Reference request: maps between moduli spaces

Question

I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with $n$ marked points to a space $X$ whose image is a cohomology class $\beta$. This has four variables, $g,n,X,\beta$; what happens as they change?

• What if I have an injective function of finite sets $m\to n$; this should induce a map ${\mathcal M}_{g,n}(X,\beta)\to{\mathcal M}_{g,m}(X,\beta)$, assuming both exist, and I want a reference for what I can expect of that map, or of the inverse limit of such maps over $n\in{\bf Fin}^{Inj}$.

• What if I have a morphism $X\to Y$ of ambient spaces. What can I know about the relationship between the associated moduli spaces?

• What if I add cohomology classes, e.g. $\beta+\beta'$? Anything to see here?

• Is there any reasonable notion of changing the genus $g$ so that a map between moduli spaces might be induced?

I'd like a single reference where I can find such answers, and I'd like it to have a category-theoretic flavor, phrasing as much as possible in terms of functorial relationships. If that can't be found, any help on the above types of questions would also be appreciated.

Answer 1

One thing that can help understand your points (a), (b) is the following picture of $M = \mathcal{M}_{g,n}(X,\beta)$: Let $T$ be an affine scheme. Then a family of objects in $M$ over $T$ is a diagram $$ \require{AMScd} \begin{CD} U @>f>> X \\ @V{\pi}VV \\ T \end{CD} $$ where the fibres over $t \in T$ are (arithmetic) genus $g$ curves together with $n$ sections $(s_i)_{1 \leq i \leq n}$ of the map $\pi$, which correspond to the marked points.

From this picture it is easy to see that the maps corresponding to injective maps $m \to n$ exist (which correspond to forgetting a collection of sections $s_{i_j}$), and further that given a map $X \to Y$ there is a map $\mathcal{M}_{g,n}(X,\beta) \to \mathcal{M}_{g,n}(Y,f_*\beta)$, at least naively.

One of course has to be concerned about the existence of such moduli spaces (as DM stacks, at least?): If you forget enough points so that you destabilize your curves, there may be problems, etc. etc. But this does give you a categorical picture of the content of these moduli spaces, and it's easy to see that the maps above are functorial constructions, which is nice.

As for (c), (d), changing the genus, it depends. If you mean the arithmetic genus, then there are gluing maps $$ gl : \mathcal{M}_{g_1, n_1 + 1}(X, \beta_1) \times \mathcal{M}_{g_2,n_2+1}(X, \beta_2) \to \mathcal{M}_{g_1+g_2,n_1+n_2}(X,\beta_1+\beta_2) $$ which satisfy a number of axioms. If you mean geometric genus, then there are boundary divisors in the moduli spaces (which correspond to certain degenerations of the curves), which may have lower geometric genus.

Much of this material can be found in some guise in "Mirror Symmetry and Algebraic Geometry" by Cox and Katz, although it is less focused on a categorical perspective. There is also chunks of this material in a series of papers by Abramovich, Graber, Vistoli on the discussion of GW invariants for orbifolds, which might help elucidate this further.