Examples of Moduli Spaces

Now we’re done constructing $\mathrm{Hilb}(\mathbb{P}^n/S)$, so it’s time to get the general Hilbert scheme done, and then to construct some other moduli spaces. Now, as $X\subseteq\mathbb{P}^n$, we can see that $Hilb(X/S)$ is a subfunctor of $Hilb(\mathbb{P}^n/S)$, so we want to try to find a subscheme of $\mathrm{Hilb}(\mathbb{P}^n/S)$ to represent it.

Now, something we didn’t mention is that there is what’s called a universal family over $\mathrm{Hilb}(\mathbb{P}^n/S)$. In fact, any time you have a moduli functor $F$ and a scheme representing it $X$, you want there to be an element $U\in F(X)$, that is, we are looking for families over the moduli space itself. Now, we know that for any family over $Z$, we get a map $Z\to X$, and now we have a family on $X$. The property we want is that by base change along this morphism, we get the family over $Z$ that we started with, so in that sense, this family is universal.

So for $\mathrm{Hilb}(\mathbb{P}^n/S)$, the universal family is $U\subset \mathbb{P}^n\times_S \mathrm{Hilb}_P(\mathbb{P}^n/S)$ which has over each point the subscheme of $\mathbb{P}^n$ parameterized by it. Now we let $V=U\cap (X\times_S \mathrm{Hilb}(\mathbb{P}^n/S)\subset \mathbb{P}^n\times_S \mathrm{Hilb}_P(\mathbb{P}^n/S)$, 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 $G_P$) we get a subscheme $H_P\to \mathrm{Hilb}_S(\mathbb{P}^n/S)$ such that for any $p:Z\to \mathrm{Hilb}_P(\mathbb{P}^n/S)$ the pullback to $V\times_{\mathrm{Hilb}_P(\mathbb{P}^n/S)}Z$ is flat if and only if it factors through $H_P$. And now this $H_P$ will represent the functor $Hilb_P(X/S)$.

Now, all the Hilbert schemes happen to be locally closed subschemes of Grassmannians, so they $\mathrm{Hilb}_P$‘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 $hom_S(X,Y)(T)$ is the collection of morphisms $X\times_S T\to Y\times_S T$ which commute with projection down to $T$ (called $T$-morphisms). Now, taking graphs of morphisms does give this as a subfunctor of the Hilbert functor of subschemes of $X\times_S Y$. But not all subfunctors of a scheme are subschemes, so there’s some work left.

Theorem: If $X,Y$ are projective over $S$ and $X$ is flat over $S$, then $hom_S(X,Y)$ is represented by a scheme $\hom_S(X,Y)$.

Proof: We have $f:X\times_S T\to Y\times_S T$, and we send it to its graph $\Gamma:X\times_S T\to X\times_S T\times_T Y\times_S T\cong X\times_S Y\times_S T$ is a closed immersion. So $\Gamma(X\times_S T)\cong X\times_S T$ is flat over $T$. This gives our map into the Hilbert functor.

For simplicity, denote $\mathrm{Hilb}=\mathrm{Hilb}_S(X\times_S Y/S)$ and $U$ the universal family. Then we have a $\mathrm{Hilb}$-morphism $\pi:U\to X\times_S \mathrm{Hilb}$, which is the restriction of the projection $X\times_S Y\times_S \mathrm{Hilb}\to X\times_S \mathrm{Hilb}$.

So now we assume that $\pi_z$ from the fiber in $U$ of a point to the fiber in $X\times_S \mathrm{Hilb}$ of that point is an isomorphism at some $z\in \mathrm{Hilb}$. That is, the fiber in $U$ is a graph. Applying a lemma which says that if $0\in T$ is the spectrum of a local ring and $U/T$ is flat and proper, $V/T$ arbitrary with $p:U\to V$ a $T$-morphism, then $p_0:U_0\to V_0$, the restriction of the morphism to the fibers over 0 is a closed immersion (or isomorphism) then $p$ is as well. So then $\pi$ is an isomorphism over an open neighborhood of $Z$. Thus, we can view the subfunctor of graphs as a subscheme, and so $hom_S(X,Y)$ is representable. QED

