## Constructing the Hilbert Scheme II

Last time we did a quick run through of how to put together the Hilbert Scheme. A few questions came up in the comments, the first being: how can we guarantee the existence of the $N$ we used, which works uniformly for ideals with a given Hilbert Polynomial? The other is in proving that the $G_P$ defined exists, and it’s called the flattening stratification. Also, to answer Todd, we do need $X$ to be flat over $S$ for the $\hom$ schemes to all work out, I missed that. So now, we’ll go about proving the lemmas needed for the construction.

We’re going to follow Mumford, as Kollár recommends. We’ll start with the existence of $N$, and see how far we get. First we define a coherent sheaf $\mathscr{F}$ on $\mathbb{P}^n$ to be $m$-regular if $H^i(\mathbb{P}^n,\mathscr{F}(m-i))=0$ for all $i>0$. This definition might seem funny at first glance, but it’s just what we need for a vanishing theorem of Castelnuovo’s.

First, though, an aside of vanishing theorems. Now, we call something a vanishing theorem whenever we get a result indicating that all the nonzero cohomology of a sheaf is zero under some circumstances. One reason we like these is that there are a lot of nice theorems that compute the Euler characteristic, which is $\chi(\mathscr{F})=\sum_{i=0}^\infty (-1)^i \dim H^i(X,\mathscr{F})$, and if everything higher vanishes, this reduces to just $\dim H^0(X,\mathscr{F})$, which we denote by $h^0(X,\mathscr{F})$. Now, way back on the first Hilbert Polynomial post, John was asking about why the Hilbert polynomial was evaluated at zero, when it only depended on data far away from zero. The cause is a vanishing theorem. The classical Hilbert function is just $h^0(X,\mathscr{F}(n))$. However, the Hilbert Polynomial happens to be equal to $\chi(\mathscr{F}(n))$. They’re equal once all the higher cohomology dies, but not until then. So then the Hilbert polynomial at zero is just $\chi(\mathscr{F})$, whose significance is a bit clearer.

Anyway, back to $m$-regularity. The Castelnuovo theorem is as follows:

Theorem: Let $\mathscr{F}$ be an $m$-regular coherent sheaf on $\mathbb{P}^n$. Then

1. We have $H^0(\mathbb{P}^n,\mathscr{F}(k))$ spanned by $H^0(\mathbb{P}^n,\mathscr{F}(k-1))\otimes H^0(\mathbb{P}^n,\mathscr{O}(1))$ for $k>m$.
2. $H^i(\mathbb{P}^n,\mathscr{F}(k))=0$ for $i>0$ and $k+1\geq m$.

This in fact tells us that if $k\geq m$, then $\mathscr{F}(m)$ is generated by its global sections.

Proof: We will proceed by induction. If $n=0$, then $\mathbb{P}^n$ is a point. Then things go kind of easily, because coherent sheaves over a point are finite dimensional vectors spaces over a field.

Next the induction. Given $\mathscr{F}$ take a generic hyperplane $H$, then we tensor the exact sequence $0\to \mathscr{O}_{\mathbb{P}^n}(-H)\to \mathscr{O}_{\mathbb{P}^n}\to \mathscr{O}_H\to 0$ with $\mathscr{F}(k)$. For any point in projective space, multiplication by the local equation for $H$ is injective, because for a generic hyperplane, the local equation will be a unit at all the associated primes of $\mathscr{F}_x$. What that means is that it will be a unit in the local ring at the points represented by those primes. Note that an associated prime to a module is just a prime ideal which is the set of elements multiplying some specified element of the module to zero.

So the injectivity of this multiplication map gives us the short exact sequence $0\to \mathscr{F}(k-1)\to \mathscr{F}(k)\to (\mathscr{F}\otimes\mathscr{O}_H)(k)\to 0$, and we’ll denote the last term by $\mathscr{F}_H(k)$. This gives us a long exact sequence on cohomology, a piece of which is $H^i(\mathscr{F}(m-i))\to H^i(\mathscr{F}_H(m-i))\to H^{i+1}(\mathscr{F}(m-i-1))$. Now, as $\mathscr{F}$ is $m$-regular, the sheaf $\mathscr{F}_H$ on $H$ will be too, and since $H\cong \mathbb{P}^{n-1}$, we use induction to claim the result for $\mathscr{F}_H$.

So we now look at the part $H^{i+1}(\mathscr{F}(m-i-1))\to H^{i+1}(\mathscr{F}(m-i))\to H^{i+1}(\mathscr{F}_H(m-i))$ of the exact sequence, and by the second part of the theorem for $\mathscr{F}_H$, the last group is zero, and by $m$-regularity for $\mathscr{F}$, the first is. By exactness, the middle group is zero, adn so $\mathscr{F}$ is $(m+1)$-regular. Continuing in this way, we get part 2 for $\mathscr{F}$. So now on to part 1.

