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28
votes
4answers
2k views

Curves which are not covers of P^1 with four branch points

The following interesting question came up in a discussion I was having with Alex Wright. Suppose given a branched cover C -> P^1 with four branch points. It's not hard to see that the field of ...
11
votes
2answers
531 views

Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces $\newcommand{\Ch}{\hat{\mathbb{C}}}$ A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...
4
votes
1answer
395 views

technical question about a paper on Belyi's theorem

I'm trying to understand Belyi's theorem, as presented here: http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf He defines a curve $X$ over a field $C$ to be a smooth projective geometrically connected ...
3
votes
1answer
252 views

When do tamely ramified Belyi maps exist in characteristic p?

Let $p$ be a prime. Let $d$ be a number. Let $\lambda_0,\lambda_1,\lambda_{\infty}$ be three partitions of $d$ whose parts are prime to $p$. Consider the following question: Does there exist a ...
0
votes
0answers
74 views

Belyi's theorem for surfaces. Does exist another generalization?

Is there a generalization of Belyi's theorem in dimension $2$ (or higher) other than this by Gabino Gonzalez Diez? I haven't found no other references on the argument. Thanks in advance