Now we’re done constructing , so it’s time to get the general Hilbert scheme done, and then to construct some other moduli spaces. Now, as , we can see that is a subfunctor of , so we want to try to find a subscheme of to represent it.
Now, something we didn’t mention is that there is what’s called a universal family over . In fact, any time you have a moduli functor and a scheme representing it , you want there to be an element , that is, we are looking for families over the moduli space itself. Now, we know that for any family over , we get a map , and now we have a family on . The property we want is that by base change along this morphism, we get the family over that we started with, so in that sense, this family is universal.
So for , the universal family is which has over each point the subscheme of parameterized by it. Now we let , where that intersection is a scheme theoretic intersection, which means we take the sums of their ideal sheaves. This is the same as taking the fibered product of the two schemes along their inclusions. (Moral: define all things by fibered products…)
So, by the existence of a flattening stratification (same thing that gave us ) we get a subscheme such that for any the pullback to is flat if and only if it factors through . And now this will represent the functor .
Now, all the Hilbert schemes happen to be locally closed subschemes of Grassmannians, so they ‘s are all of finite type and separated. So they’re ALMOST varieties, the only things missing are irreducibility, which we can forgive, and being reduced, which is much bigger. We’ll probably talk a bit about the nonreducedness later. For now, a note on properness (which would imply that Hilbert schemes are actually projective). We’ll prove this later once we’ve discussed DVRs and the valuative criterion for properness (which I’m only now really getting my appreciation for, because it seems technical and not that useful at first glance, but it proves some good things later.)
So now, we’ll put a bit more detail into the hom schemes mentioned before. We can define the functor any time, just is the collection of morphisms which commute with projection down to (called -morphisms). Now, taking graphs of morphisms does give this as a subfunctor of the Hilbert functor of subschemes of . But not all subfunctors of a scheme are subschemes, so there’s some work left.
Theorem: If are projective over and is flat over , then is represented by a scheme .
Proof: We have , and we send it to its graph is a closed immersion. So is flat over . This gives our map into the Hilbert functor.
For simplicity, denote and the universal family. Then we have a -morphism , which is the restriction of the projection .
So now we assume that from the fiber in of a point to the fiber in of that point is an isomorphism at some . That is, the fiber in is a graph. Applying a lemma which says that if is the spectrum of a local ring and is flat and proper, arbitrary with a -morphism, then , the restriction of the morphism to the fibers over 0 is a closed immersion (or isomorphism) then is as well. So then is an isomorphism over an open neighborhood of . Thus, we can view the subfunctor of graphs as a subscheme, and so is representable. QED
Ok, now onto families of divisors, also a rather important object. Take flat over and an effective Cartier divisor. One definition of this, which is pretty general, is that for each , there’s which is not a zero divisor and such that in a neighborhood of . We’ll call a relative Cartier divisor if either of the following (equivalent) things happens:
- is flat over
- For each , is not a zero divisor in , where is the structure map.
So now, for flat, we define a functor which takes to the set of relative effective Cartier divisors of .
We want to see that this is represented by an open subscheme of . To do this, we want to show that if is flat and is flat over , then the set of such that the fiber in , over is a Cartier divisor is an open set.
Let be the ideal sheaf of and let . Because is flat over , we have that . So we should take to generate locally at . Let be a local section of at that, when tensored with 1 over gives us . Now, Nakayama tells us that this is a local generator of at (note that Nakayama’s big purposes are proving that modules are zero and reducing the problem of finding generators to a simpler situation). So is relative Cartier at . What this tells us is that there is some open set on which is defined as a generator, so we have our desired result.
Other properties of are harder to check. For instance, if is smooth over , then it is universally closed. That is, the inclusion of will take closed sets to closed sets, and any base change will as well. In fact, if we have geometrically normal over a field (that is, the fiber product is normal) then is a closed subset of the Hilbert scheme.
I think we’ll be taking a slight break from the Hilbert scheme after this one, though it may come up in other places (like we’ll prove properness for it once we do the valuative criterion for properness). Not sure what, precisely, we’ll do next, though.