## Which pictures do children draw with straight lines?

Edray Goins gave a great colloquium today about his work on dessins d’enfants.  And in this talk there was a picture that surprised me.  It was one of the ones on the lower right of this poster.  Here, I’ll put in a screen shot:

Let me tell you what you’re looking at.  You are looking for elliptic curves E admitting a Belyi map f: E -> P^1, which is to say a map ramified only over 0,1, and infinity.  For each such map, the blue graph is f^{-1}([0,1]), the preimage of the line segment joining o and 1 in P^1(R).

In four of these cases, the graph is piecewise linear!  I didn’t know there were examples like this.  Don’t know if this is easy, but:  for which Belyi maps (of any genus, not just genus 1) is f^{-1}([0,1]) a union of geodesics?

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## 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.