A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of $\mathcal{M}_g$
Does this help a little?
"In a 1891 paper, Hurwitz explains how the set of degree d simple covers (all fibers consist of at least d-1 points) P1 (the projective line – Riemann sphere) has a structure of complex manifold. In this he follows a much earlier (1867) paper of Clebsch who showed the connectedness of the space of simple covers. Hurwitz's paper thereby applies to show the connectedness of the moduli space of compact surfaces of genus g."
If nothing else this may point you to folks who may know (Pierre Debes and Mike Fried).
I was able to find the resources online (6 years after this question was asked):
- 1871 https://eudml.org/doc/156527 Lüroth - 4 pages
- 1873 https://eudml.org/doc/156610 Clebsch - 16 pages
- 1891 https://eudml.org/doc/157563 Hurwitz - 61 pages
I think it's pretty clear Luroth was first, but Hurwitz developed this material much further.
I get thrown off because we way Riemann surface is just a polygon with edges glued together, and that's pretty much the picture of moduli space painted here.
Harris and Morrison point you (after stating the theorem in 1.5.4) to Clebsch's Zur Theorie der Rieman'schen Flachen Math Ann. 6 216-230, 1872. My German is not that good, but section 2 seems convincing.