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The connectedness of the moduli space $\mathcal{M}_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the line, which in turn can be shown to be connected by proving the transitivity of the the natural action of the braid group on n-tuples of transpositions in $S_n$ with product 1, which generate $S_n$: in this action, a generator $\sigma_i$ of the braid group acts as

$$(g_1, ... g_n) \to (g_1, ... g_{i+1}, g_i^{g_{i+1}}, g_{i+2}, ..., g_n)$$

This argument is often referred to as "a theorem of Clebsch (1872 or 1873), Luroth (1871), and Hurwitz (1891)." Does anyone know the history of this argument more precisely, and in particular which parts are due to Luroth, which to Clebsch, and which to Hurwitz?

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up vote 6 down vote accepted

I was able to find the resources online (6 years after this question was asked):

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.

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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).

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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.

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