# The Picard scheme

*Posted on April 24, 2016 by Dima*

## 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*

## 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)

# Grauert's criterion of ampleness

*Posted on October 1, 2015 by Dima*

Here’s a writeup of a proof of Grauert’s criterion for ampleness (here is the link to the original paper). Since often proving that a variety is algebraic is not far from proving that it is projective, this criterion can be useful in judging algebraicity of a variety.

We start with some observations on cohomology of invertible sheaves which are powers of the invertible sheaf associated to hyperplane section.

Recall that if \(D\) is a (Cartier) divisor on a variety \(X\) then it gives rise to a sheaf \[
{\cal O}(D) := \{ f \in k(X) \mid (f) + D \geq 0 \}
\] and a section \(s \in H^0(X, {\cal O}(D))\) such that \((s) = D\). Let \(V = H^0(X, L)\). Then a (generally speaking, partial) map \(\iota: X \dashrightarrow \mathbb{P}(V^\vee)\) is defined: \(x \mapsto (f \mapsto f(x))\). The value on the right, before projectivization, depends on trivialization, hence is only well-defined up to a constant, but since we projectivize, the map is well-defined. A line bundle (or a divisor \(D\)) is called *very ample* if \(\iota\) is a closed embedding, *ample* if some multiple of it is ample.

If \(D\) is ample, then \(H:=mD\) is a hyperplane section for some \(m > 0\). Then \(D_H^r\) is the degree of \(X\), hence positive.

**Theorem**. Let \(X=\mathbb{P}^d\)

- \(H^i(X, {\cal O}(n)) = 0\), if \(0 < i < d\) or \(i < 0\)
- \(H^i(X, {\cal O}(n)) \cong H^{n-i}(X, {\cal O}(-d-n-1))^\vee\)

A paranthesis on ampleness.

**Lemma**. Let \(X\) be a scheme covered by finitely many affine schemes \(X_i\) such that \(X_i\) is the locus of points \(x\) such that \(s_i\) generates \({\operatorname{\mathcal{O}}}_{X,x}\) for an \(s_i \in H^0(X, {\operatorname{\mathcal{L}}})\). Let \(s_{ij} \in H^0(X_i, {\operatorname{\mathcal{L}}})\) be sections such that \(s_{ij}/s_i\) is a base in \(H^0(X_i, {\operatorname{\mathcal{O}}}_X)\) for each \(i\). (Note that \(s_i\) generate \({\operatorname{\mathcal{L}}}\). ) Then the morphism \(X \to {\operatorname{Proj}}[s_i, s_{ij}]\) is an embedding.

**Theorem**. Let \({\operatorname{\mathcal{L}}}\) be a sheaf such that for any finitely generated quasi-coherent \({\operatorname{\mathbb{F}}}\) there exists an \(n_0\) such that \({\operatorname{\mathbb{F}}}\otimes {\operatorname{\mathcal{L}}}^n\) is generated by global sections for \(n \geq n_0\). Then there exists an \(m\) such that \({\operatorname{\mathcal{L}}}^m\) is very ample (defines an embedding into \(P^n\)).

*Proof* ([Liu, 5.1.34, p.169]). Let \(U\) be an affine neighbourhood of \(x\) such that \({\operatorname{\mathcal{L}}}|_U\) is free, and let \({\operatorname{\mathcal{I}}}\) be the sheaf of ideals that cuts out the complement of \(U\). There exists an \(n_0\) such that \({\operatorname{\mathcal{I}}}{\operatorname{\mathcal{L}}}^n\) is globally generated, so there is a section \(s \in H^0(X, {\operatorname{\mathcal{I}}}{\operatorname{\mathcal{L}}}^n) \subset H^0(X, {\operatorname{\mathcal{L}}}^n)\) that doesn’t vanish at \(x\). Since sectios of \({\operatorname{\mathcal{I}}}{\operatorname{\mathcal{L}}}^n\) that vanish on the complement to \(U\), \(X_s\) is contained in \(U\).

Now by compactness \(X\) is covered by finitely many affines of the form \(X_{s_i}\), \(s_i \in H^0(X, {\operatorname{\mathcal{L}}}^n)\) with \(H^0(X_{s_i}, {\operatorname{\mathcal{O}}})\) finitely generated. We can apply the previous lemma to conclude.

**Theorem**. Let \({\operatorname{\mathcal{L}}}\) be a line bundle such that for any sheaf of ideals \({\operatorname{\mathcal{I}}}\) there exists a number \(n\) such that \(H^1(X, {\operatorname{\mathcal{I}}}\otimes {\operatorname{\mathcal{L}}}^n)\) vanishes. Then \({\operatorname{\mathcal{L}}}\) is ample, i.e. \({\operatorname{\mathbb{F}}}\otimes {\operatorname{\mathcal{L}}}^m\) is globally generated for big enough \(m\).

# What I have learned about quantum mechanics

*Posted on September 28, 2015 by Dima*

Being confused for a long time about the main aspects of quantum mechanics and quantisation, I decided now to summarise my current knowledge.

So, dear diary, non-relativistic classical mechanics essentially equals symplectic geometry. To be totally honest, while a symplectic manifold captures all the relevant information about a mechanical system, it only appears as result of a complete understanding of the system. If one wishes to derive the equations of motion from fundamental principles then one proceeds as follows.

Take a manifold (probably a Riemannian manifold) which describes possible positions of elements of your mechanical system. Trajectories are just curves in this manifold. Knowing that your system is in a state \(x_0\) we need to understand how it will evolve in time. So we have a functional called *action* on the space of possible trajectories obtained by integrating something called *Lagrangian density* over the curve. The principle of least action says that the system will evolve according to the path which minimizes the action; the differential equations of motion are called the *Euler-Lagrange equations*. One can prove a theorem that for mechanical systems of a certain kind (? not sure what are the precise requirements here) the trajectories can be described in a particularly nice way as curves parallel with respect to Hamiltonian flow on a symplectic manifold. The translation from Riemannian to symplectic picture is called Legendre transform.