There’s no 4-branched Belyi’s theorem — right?

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

Give an example of a curve in {\mathcal{M}}_g defined over \bar{Q} 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 \bar{Q} which isn’t a 3-branched cover of P^1.”  But there isn’t — that’s Belyi’s theorem.

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6 thoughts on “There’s no 4-branched Belyi’s theorem — right?

  1. Peter says:

    Would you be willing to tag this post “PlanetMO” so that it can be found via ?

  2. Peter says:


  3. Jason Starr says:

    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

  4. JSE says:

    done, Jason.

  5. […] 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 […]

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