Let \(X\) be projective algebraic variety and fix a class \(\beta\in H_2(X,\mathbb Z)\). Then we can define a moduli functor (contravariant)
\begin{equation} \overline{\mathcal M}_{g,n}(X,\beta):(\mathbb C -\mathsf{Schemes})\to (\mathsf{Sets}) \end{equation}by sending a scheme \(S\) to the set of all isomorphism classes of families of stable maps over \(S\) of genus \(g\) and class \(\beta\).
To solve this moduli problem one must represent this functor in some way. The straightforward approach is to show there exists a course moduli space and the more sophisticated approach is to show that this moduli functor is actually and algebraic stack.
Course moduli space of stable maps
Theorem
If \(X\) is a projective variety over \(\mathbb C\) and \(\beta\) is a homology class in \(H_2(X,\mathbb Z)\), then the functor \(\overline{\mathcal M}_{g,n}(X,\beta)\) has a coarse moduli space \(\overline{M}_{g.n}(X,\beta)\) which is a projective scheme over \(\mathbb C\).
A fairly explicit constuction can be found in [FP, /Notes on stable maps and quantum cohomology/].
Algebraic stack of stable maps
Theorem
If \(X\) is a projective variety over \(\mathbb C\) and \(\beta\) is a homology class in \(H_2(X,\mathbb Z)\) then the functor \(\overline{\mathcal M}_{g,n}(X,\beta)\) is an algebraic stack which is proper over \(\mathbb C\). Furthermore, it is a smooth stack when \(g=0\).
The underlying algebraic space of \(\overline{\mathcal M}_{g,n}(X,\beta)\) is the course moduli space \(\overline{M}_{g,n}(X,\beta)\).
Natural Maps
We get the following natural maps:
which are defined as follows. Given a stable map \(f:(C,p_1,...,p_n)\to X\), we get a well-defined tuple
\begin{equation} \label{eq:full-evaluation-map} \pi_1(C,p_1,...,p_n,f) = (f(p_1),...,f(p_n)) \in X^n \end{equation}defining \(\pi_1\) set-theoretically. It shouldn’t be too unbelievable that this is indeed a morphism as well. I think of this as the “full evaluation map”, since composition with the projection onto the $i$th factor of \(X^n\) gives the map \(\operatorname{ev}_{i}\):
\begin{align*} \operatorname{ev}_{i}:\overline M_{g,n} \to X ~,~ \operatorname{ev}_{i}(C,p_1,...,p_n,f) = f(p_{i}). \end{align*}As for \(\pi_2\), it should just be the map which forgets the stable map \(f\). However, \(C\) need not be a stable curve if \(f:C\to X\) is a stable map; for instance, we could have a component \(C_i\cong \mathbb P^1\) of \(C\) which isn’t contracted by \(f\), in which case it need not have 3 specal points. However, once we contract the non-stable components of \(C\) we do get a stable curve \(\widetilde C\) – provided that \(n + 2g \geq 3\) (so that \(\overline M_{g,n}\) exists).
We define \(\pi_2\) set-theoretically by sending \(C\) to \(\widetilde C\).