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!



Is the number of ramified coverings of given degree of a curve with prescribed branch divisor finite?
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](https://www.math.upenn.edu/~harbater/RETppr.pdf), the category of finite etale covers of the algebraic curve is equivalent to the category of finite topological covers of its analytification.


