.org/10.4310/PAMQ.2014.v10.n2.a1

#### Author

#### Abstract

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.