Let ${X}$ be an abelian variety over an algebraically closed field ${k}$. If ${k = \mathbb{C}}$, then ${X}$ corresponds to a complex torus: that is, ${X}$ can be expressed complex analytically as ${V/\Lambda}$ where ${V}$ is a complex vector space of dimension ${\dim X}$ and ${\Lambda \subset V}$ is a lattice (i.e., a ${\mathbb{Z}}$-free, discrete submodule of rank ${2g}$). In this case, one can form the dual abelian variety

$\displaystyle X^{\vee} = \hom(X, S^1) = \hom_{\mathrm{cont}}(V/\Lambda, \mathbb{R}/\mathbb{Z}) \simeq \hom_{\mathbb{R}}(V, \mathbb{R})/2\pi i \hom(\Lambda, \mathbb{Z}).$

At least, ${X^{\vee}}$ as defined is a complex torus, but it turns out to admit the structure of an abelian variety.

The purpose of the next few posts is to describe an algebraic version of this duality: it turns out that ${X^{\vee}}$ can be constructed as a scheme, purely algebraically. I’d like to start with a couple of posts on Picard schemes. A useful reference here is this article of Kleiman.

1. The Picard scheme analytically

Let ${X }$ be a smooth projective variety over the complex numbers ${\mathbb{C}}$. The collection of line bundles ${\mathrm{Pic}(X)}$ is a very interesting invariant of ${X}$. Usually, it splits into two pieces: the “topological” piece and the “analytic” piece. For instance, there is a first Chern class map

$\displaystyle c_1: \mathrm{Pic}(X) \rightarrow H^2(X; \mathbb{Z}) ,$

which picks out the topological type of a line bundle. (Topologically, line bundles on a space are classified by their first Chern class.) The admissible topological types are precisely the classes in ${H^2(X; \mathbb{Z})}$ which project to ${(1,1)}$-classes in ${H^2(X; \mathbb{C})}$ under the Hodge decomposition. (more…)

Let ${M_{1, 1}}$ be the moduli stack of elliptic curves. Given a scheme ${S}$, maps ${S \rightarrow M_{1, 1}}$ are given by the groupoid of elliptic curves over ${S}$, together with isomorphisms between them. The goal of this post is to compute ${\mathrm{Pic}(M_{1, 1})}$ away from the primes ${2, 3}$. (This is done in Mumford’s paper “Picard groups of moduli problems.”)

In the previous post, we saw that ${M_{1, 1}}$ could be described as a quotient stack. Namely, consider the scheme ${B_1 = \mathrm{Spec} \mathbb{Z}[a_1, a_2, a_3, a_4, a_6]}$ and the Weierstrass equation

$\displaystyle Y^2 Z + a_1 XYZ + a_3 YZ^2 = X^3 + a_2 X^2 Z + a_4 XZ^2 + a_6 Z^3$

cutting out a subscheme ${E_1 \subset \mathbb{P}^2_{B_1}}$. This is a flat family of projective cubic curves over ${\mathbb{P}^2_{B_1}}$ with a section (the point at infinity given by ${[X: Y: Z] = [0 : 1 : 0]}$). There is an open subscheme ${B \subset B_1}$ over which the family ${E_1 \rightarrow B_1}$ is smooth, i.e., consists of elliptic curves. A little effort with cohomology and Riemann-Roch allows us to show that, Zariski locally, any elliptic curve ${X \rightarrow S}$ can be pulled back from one of these: that is, any elliptic curve locally admits a Weierstrass equation.

The Weierstrass equation was not unique, though; any change of parametrization (in affine coordinates here)

$\displaystyle x' = a^2 x + b, \quad y' = a^3 x + c + d, \ a \mathrm{\ invertible}$

preserves the form of the equation, and these are the only transformations preserving it. In other words, the map

$\displaystyle B \rightarrow M_{1, 1}$

exhibits ${B}$ as a torsor over ${M_{1,1}}$ for the group scheme ${\mathbb{G} = \mathrm{Spec} \mathbb{Z}[a^{\pm 1}, b, c, d]}$ with a multiplication law given by composing linear transformations. That is,

$\displaystyle M_{1, 1} \simeq B/\mathbb{G};$

that is, to give a map ${S \rightarrow M_{1, 1}}$, one has to choose an étale cover ${\left\{S_\alpha\right\}}$ of ${S}$ (Zariski is enough here), maps ${S_\alpha \rightarrow B}$ inducing elliptic curves over the ${S_\alpha}$, and isomorphisms (coming from maps to ${\mathbb{G}}$) over ${S_\alpha \times_S S_\beta}$. (more…)

Let ${S}$ be a scheme. An elliptic curve over ${S}$ should be thought of as a continuously varying family of elliptic curves parametrized by ${S}$.

Definition 1 An elliptic curve over ${S}$ is a proper, flat morphism ${p: X \rightarrow S}$ whose geometric fibers are curves of genus one together with a section ${0: S \rightarrow X}$.

This is a reasonable notion of “family”: observe that a morphism ${T \rightarrow S}$ can be used to pull back elliptic curves over ${S}$. The flatness condition can be thought of as “continuity.” For an algebraically closed field, this reduces to the usual notion of an elliptic curve.

A basic property of elliptic curves over algebraically closed fields is that they imbed into ${\mathbb{P}^2}$ and are cut out by (nonsingular) Weierstrass equations of the form

$\displaystyle Y^2 Z + a_1 XYZ + a_3 YZ^2 = X^3 + a_2 X^2 Z + a_4 XZ^2 + a_6 Z^3.$

This equation is unique up to an action of a certain four-dimensional group of transformations. The first goal is to show that, locally, the same is true for an elliptic curve over a base.  (more…)

I’ve been trying to learn a bit about stacks lately. This is actually going to tie in quite nicely into the homotopy-theoretic story I was talking about earlier: the moduli stack of formal groups turns out to be a close approximation to stable homotopy theory. But before that, there are other more fundamental examples of stacks.

Here is a toy example of an Artin stack I would like to understand. Let ${X}$ be a noetherian scheme; we’d like a stack ${\mathrm{Coh}(X)}$ which parametrizes coherent sheaves on ${X}$. This is definitely a “stack” rather than a scheme, because coherent sheaves are likely to have plenty of automorphisms.

Let’s be a little more precise. Given a test scheme (say, affine) ${T}$, a map

$\displaystyle T \rightarrow \mathrm{Coh}(X)$

should be a “family of coherent sheaves on ${X}$” parametrized by ${T}$. One way of saying this is that we just have a quasi-coherent sheaf ${\mathcal{F}}$ of finite presentation on ${X \times_{\mathbb{Z}} T}$. In order to impose a “continuity” condition on this family, we require that ${\mathcal{F}}$ be flat over ${T}$.

To make this more precise, we define it relative to a base:

Definition 1 Let ${S}$ be a base noetherian scheme, and let ${X \rightarrow S}$ be an ${S}$-scheme of finite type. We define a functor

$\displaystyle \mathrm{Coh}_{X/S}: \mathrm{Aff}_{/S}^{op} \rightarrow \mathrm{Grpd}$

sending an affine ${T \rightarrow S}$ to the groupoid of finitely presented quasi-coherent sheaves on ${T \times_S X}$, flat over ${T}$. The pull-back morphisms are given by pull-backs of sheaves.

In this post, we will see that this is an Artin stack under certain conditions.