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Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that $M_g$ has dimension $3g-3$.

A modern proof involves deformation theory and Riemann-Roch theorem. In particular, one needs the notion of sheaf cohomology, which was not available at Riemann's time.

How did Riemann prove that $M_g$ has dimension $3g-3$ for $g>1$?

I would appreciate if someone could provide some reference to the original proof by Riemann.

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    $\begingroup$ I guess the Riemann-Roch theorem was already known to Riemann. $\endgroup$ – ThiKu Dec 1 at 1:25
  • $\begingroup$ I take a look at the article, it seems degree-genus formula is already known to Riemann. $\endgroup$ – Bombyx mori Dec 1 at 2:49
  • $\begingroup$ @Bombyxmori That would be amazing, considering the notion of Riemann surfaces (or complex manifold) was probably not understood by the rest of the world. $\endgroup$ – Yuhang Chen Dec 1 at 15:19
  • $\begingroup$ @ThiKu I believe so. $\endgroup$ – Yuhang Chen Dec 1 at 15:22
  • $\begingroup$ @ThiKu The issue is whether the version with $H^1(X,L)$ instead of $H^0(X,K_X\otimes L^{-1})$ was known to Riemann. Riemann's proof discussed in the answers of F Zaldivar and Alexandre Ermenko uses only the second, more elementary, notion. $\endgroup$ – Will Sawin Dec 1 at 17:25
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Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays. He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere. He computes this dimension in two ways. By Riemann-Roch this dimension is $2d-g+1$, for a fixed Riemann surface. (Indeed, Riemann-Roch says that the dimension of the space of such functions with $d$ poles fixed is $d-g+1$ (when $d\geq 2g-1$ which we may assume), but these poles can be moved, so one has to add $d$ parameters).

On the other hand, such a function has $2(d+g-1)$ critical points by Riemann-Hurwitz. Generically, the critical values are distinct, and can be arbitrarily assigned, and this gives the dimension of the set of all such maps on all Riemann surfaces of genus $g$. So the space of all Riemann surfaces of genus $g$ must be of dimension $$2(d+g-1)-(2d-g+1)=3g-3.$$ Riemann-Roch is proved in section 5 of Part I and and the dimension of the moduli space is counted in section 12 of Part I of the paper cited in F. Zaldivar's answer.

Remark. Indeed, Riemann did not know about sheaves, cohomology and Serre duality. Neither he knew the general definition of a Riemann surface (which is due to Weyl). But one should take into account that all these notions were developed for the purpose to explain and digest what Riemann wrote in this paper.

Remark 2. A pair $(S,f)$, where $S$ is a Riemann surface, and $f$ a meromorphic function from $S$ to the Riemann sphere is called "a Riemann surface spread over the sphere'' (Uberlagerungsflache). All such pairs can be constructed in the following way: choose critical values of $f$ and make some cuts between them so that the remaining region on the sphere is simply connected. Then take $d$ copies of this region (they are called sheets) stack them over the sphere, and paste them together along the cuts. You obtain a surface $S$ together with a map $f$, the "vertical" projection onto the sphere. Parameters are critical values.

Riemann did not have any exact definition of "Riemann surface", he just explained this procedure of gluing as a visualization tool. For him, $S$ is a "class of algebraic curves $F(x,y)=0$ under birational equivalence". Until the work of Weyl, these pairs $(S,f)$ were called Riemann surfaces, and only Weyl defined exactly what $S$ is. Nowadays $S$ is called a Riemann surface, and a pair $(S,f)$ a "Riemann surface spread over the sphere".

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  • $\begingroup$ sixed -> fixed? $\endgroup$ – Igor Rivin Dec 1 at 19:43
  • $\begingroup$ Thanks. I corrected. $\endgroup$ – Alexandre Eremenko Dec 1 at 21:58
  • $\begingroup$ This is fantastic. Can you explain a bit why the dimension comes out as the difference between $2(d+g-1)$ and $2d-g+1$? Is $2(d+g-1)$ the number of critical points of a function on a fixed generic Riemann surface as well? $\endgroup$ – Yuhang Chen Dec 2 at 11:28
  • $\begingroup$ @Yuhang Chen: the number of critical points of generic function on generic surface is $2(d+g-1)$. When you vary these critical values, both the Riemann surface and the function change. You can translate this to the modern language of fibrations. For given critical values, the pair (surface,function) is constructed by scissors and glue, as Riemann did. $\endgroup$ – Alexandre Eremenko Dec 2 at 15:05
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    $\begingroup$ @Yuhang Chen: $z$ if a critical point when $f'(z)=0$ and $f(z)$ is the critical value. $\endgroup$ – Alexandre Eremenko Dec 2 at 17:28
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The original paper of Riemann is his celebrated "Theorie der Abel'schen Functionen" in Crelle's Journal of 1854. This paper can be found online at https://www.maths.tcd.ie/pub/HistMath/People/Riemann/AbelFn/

There is an English translation of Riemann's Collected Papers (Kendrick Press, I believe).

Modern accounts can be found in several textbooks or monographs, e.g., Harris and Morrison "Moduli of Curves" (Springer).

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    $\begingroup$ Is it possible to explain a bit the outline of Riemann's original proof? $\endgroup$ – Yuhang Chen Dec 1 at 15:10

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