# Recap of Riemann's original proof of the dimension of Moduli space $M_g$

I am trying to understand some single steps in this recap of Bernhard Riemann's original proof by Alexandre Eremenko that the moduli space $$M_g$$ of compact Riemann surfaces with genus $$g \ge 2$$ has dimension $$3g-3$$ (or in original context the $$3g-3$$ 'degrees of freedom' are what Riemann called 'moduli')

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 $$2(d+g-1)-(2d-g+1)=3g-3.$$

There are essentially two things I struggle with.

Point 1: Why the dimension of the space of holomorphic maps of degree $$d$$ from the Riemann surface $$S$$ of genus $$g$$ to the sphere equals $$2d-g+1$$?

As far as can follow we are going to apply RR. Let $$h$$ any holomorphic map $$S \to \mathbb{P}^1$$ of degree $$d$$ which we regard as meromorphic function $$h$$ which behaves compatible with degree $$d$$ in it's poles and zeros, set $$(h) :=D$$ and apply RR to it. RR tells $$l(D)= d -g+1$$ where $$l(D)$$ is the $$C$$-dimension of meromorphic functions $$f$$ on $$S$$ such that all the coefficients of the divisors $$(f) + D$$ are non-negative.

Eremenko argues then that the additional $$d$$ dimensions we obtain because 'the poles of can be moved, so one has to add $$d$$ parameters.'

I not understand why moving of the $$d$$ poles gives additionally exaxt $$d$$ parameters.

Point 2: Why the difference

$$2(d+g-1)-(2d-g+1)$$

between the number of critical points of these moromorphic functions (that's Riemann-Hurwitz-Thm) and the counted dimension of the space of holomorphic maps of degree $$d$$ from point 1 gives exactly the dimension of the moduli?

1. To specify a degree $$d$$ map $$f: X \to \mathbb{P}^1$$, one first chooses an effective divisor $$D$$ of degree $$d$$ to specify the poles of $$f$$. Since $$D$$ has support consisting of $$d$$ points, this amounts to $$d$$ choices. So this by this "the poles can be moved" comment, he means that choosing a different divisor $$D'$$ will result in a different map with poles given by the support of $$D'$$ instead. Then one computes $$\ell(D)$$ as you have described above to get the dimension of the space of degree $$d$$ maps $$f: X \to \mathbb{P}^1$$ with $$\renewcommand{\div}{\operatorname{div}} \div_\infty(f) = D$$.

2. We want to compute the dimension of $$\mathcal{M}_g$$, the moduli space of curves of genus $$g$$, but what we've really done is computed the dimension of the moduli space of pairs $$(X,f)$$ where $$X$$ is a curve of genus $$g$$ and $$f$$ is a degree $$d$$ map $$X \to \mathbb{P}^1$$. This related moduli space is called a Hurwitz space, denoted $$\DeclareMathOperator{\Hur}{Hur} \Hur_{d,g}$$. In order to remove this choice of a map $$f: X \to \mathbb{P}^1$$, we subtract off the dimension of the space of degree $$d$$ maps $$X \to \mathbb{P}^1$$ which was computed above.

You can interpret this using the projection \begin{align*} \pi: \Hur_{d,g} &\to \mathcal{M}_g\\ (X,f) &\mapsto X \, . \end{align*} We've computed the dimension of $$\Hur_{d,g}$$ and, for a given $$X$$, the dimension of the fiber $$\pi^{-1}(X)$$. Assuming $$\pi$$ is sufficiently "nice", we should have \begin{align*} \dim(\mathcal{M}_g) = \dim(\Hur_{d,g}) - \dim(\pi^{-1}(X)) = 2d + 2g - 2 - (2d - g + 1) = 3g-3 \, , \end{align*} as desired.

This is covered in greater detail in chapter 2, section 3 (p. 255) of Principles Of Algebraic Geometry by Griffiths and Harris. Jarod Alper also recently covered this in a lecture for a course he's teaching, which you can watch here. (See here for the dimension calculation you've asked about.)

• The point $2$ meanwhile presumably I understand. In summary there exist an dense open subset $U \subset \DeclareMathOperator{\Hur}{Hur} \Hur_{d,g}$ over which the restriction of the projection of $\pi$ to $\pi \vert _U$ behaves 'like a fibre bundle' from dimensional conting point of view, thus the the formula $\dim(\mathcal{M}_g) = \dim(\Hur_{d,g}) - \dim(\pi^{-1}(X))$, right? Commented Feb 23, 2021 at 1:28
• About the point $1$ I still not sure if I understand it completely. I would to explain the proble i see. We fix a Riemann surface $X$ and want to deduce that the dimension of the holom maps of degree $d$ $X \to P^1$ equals exactly $\ell(D)+ d$. As you said every such map has an associated divisor $D$ of deg $d$ which is supported in $d$ points (counted with multiplimities). And RR says $\ell(D)=d-g +1$. Then we vary these $d$ points slowly say from $D:= \{p_1,...,p_d\}$ to $D':=\{p'_1,...,p'_d\}$. Commented Feb 23, 2021 at 1:29
• Since all these points live on $X$, which is a $1$-dim manifold, so the variation of every point $p \in X$ separately contributes exactly one 'degree of freedom' =$\mathbb{C}$-dimension, as far correct? But now I see a problem: Commented Feb 23, 2021 at 1:30
• But I think now you precisely hit the essential point that answers my concern. We want to count the dimension generically, that's it's suffice to pass somehow to open dense sets $V \subset D(X,d)$ and family of open dense subsets $U_D \subset L(D)$ for $D \in V$, with property $U_D \cap U_{D'} = \emptyset$ for $D \neq D' \in V$, that was your argument, right? Commented Feb 24, 2021 at 1:20