# The Picard scheme

Posted on April 24, 2016 by Dima

tags: moduli, picard

## Functors $$Pic$$ and $$Div$$

We will denote base change with a subscript: $$X_T = X \times T$$.

If $$X$$ is a scheme, the Picard group of $$X$$ is defined to be the group of isomorphism classes of invertible sheaves on $$X$$. The relative Picard functor of an $$S$$-scheme $$X$$ is defined as ${\operatorname{Pic}}_{X/S}(T) := {\operatorname{Pic}}(X_T) / {\operatorname{Pic}}(T)$ where the embedding $${\operatorname{Pic}}(T) \hookrightarrow {\operatorname{Pic}}(X_T)$$ is given by the pullback along the structure maps $$X_T \to T$$.

Caveat: all representability results work with a sheafification of this functor in some topology, Zariski, étale, or fppf. Unless $$X \to S$$ is proper and has a section, they need not be isomorphic.

An effective divisor is a closed subscheme such that its ideal is invertible. If $$f: X \to S$$ is a morphism of schemes then a relative effective divisor on $$X$$ is an effective divisor $$D$$ such that $$D$$ is flat over $$S$$. For a morphism $$X \to S$$ define the functor of relative divisors ${\operatorname{Div}}_{X/S}(T) := {\{\ \textrm{ relative effective divisors on } X_T/T \ \}}$

We are interested in representability of this functor, so to this end we prove a little lemma.

Lemma. Let $$X \to S$$ be a flat morphism. Let $$D$$ be a closed subscheme of $$X$$ flat over $$S$$. Then $$D$$ is relative effective divisor in a neighbourhood of $$x \in X$$ if and only if $$D_s$$ is cut out in a neighbourhood of $$x$$ in $$X_s$$, where $$s$$ is the image of $$x$$, by a single non-zero element of $${\operatorname{\mathcal{O}}}_{X_s,x}$$ (which amounts to being an effective divisor in $$X_s$$ but we won’t prove it).

# Construction of Hilb and Quot

Posted on February 16, 2016 by Dima

tags: moduli, hilb, quot

## 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. Read more (comments)