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 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, 'etale, or fppf. Unless \(X \to S\) is proper and has a section, they need not be isomorphic.

An effective divisor is a closed subscheme of codimension 1. If \(f: X \to S\) be a morphism of schemes then a relative effective divisor 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 \(X\). Then \(D\) is codimension 1 in a neighbourhood of \(x \in X\) if and only if \(D_s\) is codimension 1 in the fibre \(X_s\) where \(s\) is the image of \(x\).

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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)

Grauert's criterion of ampleness

Posted on October 1, 2015 by Dima

tags: complex geometry, projectivity

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 \(X=\mathbb{P}^d\)

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What I have learned about quantum mechanics

Posted on September 28, 2015 by Dima

tags: physics

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.

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