Basic Properties of Moduli Spaces


I realized I left off in really strange place last time. Sorry about that. There should be a burning question in everyone’s mind. To recap, we’ve seen that sometimes coarse moduli spaces don’t exist. When they do, sometimes there is no universal family. When there is, sometimes it is not a fine moduli space. Despite all these examples I’ve shown, I forgot to show you an example where a fine moduli space exists! So the question should be: is fine too strong a condition to ever exist?

Of course not. We can sort of cheat to make an example by working backwards. Our moduli functor {\mathcal{F}} is fine if it is naturally isomorphic to {h_M} for some scheme {M/k}. Thus take any scheme {M/k} and make the moduli functor {h_M}. This is the moduli space of points on {M}, so it is not surprising that {M} represents this functor. Still, this is a good example to understand as sort of the easiest of all possible moduli problems.

Lots of other schemes are defined to be the scheme whose functor of points is some moduli problem. The Hom scheme, the Quot scheme, the Hilbert scheme, and the Picard scheme are all examples of schemes (when they are schemes) that are fine moduli spaces for less trivial moduli problems.

Let’s look at some standard properties of moduli spaces. The moduli functor is called bounded if there is some finite type scheme {S/k} and a family {X\in \mathcal{F}(S)} such that for any object {Z\in \mathcal{F}(k)} there is some fiber {X_s} such that {Z\simeq X_s}. This is just saying that the objects you are trying to make into a moduli space fit into some finite type scheme.

The moduli functor is called separated if for any nonsingular curve {S/k} and a fixed {s_0\in S} if {X, X'\in \mathcal{F}(S)} with all fibers {X_s, X_s'} isomorphic for all {s\neq s_0} then {X_{s_0}\simeq X_{s_0}'}. This is essentially the functor of points translation of the valuative criterion for separatedness, so the term makes sense. Intuitively this is just saying that if you have a family of objects over a punctured curve, there is at most one way to fill in an object over that point to make a family over the whole curve.

The moduli functor is called complete if you can always fill in a family over a punctured curve. If there is a fine moduli space {M} associated to {\mathcal{F}}, then these properties translate exactly to the corresponding properties for the scheme {M}. Namely, if {M/k} is of finite type, then {\mathcal{F}} is bounded. Also, {M} is separated if and only if {\mathcal{F}} is separated, and {M} is proper if and only if {\mathcal{F}} is complete. The proofs are that these are exactly the respective valuative criteria.

Fix a positive integer {N} and a Hilbert polynomial {p}. Let’s do a different type of example for today. The Hilbert functor assigns to {S\in Sch_k} the set of subschemes {Y\subset \mathbb{P}^N_S} flat over {S} whose fibers all have Hilbert polynomial {p}. We won’t prove that there is a fine moduli space for this problem, since this is a fairly long proof due to Grothendieck.

One way to prove that the functor is bounded is to convert the moduli problem to one that involves the ideal sheaves of the closed subsets. Once this is done, bounded becomes equivalent to finding a single {m_0} such that all coherent sheaves in the problem are {m_0}-regular. This can be found using Castelnuovo-Mumford regularity. Thus the Hilbert scheme {Hilb_p(n)} is of finite type.

The other two conditions we talked about hold for more trivial reasons. A family over a punctured curve {S_0} is a closed subscheme of {Y\subset\mathbb{P}^N_{S_0}}, flat over {S_0}. Thus we can always fill in the punctured point by taking the scheme theoretic closure of {S} in {\mathbb{P}^N_S} where {S} is the filled in curve. This is the unique way of doing it. This shows that {Hilb_p(n)} is proper. Actually, it is projective, but this takes more work.

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