# Moduli of Riemann surfaces (genus g curves) is a variety.

I often see the moduli spaces $\mathcal{M}_g$, or at least the coarse moduli space, of Riemann surfaces of genus $g$ described as the set of isomorphism classes of Riemann surfaces of genus $g$.

Obviously, the moduli space has more structure than a set. However, the book I have been following immediately goes on to call $\mathcal{M}_g$ a variety.

How do we go from $\mathcal{M}_g$ being a set to $\mathcal{M}_g$ being a variety?

• Are you reading Rick Miranda's book? – Tanner Strunk Apr 6 '17 at 13:39

Your book is being extremely sneaky! This in fact a highly nontrivial construction. One good source to read about it is the end of Chapter I in the book Moduli of Curves by Joe Harris and Ian Morrison.

Curiously, long before anyone constructed $M_g$, Riemann was happily computing that its dimension equals $3g-3$. Riemann's argument is very nicely explained by user Brenin in another question on this site, which I can't find just at the moment.

The first construction of $M_g$, if I remember correctly, is due to Bers around 1950. This construction is analytic: it shows that the Teichmueller space $T_g$ is an open subset of $\mathbf C^{3g-3}$, and $M_g$ is then a quotient of $T_g$ by a discrete group action.

The first purely algebraic construction of $M_g$ was then given by Mumford in the early 1960s, using Geometric Invariant Theory. The idea here is that all smooth curves of genus $g$ can be embedded in a projective space $\mathbf P^N$ of fixed dimension (depending on $g$). So one can consider the corresponding component of the Hilbert scheme of $\mathbf P^N$; the moduli space $M_g$ is then the quotient of (an open subset of) this component by the action of the group $SL(N+1)$.

Anyway, the moral is: $M_g$ is not easy!

• Thank you! Knowing that this fact is nontrivial is already very useful! – user7090 Feb 23 '17 at 0:29
• @bertram: Right, except the first was not Bers, but Rauch. Rauch used periods of abelian differentials, unlike Bers who came in a bit later. One more curiosity: Bers was so impressed by Mumford's work that he found an alternative way to prove that $M_g$ is quasiprojective in the early 70s. – Moishe Kohan Feb 23 '17 at 4:54
• @MoisheCohen: thanks, that's interesting to know. Can you give a reference for Rauch's work? – bertram Feb 23 '17 at 5:47
• Rauch's work is discussed in great detail in Earl's article in: Hershel M. Farkas, Isaac Chavel (eds.): Differential geometry and complex analysis: a volume dedicated to the memory of Harry Ernest Rauch, Springer, 1985. Rauch's papers: On the transcendental moduli of algebraic Riemann surfaces, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 42–49. On the moduli of Riemann surfaces, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 236–238. A more detailed version is in: Weierstrass points, branch points, and moduli of Riemann surfaces, Comm. Pure Appl. Math. 12 (1959), 543–560. – Moishe Kohan Feb 23 '17 at 17:47
• Here's a link to the question you mention: math.stackexchange.com/q/513970 – Richard D. James Jan 18 '19 at 4:50