# References for a standard result about coverings of Riemann surfaces

I my thesis I have to cite the following standard result:

Let $Y$ be a compact Riemann surface and let $B\subseteq Y$ be a finite subset. Given a natural number $d$, there are only finitely many isomorphism classes of (holomorphic) coverings $f:X\longrightarrow Y$ of degree $d$ and with branch locus contained in $B$.

I need a reference (also a paper), different from Rick Miranda's book, in which this theorem is proved.

Remark: I think that Miranda's book is a beautiful reference for Riemann surfaces, simply I don't like how the above theorem is presented. In general I'm not comfortable with theorems whose statements are given after the proofs.

Thanks in advance.

• Have you looked in Farkas/Kra? (If you have, there's no point for me to look whether it's in there.) May 21 '14 at 15:16
• I've looked in Farkas/Kra and there isn't the theorem May 21 '14 at 15:26
• The proof is short enough that you can just give it, can't you? If you believe that such a thing is a covering map away from the branch locus then this follows once you believe that $Y \setminus B$ has finitely generated fundamental group, which is clear from Seifert-van Kampen. May 21 '14 at 18:07
• @QiaochuYuan I know the proof, but I'm a bit undecided on writing it in my thesis. In general I can't distinguish between theorems that must be only cited and those that need an explicit proof. But this is another kind of problem. May 21 '14 at 18:31
• Dear fair-coin: Can you say what the proof in Miranda's book is, so we know what not to answer? May 28 '14 at 0:36

## 1 Answer

As requested:

A. D. Mednykh, "Nonequivalent coverings of Riemann surfaces with a prescribed ramification type", Siberian Mathematical Journal, 1984, Volume 25, Issue 4, pp 606-625. (English translation)

All in all the problem of computing Hurwitz numbers, is a very active area of mathematics, see for instance this paper by A.Okounkov and R.Pandaripande and this thesis by A.Zvonkine. See also here for further references.