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 \input{preamble} % OK, start here. % \begin{document} \title{Picard Schemes of Curves} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we do just enough work to construct the Picard scheme of a projective nonsingular curve over an algebraically closed field. See \cite{Kleiman-Picard} for a more thorough discussion as well as historical background. \medskip\noindent Later in the Stacks project we will discuss Hilbert and Quot functors in much greater generality. \section{Hilbert scheme of points} \label{section-hilbert-scheme-points} \noindent Let $X \to S$ be a morphism of schemes. Let $d \geq 0$ be an integer. For a scheme $T$ over $S$ we let $$\text{Hilb}^d_{X/S}(T) = \left\{ \begin{matrix} Z \subset X_T\text{ closed subscheme such that }\\ Z \to T\text{ is finite locally free of degree }d \end{matrix} \right\}$$ If $T' \to T$ is a morphism of schemes over $S$ and if $Z \in \text{Hilb}^d_{X/S}(T)$, then the base change $Z_{T'} \subset X_{T'}$ is an element of $\text{Hilb}^d_{X/S}(T')$. In this way we obtain a functor $$\text{Hilb}^d_{X/S} : (\Sch/S)^{opp} \longrightarrow \textit{Sets},\quad T \longrightarrow \text{Hilb}^d_{X/S}(T)$$ In general $\text{Hilb}^d_{X/S}$ is an algebraic space (insert future reference here). In this section we will show that $\text{Hilb}^d_{X/S}$ is representable by a scheme if any finite number of points in a fibre of $X \to S$ are contained in an affine open. If $\text{Hilb}^d_{X/S}$ is representable by a scheme, we often denote this scheme by $\underline{\text{Hilb}}^d_{X/S}$. \begin{lemma} \label{lemma-hilb-d-sheaf} Let $X \to S$ be a morphism of schemes. The functor $\text{Hilb}^d_{X/S}$ satisfies the sheaf property for the fpqc topology (Topologies, Definition \ref{topologies-definition-sheaf-property-fpqc}). \end{lemma} \begin{proof} Let $\{T_i \to T\}_{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_i = X_{T_i} = X \times_S T_i$. Note that $\{X_i \to X_T\}_{i \in I}$ is an fpqc covering of $X_T$ (Topologies, Lemma \ref{topologies-lemma-fpqc}) and that $X_{T_i \times_T T_{i'}} = X_i \times_{X_T} X_{i'}$. Suppose that $Z_i \in \text{Hilb}^d_{X/S}(T_i)$ is a collection of elements such that $Z_i$ and $Z_{i'}$ map to the same element of $\text{Hilb}^d_{X/S}(T_i \times_T T_{i'})$. By effective descent for closed immersions (Descent, Lemma \ref{descent-lemma-closed-immersion}) there is a closed immersion $Z \to X_T$ whose base change by $X_i \to X_T$ is equal to $Z_i \to X_i$. The morphism $Z \to T$ then has the property that its base change to $T_i$ is the morphism $Z_i \to T_i$. Hence $Z \to T$ is finite locally free of degree $d$ by Descent, Lemma \ref{descent-lemma-descending-property-finite-locally-free}. \end{proof} \begin{lemma} \label{lemma-hilb-d-limit-preserving} Let $X \to S$ be a morphism of schemes. If $X \to S$ is of finite presentation, then the functor $\text{Hilb}^d_{X/S}$ is limit preserving (Limits, Remark \ref{limits-remark-limit-preserving}). \end{lemma} \begin{proof} Let $T = \lim T_i$ be a limit of affine schemes over $S$. We have to show that $\text{Hilb}^d_{X/S}(T) = \colim \text{Hilb}^d_{X/S}(T_i)$. Observe that if $Z \to X_T$ is an element of $\text{Hilb}^d_{X/S}(T)$, then $Z \to T$ is of finite presentation. Hence by Limits, Lemma \ref{limits-lemma-descend-finite-presentation} there exists an $i$, a scheme $Z_i$ of finite presentation over $T_i$, and a morphism $Z_i \to X_{T_i}$ over $T_i$ whose base change to $T$ gives $Z \to X_T$. We apply Limits, Lemma \ref{limits-lemma-descend-closed-immersion-finite-presentation} to see that we may assume $Z_i \to X_{T_i}$ is a closed immersion after increasing $i$. We apply Limits, Lemma \ref{limits-lemma-descend-finite-locally-free} to see that $Z_i \to T_i$ is finite locally free of degree $d$ after possibly increasing $i$. Then $Z_i \in \text{Hilb}^d_{X/S}(T_i)$ as desired. \end{proof} \noindent Let $S$ be a