Home > algebraic geometry > The Picard scheme of a projective curve

## The Picard scheme of a projective curve

Today we study the objects appearing on the opposite side of the class field theory isomorphism from the fundamental group, namely line bundles on our curve. Ultimately we are interested in the Picard group, which is the abstract abelian group of isomorphism classes of line bundles, but in order to actually prove the main theorem we must reformulate it in a still more geometrical way. To this end, we view the Picard group as a “moduli problem,” meaning we allow line bundles on the curve to vary in algebraic families and try to find some space which parameterizes them. It turns out that in our situation, general theorems of Grothendieck guarantee that this space is actually a scheme, and a pretty reasonable one at that. Be warned that this is a special and pleasant state of affairs, as many moduli problems of interest are not represented by a scheme: in particular, this is not the case for vector bundles of higher rank.

This post will be more technical than usual, so readers who are mostly interested in applications to class field theory should feel free to skim over the details. For background material on Grothendieck topologies and descent, see Vistoli’s article in FGA Explained or Chapter 6 of Néron Models by Bosch, Lütkebohmert, and Raynaud. General theorems about the Picard scheme can be found in Kleiman’s article in the former book or Chapters 8 and 9 in the latter.

Let $X$ be a scheme over a field $k$. We can define a contravariant functor from the category $\text{Sch}_k$ of schemes over $k$ to the category $\text{Ab}$ of abelian groups by sending a $k$-scheme $S$ to the Picard group $\text{Pic}(X \times S)$. Here we can think of a line bundle on $X \times S$ as a family of line bundles on $X$ parameterized by $S$. This is the first functor which we might naïvely hope is representable: however, this is never the case if $X$ is nonempty. To see why, take $S = \mathbb{P}^1$, so $\mathbb{P}^1_X = X \times \mathbb{P}^1$ carries the nontrivial line bundle $\mathcal{O}(1)$.  But if we pull back along the standard Zariski covering $\mathbb{A}^1 \coprod \mathbb{A}^1 \to \mathbb{P}^1$, the bundle $\mathcal{O}(1)$ trivializes, so that in particular the induced map $\text{Pic}(\mathbb{P}^1_X) \to \text{Pic}(\mathbb{A}^1_X \coprod \mathbb{A}^1_X)$ is not injective. Since a representable functor is necessarily a sheaf with respect to the Zariski (Grothendieck) topology on $\text{Sch}_k$, it follows that $S \mapsto \text{Pic}(X \times S)$ is not representable.

This leads us to the following definition, in which we simply declare the bundles which arise by pullback along $X \times S \to S$ to be trivial.

Definition The Picard functor $\text{Pic}_X : \text{Sch}_k^{\text{op}} \to \text{Ab}$ is defined by the formula

$\text{Pic}_X(S) = \text{coker}(\text{Pic}(S) \to \text{Pic}(X \times S))$.

If $\text{Pic}_X$ is representable by a scheme, we call it the Picard scheme of $X$.

In particular we have $\text{Pic}_X(k) = \text{Pic}(X)$, so this definition is consistent with the notation from two posts ago (here and in the sequel, when $A$ is a $k$-algebra and $F$ is a functor on $k$-schemes we commit the standard abuse of notation $F(A) = F(\text{Spec } A)$). Note that if the Picard scheme exists, it is a commutative group scheme by construction.

Ours is not the most refined version of the Picard functor, since there are situations where $\text{Pic}_X$ is not representable but its fppf-sheafification is. The next theorem shows that $\text{Pic}_X$ suffices for our purposes, so we will not linger on this technical point.

Theorem If $X$ is proper, geometrically integral, and admits a degree 1 zero-cycle (i.e. a finite collection of closed points whose degrees generate the unit ideal in $\mathbb{Z}$), then $\text{Pic}_X$ is represented by a scheme which is locally of finite type over $k$.

