# 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.

• Have you looked in Farkas/Kra? (If you have, there's no point for me to look whether it's in there.) – Daniel Fischer May 21 '14 at 15:16
• I've looked in Farkas/Kra and there isn't the theorem – Dubious 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. – Qiaochu Yuan 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. – Dubious 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? – Bruno Joyal May 28 '14 at 0:36