The Picard scheme

Posted on April 24, 2016 by Dima

tags: moduli, Abelian varieties

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

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

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

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 \(m\).

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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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