Proof. Theorem 3 in Section 8.2 of Néron Models says that the fppf-sheafification $\widetilde{\text{Pic}}_X$ of $\text{Pic}_X$ is representable. Our statement can be deduced from this and Proposition 4 in Section 8.1 as follows. The hypothesis that $\pi_*\mathcal{O}_{X \times S} = \mathcal{O}_S$ for any $k$-scheme $S$ is satisfied because $X$ is proper and geometrically integral. Thus the proposition yields an exact sequence

$0 \to \text{Pic}(S) \to \text{Pic}(X \times S) \to \widetilde{\text{Pic}}_X(S) \to \text{Br}(S) \to \text{Br}(X \times S)$,

where $\text{Br}$ is the Brauer group, i.e. the étale cohomology group $\text{Br}(S) = H^2(S,\mathbb{G}_m)$. We must prove that $\text{Br}(S) \to \text{Br}(X \times S)$ is injective: in fact, we will produce a retraction of this map. Let $x_1,\cdots,x_r \in X$ be closed points of degrees $d_1,\cdots,d_r$ respectively, and choose $m_1,\cdots,m_r \in \mathbb{Z}$ such that $\sum_i m_id_i = 1$. For each $1 \leq i \leq r$, the morphism $\text{Spec } k(x_i) \to X$ induces another $\text{Spec } k(x_i) \times S \to X \times S$, and we denote by $\varphi_i : \text{Br}(X \times S) \to \text{Br}(S)$ the composition of pullback along this map with the corestriction $\text{Br}(\text{Spec } k(x_i) \times S) \to \text{Br}(S)$. A standard property of restriction and corestriction maps says that the composition

$\text{Br}(S) \to \text{Br}(X \times S) \stackrel{\varphi_i}{\to} \text{Br}(S)$

is multiplication by $d_i$, so $\sum_i m_i\varphi_i$ is the desired retraction (thanks to Olivier Benoist and Jason Starr for explaining this to me over at MO).

$\Box$

We will prove soon that a projective and geometrically integral curve over a finite field admits a degree 1 zero-cycle. In fact, this is true for an arbitrary geometrically integral variety over a finite field: the general statement can be deduced from the Lang-Weil estimates, but our proof for projective curves will be less advanced and hopefully more understandable.

Finally, let us give some more refined information about the Picard scheme when $X$ is a projective and geometrically integral curve. Then there is a natural degree map $\text{Pic}_X \to \mathbb{Z}_k$, where $\mathbb{Z}_k$ is the constant group scheme over $k$ associated with the abstract group $\mathbb{Z}$. We write $\text{Pic}^d_X$ for the fiber over $d$, which parameterizes line bundles of degree $d$.

Proposition The kernel $\text{Pic}^0_X$ is a smooth, connected, commutative algebraic group over $k$, called the generalized Jacobian of $X$. If we assume also that $k$ is perfect, then the normalization $\widetilde{X} \to X$ induces an epimorphism $\text{Pic}^0_X \to \text{Pic}^0_{\widetilde{X}}$ whose kernel is a smooth connected affine algebraic group, and $\text{Pic}^0_{\widetilde{X}}$ is an abelian variety of dimension equal to the genus of $\widetilde{X}$.

This is essentially Corollary 11 in Section 9.2 of Néron Models, except for the assertion about dimension, which is Theorem 1(b) in Section 8.4. There the interested reader can also find more details regarding the linear algebraic group $\text{ker}(\text{Pic}^0_X \to \text{Pic}^0_{\widetilde{X}})$, which encodes information about the singularities of $X$. For instance, this group is a split extension of a unipotent group by a torus: the former comes from cusps and the latter from nodes (we remind the reader that here $k$ is perfect).

Even if we are only interested a priori in smooth curves, these generalized Jacobians of singular curves enter naturally when considering ramified covers. However, we will focus on the unramified case for a while, and there we really only need to consider smooth curves.

Since this post has already gotten out of hand, we’ll put off the discussion of idèles until next time. While not logically necessary, this will establish the connection with the more classical arithmetic formulation of class field theory.