## 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.

## 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 mathblogging.org/planetmo ?

2. JSE says:

done!

3. Peter says:

Thanks!

4. 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

5. JSE says:

done, Jason.

6. […] 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 […]