What is commonly meant by Hurwitz's construction of simple covers? 



Hurwitz's construction of simple covers
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 $n-1$ distinct points.


