## Low Genus Moduli of Curves

Pretty much everything in this post is in Mumford’s “Curves and their Jacobians,” but I do a couple of things slightly differently, and I intend to supply a bit more detail in some places.  The goal here is to construct the moduli of curves of genus $g$ for small $g$.

Let’s start with some definitions.  The set $\mathcal{M}_{g,n}$ is the set of $(C,x_1,\ldots,x_n)$ where $C$ is a genus $g$ curve and $x_1,\ldots,x_n\in C$ are distinct.  Now, sadly, $\mathcal{M}_{g,n}$ is not a variety.  It can be realized as a smooth stack, however, but that’s a bit more than we are going to even try to handle.  Instead, we’ll quote a very good, powerful and classical theorem, which is true (and clearly so, because I am using a boldface for the word theorem):

Theorem: There exists a unique normal quasiprojective variety $M_{g,n}$ such that it is in bijection with $\mathcal{M}_{g,n}$ and if $\pi:X\to S$ is a proper smooth map whose fibers are genus $g$ curves and $\sigma_1,\ldots,\sigma_n:S\to X$ are $n$ disjoint sections, then the induced map of sets $\phi:S\to M_{g,n}$ is a morphism of varieties.

Rather, the above is true when $3g-3+n\geq 0$.  When it’s smaller, then the space has expected dimension negative (this only happens for genus 0 with at most 2 points, and genus 1 with no points) and we really need the stacky interpretation to make sense of things.

So why is this the dimension? The computation is fairly quick: first off, each point is a degree of freedom, so $\dim M_{g,n}=n+\dim M_{g,0}$, and $M_{g,0}$ is just deformations of a curve.  It can be shown that the tangent space to $M_{g,0}$ at a curve $C$ is just $H^1(C,T_C)$, so we must compute this dimension, which by Riemann-Roch is $3g-3$ for $g\geq 2$.  For $g=0,1$, we need to check things carefully because of the aforementioned stacky problems (it amounts to positive dimensional families of automorphisms for these curves), but the result works out.

We’re really only going to have one example, and then the others will follow from it.  We’ll directly compute $M_{0,n}$ for $n\geq 3$.  There’s only one genus zero curve, and that’s $\mathbb{P}^1$, so we’re looking at sets of $n$ ordered points in $\mathbb{P}^1$.  The automorphisms let us take the first three to $0,1,\infty$, so we have $n-3$ points to pick, and there are no automorphisms left once we’ve done this, so $M_{0,n}\cong (\mathbb{P}^1)^{n-3}\setminus\mbox{diagonals}$, because the points are distinct.

Next, we’ll look at $M_{1,1}$, as this is the next simplest that has positive dimension.  There are a few ways we can attack this.  The simplest is by using Weierstrass form and the $j$-invariant, and this works for characteristic not $2,3$, and we’re going to imagine ourselves over $k=\mathbb{C}$, so it’s fine.  Every elliptic curve can be written as $y^2=x(x-1)(x-\lambda)$ for some $\lambda\in k$, with $\lambda\neq 0,1$.  However, this representation isn’t unique.  So whatever our curve is, it’s covered by $\mathbb{P}^1\setminus\{0,1,\infty\}$.  There’s a function $j(\lambda)=256\frac{(\lambda^2-\lambda+1)^3}{\lambda^2(\lambda-1)^2}$, and it takes the same values at $\lambda$, $1-\lambda$, $\frac{1}{\lambda}$, $\frac{\lambda-1}{\lambda}$, $\frac{\lambda}{\lambda-1}$, and $\frac{1}{\lambda-1}$, which all give the same elliptic curve! Even better, no other $\lambda$ will give the same curve, so $j:\mathbb{P}^1\setminus\{0,1,\infty\}\to M_{1,1}$ is a surjective map.  And because $j$ can take any value, it shows that $M_{1,1}\cong \mathbb{A}^1$.

A slightly different presentation is that every elliptic curve can be written as a double cover of $\mathbb{P}^1$ ramified at four points.  These points are unordered, and so we can look at quartic polynomials on $\mathbb{P}^1$ with no repeated roots $f(x,y)$.  However, automorphisms of $\mathbb{P}^1$ let us take the roots to $0,1,\infty,\lambda$, so we have $f(x,y)=xy(x-y)(x-\lambda y)=0$ and its discriminant is nonzero, and we have to identify the same $\lambda$‘s as above.

