We’ve previously discussed moduli spaces a bit. The point of this is that we’ll need to understand a certain moduli space, the moduli of stable maps, in order to continue. So we’ll be starting out with moduli of stable curves, and then continuing on.

So back on the moduli spaces and base change post, I asserted the existence of moduli spaces $M_{g,n}$, and stated the the dimension of this space is $3g-3+n$. Now, this is the moduli space of smooth, irreducible curves of genus $g$ with $n$ marked points. Strictly, these are only coarse moduli spaces, but they are quasi-projective. To do anything useful for counting curves, we need to describe a compactification: $\bar{M}_{g,n}$.

We define an $n$-pointed stable curve of genus $g$, $C$, to be an $n+1$-tuple $(C,p_1,\ldots,p_n)$ with $p_i\in C$, all the $p_i$ distinct smooth points, such that $C$ is a complete connected curve with at worst nodal singularities and arithmetic genus $g$, such that there are only finitely many automorphisms of $C$ fixing the $p_i$ and nodes. The last condition can be rephrased to be that every rational component of the normalization
has at least three points lying over special points (that is, nodes or marked points). If this last condition is not necessarily satisfied, we call the curve quasi-stable.

So now we define a family of stable curves over a base $B$ to be $\mathscr{C}\to B$ a flat morphism with fibers stable curves and $n$ sections $\sigma_i:B\to \mathscr{C}$. The functor taking $B$ to the set of families of stable curves over $B$ is not representable, sadly. However, it is coarsely representable. That is, there’s a coarse moduli space of stable curves. We denote it by $\bar{M}_{g,n}$.

So this is a moduli space compactifying $M_{g,n}$ where the marked points can’t coincide. If two points are coming together in the moduli space, then the limit will be non-irreducible. It will “grow” a new rational component, which will intersect the point the two marked points were converging to, and the two marked points will move onto it.

Now, an interesting thing is to look at $\bar{M}_{0,n}\setminus M_{0,n}=D_{0,n}$. This is actually a divisor, called the boundary divisor, and it behaves rather well. In fact, there is a conjecture of Fulton’s that connects this up to toric geometry via a strong analogy, but I’m not going to go into this.

Now we move on to families of maps. We’ll first talk about quasi-stable things, before going on to stable. A family of maps of $n$-points genus $g$ quasi-stable curves to $X$ over $B$ is a family of quasi-stable curves $\mathscr{C}\to B$ along with a map $\mu:\mathscr{C}\to X$. The members of the family are the restriction of $\mu$ to the fibers. We call two families of maps isomorphic if there’s an isomorphism $\tau:\mathscr{C}\to\mathscr{C}'$, commuting with the maps to $B$, the maps to $X$, and the sections.

A family of maps will be a family of stable maps if the map restricted to each fiber is stable. So all that’s left is to define what it means for a map $\mu:C\to X$ from a quasi-stable curve to $X$ to be stable. It will just be a condition on the components: let $E\subset C$ be an irreducible component. Then the map is stable if the two following conditions hold:

1. If $E$ is isomorphic to $\mathbb{P}^1$ and $E$ is mapped to a point by $\mu$, then $E$ contains at least three special points.
2. If $E$ is of arithmetic genus 1 and is mapped to a point, then it contains at least one marked point.

That is, the map restricted to each component is constant only if that component is stable.

The last point in defining the moduli space is breaking it up into components. Let $\beta\in H_2(X,\mathbb{Z})$ a homology class on $X$. Then we say that $\mu:C\to X$ represents $\beta$ if $\mu_*[C]=\beta$.

So now, we have a few big theorems and small definitions that we’ll need, but won’t prove:

Theorem: There exists a projective coarse moduli space $\bar{M}_{g,n}(X,\beta)$ of stable maps representing $\beta$.

Now we define $\bar{M}_{g,n}^*(X,\beta)$ to be the subset of maps with no automorphisms, and we say that a variety $X$ is convex if for all $\mu:\mathbb{P}^1\to X$, we have $H^1(\mathbb{P}^1,\mu^*T_X)=0$. We also define the boundary of $\bar{M}_{g,n}(X,\beta)$ to be the locus of non-irreducible domains. Then we have the following theorem:

Theorem: Let $X$ be a projective, smooth, convex variety

1. $\bar{M}_{0,n}(X,\beta)$ is a normal projective variety of pure dimension $\dim X+\int_\beta c_1(T_X)+n-3$
2. $\bar{M}_{0,n}(X,\beta)$ is locally the quotient of a smooth variety by a finite group. That is, it’s an orbifold.
3. $\bar{M}^*_{0,n}(X,\beta)$ is a smooth, fine moduli space.
4. The boundary of $\bar{M}_{0,n}(X,\beta)$ is a divisor with normal crossings, up to finite group quotients.

Before moving on, we will note that there’s quite a bit of structure floating around that we haven’t mentioned yet. For instance, each marked point gives a map $ev_i:\bar{M}_{g,n}(X,\beta)\to X$ by taking the stable map $\mu:C\to X$ to the point $\mu(p_i)$. Similarly, we can forget marked points (possibly contracting components) to get a map $\bar{M}_{g,n}(X,\beta)\to \bar{M}_{g,m}(X,\beta)$ whenever $m\leq n$. And finally, by mapping $\mu:C\to X$ to $C$, we get a map $\bar{M}_{g,n}(X,\beta)\to \bar{M}_{g,n}$.

From here on out, we’re pretty much going to be using only curves of genus zero. Next up: Gromov-Witten invariants.