scheme. Let $i : X \to Y$ be a closed immersion of schemes over $S$. Then there is a transformation of functors $$\text{Hilb}^d_{X/S} \longrightarrow \text{Hilb}^d_{Y/S}$$ which maps an element $Z \in \text{Hilb}^d_{X/S}(T)$ to $i_T(Z) \subset Y_T$ in $\text{Hilb}^d_{Y/S}$. Here $i_T : X_T \to Y_T$ is the base change of $i$. \begin{lemma} \label{lemma-hilb-d-of-closed} Let $S$ be a scheme. Let $i : X \to Y$ be a closed immersion of schemes. If $\text{Hilb}^d_{Y/S}$ is representable by a scheme, so is $\text{Hilb}^d_{X/S}$ and the corresponding morphism of schemes $\underline{\text{Hilb}}^d_{X/S} \to \underline{\text{Hilb}}^d_{Y/S}$ is a closed immersion. \end{lemma} \begin{proof} Let $T$ be a scheme over $S$ and let $Z \in \text{Hilb}^d_{Y/S}(T)$. Claim: there is a closed subscheme $T_X \subset T$ such that a morphism of schemes $T' \to T$ factors through $T_X$ if and only if $Z_{T'} \to Y_{T'}$ factors through $X_{T'}$. Applying this to a scheme $T_{univ}$ representing $\text{Hilb}^d_{Y/S}$ and the universal object\footnote{See Categories, Section \ref{categories-section-opposite}} $Z_{univ} \in \text{Hilb}^d_{Y/S}(T_{univ})$ we get a closed subscheme $T_{univ, X} \subset T_{univ}$ such that $Z_{univ, X} = Z_{univ} \times_{T_{univ}} T_{univ, X}$ is a closed subscheme of $X \times_S T_{univ, X}$ and hence defines an element of $\text{Hilb}^d_{X/S}(T_{univ, X})$. A formal argument then shows that $T_{univ, X}$ is a scheme representing $\text{Hilb}^d_{X/S}$ with universal object $Z_{univ, X}$. \medskip\noindent Proof of the claim. Consider $Z' = X_T \times_{Y_T} Z$. Given $T' \to T$ we see that $Z_{T'} \to Y_{T'}$ factors through $X_{T'}$ if and only if $Z'_{T'} \to Z_{T'}$ is an isomorphism. Thus the claim follows from the very general More on Flatness, Lemma \ref{flat-lemma-Weil-restriction-closed-subschemes}. However, in this special case one can prove the statement directly as follows: first reduce to the case $T = \Spec(A)$ and $Z = \Spec(B)$. After shrinking $T$ further we may assume there is an isomorphism $\varphi : B \to A^{\oplus d}$ as $A$-modules. Then $Z' = \Spec(B/J)$ for some ideal $J \subset B$. Let $g_\beta \in J$ be a collection of generators and write $\varphi(g_\beta) = (g_\beta^1, \ldots, g_\beta^d)$. Then it is clear that $T_X$ is given by $\Spec(A/(g_\beta^j))$. \end{proof} \begin{lemma} \label{lemma-hilb-d-separated} Let $X \to S$ be a morphism of schemes. If $X \to S$ is separated and $\text{Hilb}^d_{X/S}$ is representable, then $\underline{\text{Hilb}}^d_{X/S} \to S$ is separated. \end{lemma} \begin{proof} In this proof all unadorned products are over $S$. Let $H = \underline{\text{Hilb}}^d_{X/S}$ and let $Z \in \text{Hilb}^d_{X/S}(H)$ be the universal object. Consider the two objects $Z_1, Z_2 \in \text{Hilb}^d_{X/S}(H \times H)$ we get by pulling back $Z$ by the two projections $H \times H \to H$. Then $Z_1 = Z \times H \subset X_{H \times H}$ and $Z_2 = H \times Z \subset X_{H \times H}$. Since $H$ represents the functor $\text{Hilb}^d_{X/S}$, the diagonal morphism $\Delta : H \to H \times H$ has the following universal property: A morphism of schemes $T \to H \times H$ factors through $\Delta$ if and only if $Z_{1, T} = Z_{2, T}$ as elements of $\text{Hilb}^d_{X/S}(T)$. Set $Z = Z_1 \times_{X_{H \times H}} Z_2$. Then we see that $T \to H \times H$ factors through $\Delta$ if and only if the morphisms $Z_T \to Z_{1, T}$ and $Z_T \to Z_{2, T}$ are isomorphisms. It follows from the very general More on Flatness, Lemma \ref{flat-lemma-Weil-restriction-closed-subschemes} that $\Delta$ is a closed immersion. In the proof of Lemma \ref{lemma-hilb-d-of-closed} the reader finds an alternative easier proof of the needed result in our special case. \end{proof} \begin{lemma} \label{lemma-hilb-d-An} Let $X \to S$ be a morphism of affine schemes. Let $d \geq 0$. Then $\text{Hilb}^d_{X/S}$ is representable. \end{lemma} \begin{proof} Say $S = \Spec(R)$. Then we can