Ben McReynolds and I have just arXived a retitled and substantially revised version of our paper “Every curve is a Teichmuller curve,” previously blogged about here. If you looked at the old version, you probably noticed it was very painful to try to read. My only defense is that it was even more painful to try to write.
With the benefit of a year’s perspective and some very helpful comments from the anonymous referee at Duke, we more or less completely rewrote the paper, making it much more readable and even a bit shorter.
The 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 question. The 4-branched Belyi question essentially asks whether every curve C in M_g is a Hurwitz space of 4-branched covers. (Surely not!) The DDH theorem shows that if you’re going to prove C is not a Hurwitz curve, you can’t do it by means of the birational isomorphism class of C alone; every 1-dimensional function field appears as the function field of a Hurwitz curve (though probably in very high genus.)