Let Y be a smooth projective curve over C and prescribe a branch divisor B on Y. I want to know if the number of coverings of Y of fixed degree and branched along B is finite. If so, why? Or, where is the reference for this? I was told it is claimed in an article by Manin ("On branched coverings of algebraic curves", Izv. Akad. Nauk SSSR Ser. Mat., 1961, vol. 25, 789-796), in the last sentence of the paper. My problem is that the article is in Russian (I think it is not translated) and, unfortunately, I can't speak Russian, so I can't check it. Also, I don't know for what characteristic Manin's paper results are. I assume the claim should follow from the theorem stated in the first page (in Section 1), which is about field extensions, but, as I said, I am not sure about the field characteristic. If someone knowing Russian can at least tell me what the statement of the theorem and the last sentence of the article say, I think it would already help me a lot!
This can be understood by looking at fundamental groups.
Covers of $Y$ branched only along $B$ of degree $n$ correspond to homomorphisms $\pi_1(Y-B)\rightarrow S_n$ ($S_n$ is the symmetric group on $n$ letters). (The cover is connected if and only if the image is a transitive subgroup, and the cover is Galois iff the image is transitive of order $n$)
Here, isomorphism classes of covers corresponds to conjugacy classes of the homomorphism by elements of $S_n$.
(This follows from standard results which you can find in Hatcher's "Algebraic Topology" online book)
It's well known that the fundamental groups of curves are finitely generated - if $g$ is the genus of $Y$, then $\pi_1(Y-B)$ is can be generated by $2g + |B|$ elements. (Roughly speaking, the $2g$ generators come from the global topology of $Y$, and the $|B|$ additional generators come from small loops around the points in $B$)
Thus, covers of $Y$ branched only along $B$ of degree $n$ are in bijection with conjugacy classes of homomorphisms from a finitely generated group to a finite group, of which there can be only finitely many.
In the above, technically a "cover" refers to a covering space in the sense of topology, but by the Riemann Existence Theorem, the category of finite etale covers of the algebraic curve is equivalent to the category of finite topological covers of its analytification.