choose a closed immersion of $X$ into the spectrum of $R[x_i; i \in I]$ for some set $I$ (of sufficiently large cardinality. Hence by Lemma \ref{lemma-hilb-d-of-closed} we may assume that $X = \Spec(A)$ where $A = R[x_i; i \in I]$. We will use Schemes, Lemma \ref{schemes-lemma-glue-functors} to prove the lemma in this case. \medskip\noindent Condition (1) of the lemma follows from Lemma \ref{lemma-hilb-d-sheaf}. \medskip\noindent For every subset $W \subset A$ of cardinality $d$ we will construct a subfunctor $F_W$ of $\text{Hilb}^d_{X/S}$. (It would be enough to consider the case where $W$ consists of a collection of monomials in the $x_i$ but we do not need this.) Namely, we will say that $Z \in \text{Hilb}^d_{X/S}(T)$ is in $F_W(T)$ if and only if the $\mathcal{O}_T$-linear map $$\bigoplus\nolimits_{f \in W} \mathcal{O}_T \longrightarrow (Z \to T)_*\mathcal{O}_Z,\quad (g_f) \longmapsto \sum g_f f|_Z$$ is surjective (equivalently an isomorphism). Here for $f \in A$ and $Z \in \text{Hilb}^d_{X/S}(T)$ we denote $f|_Z$ the pullback of $f$ by the morphism $Z \to X_T \to X$. \medskip\noindent Openness, i.e., condition (2)(b) of the lemma. This follows from Algebra, Lemma \ref{algebra-lemma-cokernel-flat}. \medskip\noindent Covering, i.e., condition (2)(c) of the lemma. Since $$A \otimes_R \mathcal{O}_T = (X_T \to T)_*\mathcal{O}_{X_T} \to (Z \to T)_*\mathcal{O}_Z$$ is surjective and since $(Z \to T)_*\mathcal{O}_Z$ is finite locally free of rank $d$, for every point $t \in T$ we can find a finite subset $W \subset A$ of cardinality $d$ whose images form a basis of the $d$-dimensional $\kappa(t)$-vector space $((Z \to T)_*\mathcal{O}_Z)_t \otimes_{\mathcal{O}_{T, t}} \kappa(t)$. By Nakayama's lemma there is an open neighbourhood $V \subset T$ of $t$ such that $Z_V \in F_W(V)$. \medskip\noindent Representable, i.e., condition (2)(a) of the lemma. Let $W \subset A$ have cardinality $d$. We claim that $F_W$ is representable by an affine scheme over $R$. We will construct this affine scheme here, but we encourage the reader to think it trough for themselves. Choose a numbering $f_1, \ldots, f_d$ of the elements of $W$. We will construct a universal element $Z_{univ} = \Spec(B_{univ})$ of $F_W$ over $T_{univ} = \Spec(R_{univ})$ which will be the spectrum of $$B_{univ} = R_{univ}[e_1, \ldots, e_d]/ (e_ke_l - \sum c_{kl}^m e_m)$$ where the $e_l$ will be the images of the $f_l$ and where the closed immersion $Z_{univ} \to X_{T_{univ}}$ is given by the ring map $$A \otimes_R R_{univ} \longrightarrow B_{univ}$$ mapping $1 \otimes 1$ to $\sum b^le_l$ and $x_i$ to $\sum b_i^le_l$. In fact, we claim that $F_W$ is represented by the spectrum of the ring $$R_{univ} = R[c_{kl}^m, b^l, b_i^l]/\mathfrak a_{univ}$$ where the ideal $\mathfrak a_{univ}$ is generated by the following elements: \begin{enumerate} \item multiplication on $B_{univ}$ is commutative, i.e., $c_{lk}^m - c_{kl}^m \in \mathfrak a_{univ}$, \item multiplication on $B_{univ}$ is associative, i.e., $c_{lk}^m c_{m n}^p - c_{lq}^p c_{kn}^q \in \mathfrak a_{univ}$, \item $\sum b^le_l$ is a multiplicative $1$ in $B_{univ}$, in other words, we should have $(\sum b^le_l)e_k = e_k$ for all $k$, which means $\sum b^lc_{lk}^m - \delta_{km} \in \mathfrak a_{univ}$ (Kronecker delta). \end{enumerate} After dividing out by the ideal $\mathfrak a'_{univ}$ of the elements listed sofar we obtain a well defined ring map $$\Psi : A \otimes_R R[c_{kl}^m, b^l, b_i^l]/\mathfrak a'_{univ} \longrightarrow \left(R[c_{kl}^m, b^l, b_i^l]/\mathfrak a'_{univ}\right) [e_1, \ldots, e_d]/(e_ke_l - \sum c_{kl}^m e_m)$$ sending $1 \otimes 1$ to $\sum b^le_l$ and $x_i \otimes 1$ to $\sum b_i^le_l$. We need to add some more elements to our ideal because we need \begin{enumerate} \item[(5)] $f_l$ to map to $e_l$ in $B_{univ}$. Write $\Psi(f_l) - e_l = \sum h_l^me_m$ with $h_l^m \in R[c_{kl}^m, b^l, b_i^l]/\mathfrak a'_{univ}$ then we need to set $h_l^m$ equal to