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It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus in $\mathbb P^1(\mathbb C)$ is finite (up to isomorphism).

Let $X$ be a curve of genus at least two. Do there exist an integer $n\geq 2$ and a finite set of finite places $R$ of $X$ such that the set of degree $n$ finite morphisms $X\to \mathbf{P}^1(\mathbf{C})$ whose ramification locus is contained in $R$ is infinite, up to equivalence?

If $f$ and $g$ are finite morphisms from $X$ to $\mathbb P^1$, they are equivalent if $f$ equals $g$ up to an automorphism of $\mathbb P^1$.

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    @Harry, to be clear, you want to prove that for all $n, X, R$, there are only finitely many $X\to P^1$ of degree $\le n$ and étale outside $R$ ?2012-12-03

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