What is commonly meant by Hurwitz's construction of simple covers?
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I think most people understand under this term the following: It is a branched cover $X\to\mathbb P^1(\mathbb C)$ of a compact connected Riemann surface $X$ to the Riemann sphere $\mathbb P^1(\mathbb C)$ such that the monodromy generators belonging to the branch points are transpositions. Or equivalently: If $n$ is the degree of the cover $f$, then each fiber $f^{1}(x)$ for $x\in \mathbb P^1(\mathbb C)$ has at least $n1$ distinct points. 