This approach is fruitful for identifying a certain subvariety of $M_g(=M_{g,0})$ for $g\geq 2$.  This is the locus of hyperelliptic curves.  These are the curve which double cover $\mathbb{P}^1$ and thus are uniquely determined by their branch points.  For a genus $g$ hyperelliptic curve, we have $2g+2$ ramification points.  The second approach above is predicated on the fact that for $g=1$, every curve is hyperelliptic.  We’ll denote the hyperelliptic locus by $H_g$.

Now, it’s a fairly easy theorem that $H_2=M_2$, so we should be able to handle finding a description of this family of curves.  By the above, they branch at six points, so we have to look at six distinct points in $\mathbb{P}^1$, which is an open subset $U\subset\mathbb{P}^6$ of the sextic polynomials.  Then, we need to take $U/\mathrm{Aut}(\mathbb{P}^1)$.  We can achieve this by looking at ordered sextuples of points on $\mathbb{P}^1$, and having an extension of $\mathrm{Aut}(\mathbb{P}^1)$ by $S_6$ (or perhaps extension in the other direction? I never quite got that bit of language) act by first permuting the points, and then transforming $\mathbb{P}^1$ so that the first three are $0,1,\infty$.  This can actually be calculated directly, and it’s a quotient of $\mathbb{A}^3$.  The points that need to be identified are the ones where, if $\zeta$ is a primitive fifth root of unity, we have $(x,y,z)$ gives the same curve as $(\zeta x, \zeta^2 y, \zeta^3 z)$.  It’s very straightforward to compute the invariants, and they are $x^5,x^3y,xy^2,y^5,x^2z,xz^3, z^5, yz$, and give a map $\mathbb{A}^3\to \mathbb{A}^8$ which factors through the quotient, thus giving us $M_2$ inside of $\mathbb{A}^8$.

In general, we have that $H_g$ can be written as a quotient of $M_{0,2g+2}$, though computing this quotient is not always easy.  This is also less useful in higher genus, because starting in genus 3, there are nonhyperelliptic curves.  In fact, the moduli space $M_3$ can be described in two pieces: the hyperelliptic locus $H_3$ can be worked out as above.  For the nonhyperelliptic curves, the canonical map gives them as quartic plane curves, so we have $U\subset |4H|$ the open subset of the linear system of quartics consisting of smooth curves, and we have to take $U/PGL(3)$, the automorphism group of the plane.  This is plainly unirational (for those who know the term), and I vaguely recall seeing a paper proving that it’s actually rational, but I can’t find it at the moment.  So starting at $g$, these moduli spaces can be very hard to construct directly, and that’s a big part of why this subject is interesting, and why it is so difficult.

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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### 4 Responses to Low Genus Moduli of Curves

1. hilbertthm90 says:

Speaking of moduli of curves, in our student AG seminar it came up that we needed to take (sheaf) cohomology on a stack of moduli of curves.

Now our guess was that we could choose a suitably nice hypercover since it was a DM stack and compute that way and get the right answer.

I’ve been asking people and searching the internet for any mention of this, but I can’t seem to find anyone or anywhere that says this can be done. Do you know of any references, or off the top of your head whether there are nice conditions that allow you to do this?

• I don’t, actually. I mostly deal with the coarse moduli varieties involved, and sometimes I deal with them as Teichmuller space modulo the mapping class group. This seems like a good thing to ask on MO.

2. John says:

Don’t you need to divide out by the S_3 action on the points x,y,z in you genus 2 computation?

3. lap top says:

I want to know, how do differential geometers think of moduli spaces? I have read about moduli space of (Kahler-Einstein) metrics which are just set of such metrics modulo isomorphisms. What use the moduli space is of in differential geometry? Is it possible to investigate any differential geometric or analytic properties of the moduli space? Of course, in the analytic setting, the moduli space is an orbifold. But you know what I am meaning by diff….eometric properties.