Much discussion on Math Overflow has not resolved the following should-be-easy question:

Give an example of a curve in defined over which is not a family of 4-branched covers of P^1.

Surely there is one! But then again, you’d probably say “surely there’s a curve over which isn’t a 3-branched cover of P^1.” But there isn’t — that’s Belyi’s theorem.

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Would you be willing to tag this post “PlanetMO” so that it can be found via mathblogging.org/planetmo ?

done!

Thanks!

Dear Jordan — I think you should remove the overline from Mgbar. I am sure there are curves completely contained in the boundary which are not families of 4-branched covers (for any of the usual extensions of this notion to the boundary). Best — Jason

done, Jason.

[...] paper is related to the question I discussed last week about “4-branched Belyi” — or rather the theorem of Diaz-Donagi-Harbater that inspired our paper is related to that [...]