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).

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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)

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