Ok, now onto families of divisors, also a rather important object. Take $X$ flat over $S$ and $D\subset X$ an effective Cartier divisor. One definition of this, which is pretty general, is that for each $x\in X$, there’s $f_x\in \mathscr{O}_{X,x}$ which is not a zero divisor and such that $D=\mathrm{Spec}(\mathscr{O}_{X,x}/(f_x)$ in a neighborhood of $x$. We’ll call $D$ a relative Cartier divisor if either of the following (equivalent) things happens:

1. $D$ is flat over $S$
2. For each $x\in X$, $f_x$ is not a zero divisor in $\mathscr{O}_{X,x}\otimes_{\mathscr{O}_S} k(f(x))$, where $f:X\to S$ is the structure map.

So now, for $X/S$ flat, we define a functor $CDiv(X/S)$ which takes $Z$ to the set of relative effective Cartier divisors $V$ of $X\times_S Z$.

We want to see that this is represented by an open subscheme of $\mathrm{Hilb}(X/S)$. To do this, we want to show that if $g:Y=X\times_S Z\to Z$ is flat and $U\subset Y$ is flat over $Z$, then the set of $z\in Z$ such that the fiber in $U$, $U_z$ over $z$ is a Cartier divisor is an open set.

Let $I\subset\mathscr{O}_Y$ be the ideal sheaf of $U$ and let $y\in f^{-1}(z)$. Because $U$ is flat over $Z$, we have that $I_z=I\otimes_{\mathscr{O}_Z} k(z)$. So we should take $f_{z,y}$ to generate $I_z$ locally at $y$. Let $f_y$ be a local section of $I$ at $z$ that, when tensored with 1 over $\mathscr{O}_Z$ gives us $f_{z,y}$. Now, Nakayama tells us that this is a local generator of $I$ at $y$ (note that Nakayama’s big purposes are proving that modules are zero and reducing the problem of finding generators to a simpler situation). So $U$ is relative Cartier at $y$. What this tells us is that there is some open set on which $f_y$ is defined as a generator, so we have our desired result.

Other properties of $\mathrm{CDiv}(X/S)$ are harder to check. For instance, if $X$ is smooth over $S$, then it is universally closed. That is, the inclusion of $\mathrm{CDiv}\to \mathrm{Hilb}$ will take closed sets to closed sets, and any base change will as well. In fact, if we have $X$ geometrically normal over a field (that is, the fiber product $X\times_k \bar{k}$ is normal) then $\mathrm{CDiv}$ 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.

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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3 Responses to Examples of Moduli Spaces

1. Greg Stevenson says:

Good post. You could always consider representability of the relative picard functor by a scheme next. Once you have reprensentability of the functor of effective Cartier divisors you just need to say a little bit about proper flat equivalence relations to at least give a nice sketch.

Keep up the good work anyway!

2. Charles says:

Well, I was thinking of going in a different direction and doing some less technical stuff for a big, probably stuff more closely related to my orals. Was considering some stuff about curves and Jacobians, or maybe just general stuff about curves. Was also contemplating doing some computational geometry stuff, like Groebner bases and resultants. Definitely going to come back to moduli spaces and such (I very much want to talk about the coarse moduli space of curves, for instance) but I think some lower tech stuff is coming up in the immediate future.

3. Jason Starr says:

Another nice post. You could construct the Jacobian of a curve using the Hilbert scheme. The strategy Greg Stevenson suggests can be used, but it turns out to be slightly easier because of things like Riemann-Roch (for d > 2g – 2, Pic^d(C) equals the image of the morphism from C_d to Hilb(C_d) given by the closed subscheme of C_d x C_d parameterizing linearly equivalent pairs of divisors, i.e., the image in C_d x C_d of the natural embedding of the projective bundle P over C_d associated to the vector bundle [D] –> H^0(C,O(D)), . . .)