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…)

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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.

(more…)