We need to look at the commutative diagram

$\begin{array}{ccc}H^0(\mathscr{F}(k-1))\otimes H^0(\mathscr{O}_{\mathbb{P}^n}(1))\stackrel{\sigma}{\to}&H^0(\mathscr{F}_H(k-1))\\\searrow\mu&\downarrow\tau\\H^0(\mathscr{F}(k-1))\to H^0(\mathscr{F}(k)\stackrel{\nu}{\to}&H^0(\mathscr{F}_H(k))\end{array}$

(I apologize for the diagram, it was the best I could do…don’t really know yet a better way to do commutative diagrams for wordpress)

Now, $\sigma$ is surjective if $k>m$, because $H^1(\mathscr{F}(k-2))=0$ (because then the first term in the tensor product map has vanishing cokernel, so it’s surjective, and the second part is by definition.) Even better, $\tau$ is surjective when $k>m$, by part 1 for $\mathscr{F}_H$. Thus, the image in $\nu$ of the image of $\mu$ is all of $H^0(\mathscr{F}_H(k))$, by the diagrams commutativity. So then $H^0(\mathscr{F}(k))$ is spammed by the image of $\mu$ and by $H^0(\mathscr{F}(k-1))$.

Now we take $h\in H^0(\mathbb{P}^n,\mathscr{O}_{\mathbb{P}^n}(1))$ to be the global equation for $H$. Then the image $H^0(\mathscr{F}(k-1))\to H^0(\mathscr{F}(k))$ is $h\otimes H^0(\mathscr{F}(k-1))$. So this is in the image of $\mu$ already! Thus, $\mu$ is surjective and we have part 1 for $\mathscr{F}$, and the theorem is proved. QED

Now, why does this say generated by global sections? Well, there’s a theorem of Serre’s which characterizes ample line bundles and tells us that $\mathscr{F}(k)$ will be generated by global sections for large enough $k$, so we just need to show that $m$ works.

So then with part 1, we have that $H^0(\mathscr{F}(m))\otimes H^0(\mathscr{O}_{\mathbb{P}^n}(k-m))$ generates $\mathscr{F}(ll)$. But for each point of projective space, we can choose an isomorphism at that point of $\mathscr{O}_{\mathbb{P}^n}(1)$ and $\mathbb{O}_{\mathbb{P}^n}$, because both are line bundles. This gives isomorphisms at each point (note, they don’t glue together to a real isomorphism!) of $\mathscr{O}_{\mathbb{P}^n}(k-m)$ and $\mathscr{O}_{\mathbb{P}^n}$, and $\mathscr{F}(k)$ with $\mathscr{F}(m)$. Then $H^0(\mathscr{O}_{\mathbb{P}^n}(k-m))$ can be treated as just the vecotr space of elements of the local ring $\mathscr{O}_{\mathbb{P}^n,x}$, and being generated by global sections reduces to $H^0(\mathscr{F}(m))\otimes\mathscr{O}_{\mathbb{P}^n,x}$ generates the stalk $\mathscr{F}(m)_x$.

So anyway, we’ve said some stuff about $m$-regular sheaves, but what about coherent sheaves of ideals? Well, here’s another theorem:

Theorem: For all $n$, there is a polynomial $F_n(x_0,\ldots,x_n)$ such that for all coherent sheaves of ideals $\mathscr{I}$ on $\mathbb{P}^n$, if $a_0,\ldots,a_n$ are defined by $\chi(\mathscr{I}(m))=\sum_{i=0}^n a_i\binom{m}{i}$ (that is, they’re pulled out of the Hilbert polynomial of the sheaf) then $\mathscr{I}$ is $F_n(a_0,\ldots,a_n)$-regular.

So, the $a_i$ depend only on the Hilbert Polynomial, and the $n$ depends only on which projective space we’re working in. This combines with the first theorem to tell us that

Theorem: For every polynomial $P$, there is an integer $N(P)$ with the following property: if
$\mathbb{P}$ is a projective space over a field $k$ and $\mathscr{I}\subset\mathscr{O}_{\mathbb{P}}$ is a subsheaf with Hilbert polynomial $P$, then for all $n\geq N(P)$, we have

1. $h^i(\mathbb{P},\mathscr{I}(n))=0$ for all $i\geq 1$
2. $\mathscr{I}(n)$ is generated by global sections.
3. $H^0(\mathbb{P},\mathscr{I}(n))\otimes H^0(\mathbb{P},\mathscr{O}(1))\to H^0(\mathbb{P},\mathscr{I}(n+1))$ is surjective.

The proof of the second theorem is more technical than that of the first, but the statement tells us that what’s really necessary here is a nice form of $m$-regularity. This post has gone on a bit, so we’ll stop here, and pick up tomorrow with discussing flattening stratifications, and then we’ll complete the proof of the existence of the Hilbert scheme.