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)

Site generated by Hakyll