Construction of Hilb and Quot

Posted on February 16, 2016 by Dima

tags: moduli

Hilbert polynomial

If $$X$$ is a projective variety and $${\mathcal{F}}$$ is a sheaf on $$X$$ then the function $\chi(\mathcal F, n) = \sum (-1)^i H^i(X, \mathcal F(n))$ is called the Hilbert function of $$\mathcal F$$.

Lemma (Snapper lemma). It is a polynomial.

Proof. Induction on the dimension of support of $${\mathcal{F}}$$. Notice that $$\chi$$ is additive in extensions, i.e. $$\chi({\mathcal{F}}/{\mathcal{G}}) = \chi({\mathcal{F}}) - \chi({\mathcal{G}})$$, as follows from long exact sequence of cohomology.

Recall that for a divisor $$D$$ the sheaf $${\mathcal{O}}(D)$$ is defined as ${\mathcal{O}}(D)(U) := {\{\ f \in \mathcal{K}(U) \ \mid\ (f) + D \geq 0 \ \}}$ where $$\mathcal{K}$$ is the constant sheaf of rational functions on $$X$$. Let $$H$$ be a hyperplane in $${\mathbb{P}}^n$$, one has a natural short exact sequence $0 \to {\mathcal{O}}_{{\mathbb{P}}^n}(-H) \to {\mathcal{O}}_{{\mathbb{P}}^n} \to {\mathcal{O}}_H \to 0$ Assume that $$H$$ is chosen so that $$\dim {\operatorname{supp}}{\mathcal{F}}\cap H < \dim {\operatorname{supp}}{\mathcal{F}}$$. Tensoring with $${\mathcal{F}}$$, taking into account that $${\mathcal{O}}_{{\mathbb{P}}^n}(-H) \cong {\mathcal{O}}_{{\mathbb{P}}^n}(-1)$$, and passing to the long exact sequence we get $0 \to Tor_1({\mathcal{F}}, {\mathcal{O}}_H) \to {\mathcal{F}}(-1) \to {\mathcal{F}}\to {\mathcal{F}}\otimes {\mathcal{O}}_H \to 0$ Note that by looking at long exact cohomolgy sequences one establishes that Hilbert polynomial is an additive invariant. By choice of $$H$$ and by induction hypothesis, since $$\dim {\operatorname{supp}}{\mathcal{F}}\otimes {\mathcal{O}}_H < \dim {\operatorname{supp}}{\mathcal{F}}$$, $$\chi({\mathcal{F}}\otimes {\mathcal{O}}_H)$$ is a polynomial. So is the Hilbert polynomial of the first term. By additivity of $$\chi$$, we conclude.

Moreover, by Serre vanishing there is an $$n_0$$ such that higher cohomologies of $$\mathcal F(n_0)$$ vanish, so for $$n \geq n_0$$, $$\chi({\mathcal{F}}, n) = \dim H^0(X, {\mathcal{F}}(n))$$.

Hilb and Quot, and Grassmanian

Since Grothendieck taught us so, we work in the relative setting, that is, we fix a scheme $$S$$ and work in the category of schemes endowed with a morphism to $$S$$ — the $$S$$-schemes.

Let $$X$$ be an $$S$$-scheme and $$E$$ be a coherent sheaf on $$X$$. For an $$S$$-scheme $$T$$ denote $$X_T$$ the fibre product $$X \times_S T$$. Define the functor that classifies quotients of a given coherent sheaf $$E$$ over a given $$S$$-scheme $$X$$ ${\operatorname{Quot}}_{E/X/S}(T) := {\{\ \textrm{ morphisms } E_T \to {\mathcal{F}}\textrm{ on } X_T\ \mid\ {\mathcal{F}}\textrm{ is flat over } T\ \}}$

The Hilbert scheme of $$X/S$$ is the scheme that classifies closed subschemes of $$X$$, i.e. a schem that represents the functor ${\operatorname{Hilb}}_{X/S}(T) := {\{\ \textrm{ closed subschemes } Z \textrm{ of } X_T\ \mid\ Z \textrm{ is flat over } T\ \}}$ which amounts to classifying quotient sheaves of $${\mathcal{O}}_X$$: $${\operatorname{Hilb}}_{X/S}(T) = {\operatorname{Quot}}_{{\mathcal{O}}_X/X/S}$$.

If $$X$$ is projective, both functors are naturally decomposed into disjoint union of components according to the Hilbert polynomial. The component with Hilbert polynomail $$\Phi$$ with respect to an ample line bundle $$L$$ is denoted $${\operatorname{Quot}}_{E/X/S}^{\Phi,L}$$.

The Grassmanian functior is defined as ${\operatorname{Grass}}(n,r) := {\operatorname{Quot}}^{r,{\mathcal{O}}_S}_{{\mathcal{O}}_S^{\oplus n}/S/S}$

$$m$$-regularity

First, the definition: a coherent sheaf on a projective space $${\mathbb{P}}^n$$ is $$m$$-regular if $H^i({\mathbb{P}}^n, {\mathcal{F}}(m-i)) = 0, \textrm{ for all } i > 0$

Given a sheaf of $${\mathcal{O}}_X$$-modules $${\mathcal{F}}$$ the set of associated primes is defined to be $\mathrm{Ass}({\mathcal{F}}) := {\{\ x \in X\ \mid\ \exists s \in {\mathcal{F}}_X \,\mathrm{supp} s = \{ x \} \ \}}$

Lemma. If $$H$$ is sufficiently generic, i.e. $$\mathrm{Ass}({\mathcal{F}}) \cap H = \emptyset$$ and $${\mathcal{F}}$$ is $$m$$-regular then $${\mathcal{F}}\otimes {\mathcal{O}}_H$$ is also $$m$$-regular.

Proof. Tensor the long exact sequence associated to the divisor $$H$$ with $${\mathcal{F}}$$ $0 \to {\mathcal{F}}(-1) \to {\mathcal{F}}\to {\mathcal{O}}_H \otimes {\mathcal{F}}\to 0$ The exactness is preserved, becase the condition on associated primes says precisely that the map $$a \otimes s \mapsto a \cdot s, a \in H^0(U, {\mathcal{O}}_{P^n}), s \in H^0(U, {\mathcal{F}})$$ is injective. Further we can tensor with $${\mathcal{O}}(n)$$, which is a flat sheaf, and get $0 \to {\mathcal{F}}(n-1) \to {\mathcal{F}}(n) \to {\mathcal{O}}_H \otimes {\mathcal{F}}(n) \to 0$ The associated long exact sequence is $\ldots \to H^i({\mathbb{P}}^n, {\mathcal{F}}(m-i)) \to H^i({\mathbb{P}}^n, {\mathcal{F}}(m-i) \otimes {\mathcal{O}}_H) \to\\ \to H^{i+1}({\mathbb{P}}^n, {\mathcal{F}}(m-i-1)) \to \ldots$ so if $${\mathcal{F}}$$ is $$m$$-regular, the first and third term vanish, which implies that the second term vanishes, and $${\mathcal{F}}\otimes {\mathcal{O}}_H$$ is $$m$$-regular. $$\square$$

Here are some basic properties of $$m$$-regularity.

Proposition. Let $${\mathcal{F}}$$ be an $$m$$-regular sheaf on $${\mathbb{P}}^n$$. Then

• $${\mathcal{F}}$$ is $$r$$-regular for $$r \geq m$$
• the map $$H^0({\mathbb{P}}^n, {\mathcal{O}}(1)) \otimes H^0({\mathbb{P}}^n, {\mathcal{F}}(r)) \to H^0({\mathbb{P}}^n, {\mathcal{F}}(r+1))$$ is surjective for $$r \geq m$$
• $${\mathcal{F}}$$ is generated by global sections, i.e. the map $$H^0({\mathbb{P}}^n, {\mathcal{F}}) \otimes {\mathcal{O}}\to {\mathcal{F}}$$ is surjective

The proof is by induction on the dimension $$n$$ of the projective space.

Embedding Quot into a Grassmanian

Let $$X$$ be a scheme, and let $${\mathcal{F}}$$ be a (coherent) sheaf on $$X$$. If $$H^0(X, {\mathcal{F}}) \otimes O_X \to {\mathcal{F}}$$ is a surjective morphism then one says that $${\mathcal{F}}$$ is generated by global sections.

Crucial observation. All subscheaves of $${\mathcal{F}}$$ that are generated by global sections are in bijective correspondence with subsheaves of $$H^0(X, {\mathcal{F}}) \otimes O_X$$.

The following theorem, due to Mumford, shows how $$m$$-regularity can be applied to representation of Quot functor. Roughly speaking, its idea is that one can bound below the degree of regularity of a sheaf only in terms of its Hilber polynomial.

Theorem. Let $${\mathcal{F}}$$ be a sheaf which is a quotient sheaf of $${\mathcal{O}}_{{\mathbb{P}}^n}^p$$, with Hilbert polynomial of the form $\chi({\mathcal{F}}(r)) = \sum a_i {r \choose i}$

Then there exists a polynomial $$F_{n,p}$$, which does not depend on $${\mathcal{F}}$$, such that any such $${\mathcal{F}}$$ is $$F_{n,p}(a_0, \ldots, a_n)$$-regular.

The general problem of representability of $$Quot$$ can be redused to representability of $$Quot_{{\mathcal{O}}^n / {\mathbb{P}}^n /S}$$. Let us look into the case where $$S={\operatorname{Spec}}k$$, for simplicity, and we want to represent $${\operatorname{Quot}}^{\Phi}$$ for a particular polynomial $$\Phi$$. Let $$m=F_{n,p}(a_0, \ldots, a_n)$$. Any particular family over $$T$$ of quotient sheaves with Hilbert polynomial $$\xi$$ is then given by a morphism $$T \to {\operatorname{Gr}}(H^0({\mathbb{P}}^n, {\mathcal{O}}^p(r), \Phi(r))$$.

Now every quotient sheaf with the Hilbert polynomial $$\Phi$$ gives a point of the Grassmanian. Consider the family of all such sheaves fibred over all of the Grassmanian. This will not be a universal family because it is not flat. Using flattening stratification we obtain a locally closed subscheme of $${\operatorname{Gr}}(H^0({\mathbb{P}}^n, {\mathcal{O}}^p(r), \Phi(r))$$ which is the base for the universal family.

(References: 1 2)