zero. \end{enumerate} Thus setting $\mathfrak a_{univ} \subset R[c_{kl}^m, b^l, b_i^l]$ equal to $\mathfrak a'_{univ} +$ ideal generated by lifts of $h_l^m$ to $R[c_{kl}^m, b^l, b_i^l]$, then it is clear that $F_W$ is represented by $\Spec(R_{univ})$. \end{proof} \begin{proposition} \label{proposition-hilb-d-representable} Let $X \to S$ be a morphism of schemes. Let $d \geq 0$. Assume for all $(s, x_1, \ldots, x_d)$ where $s \in S$ and $x_1, \ldots, x_d \in X_s$ there exists an affine open $U \subset X$ with $x_1, \ldots, x_d \in U$. Then $\text{Hilb}^d_{X/S}$ is representable by a scheme. \end{proposition} \begin{proof} Either using relative glueing (Constructions, Section \ref{constructions-section-relative-glueing}) or using the functorial point of view (Schemes, Lemma \ref{schemes-lemma-glue-functors}) we reduce to the case where $S$ is affine. Details omitted. \medskip\noindent Assume $S$ is affine. For $U \subset X$ affine open, denote $F_U \subset \text{Hilb}^d_{X/S}$ the subfunctor parametrizing closed subschemes of $U$. We will use Schemes, Lemma \ref{schemes-lemma-glue-functors} and the subfunctors $F_U$ to conclude. \medskip\noindent Condition (1) is Lemma \ref{lemma-hilb-d-sheaf}. \medskip\noindent Condition (2)(a) follows from the fact that $F_U = \text{Hilb}^d_{U/S}$ and that this is representable by Lemma \ref{lemma-hilb-d-An}. \medskip\noindent Let $Z \in \text{Hilb}^d_{X/S}(T)$ for some scheme $T$ over $S$. Let $$B = (Z \to T)\left((Z \to X_T \to X)^{-1}(X \setminus U)\right)$$ This is a closed subset of $T$ and it is clear that over the open $T_{Z, U} = T \setminus B$ the restriction $Z_{t'}$ maps into $U_{T'}$. On the other hand, for any $b \in B$ the fibre $Z_b$ does not map into $U$. Thus we see that given a morphism $T' \to T$ we have $Z_{T'} \in F_U(T')$ $\Leftrightarrow$ $T' \to T$ factors through the open $T_{Z, U}$. This proves condition (2)(b). \medskip\noindent Condition (2)(c) follows from our assumption on $X/S$. All we have to do is show the following: If $T$ is the spectrum of a field and $Z \subset X_T$ is a closed subscheme, finite flat of degree $d$ over $T$, then $Z \to X_T \to X$ factors through an affine open $U$ of $X$. This is clear because $Z$ will have at most $d$ points and these will all map into the fibre of $X$ over the image point of $T \to S$. \end{proof} \begin{remark} \label{remark-when-proposition-applies} Let $f : X \to S$ be a morphism of schemes. The assumption of Proposition \ref{proposition-hilb-d-representable} and hence the conclusion holds in each of the following cases: \begin{enumerate} \item $X$ is quasi-affine, \item $f$ is quasi-affine, \item $f$ is quasi-projective, \item $f$ is locally projective, \item there exists an ample invertible sheaf on $X$, \item there exists an $f$-ample invertible sheaf on $X$, and \item there exists an $f$-very ample invertible sheaf on $X$. \end{enumerate} Namely, in each of these cases, every finite set of points of a fibre $X_s$ is contained in a quasi-compact open $U$ of $X$ which comes with an ample invertible sheaf, is isomorphic to an open of an affine scheme, or is isomorphic to an open of $\text{Proj}$ of a graded ring (in each case this follows by unwinding the definitions). Thus the existence of suitable affine opens by Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}. \end{remark} \section{Moduli of divisors on smooth curves} \label{section-divisors} \noindent For a smooth morphism $X \to S$ of relative dimension $1$ the functor $\text{Hilb}^d_{X/S}$ parametrizes relative effective Cartier divisors as defined in Divisors, Section \ref{divisors-section-effective-Cartier-morphisms}. \begin{lemma} \label{lemma-divisors-on-curves} Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D \subset X$ be a closed subscheme. Consider the following conditions \begin{enumerate} \item $D \to S$ is finite locally free, \item $D$ is a relative effective Cartier divisor on $X/S$, \item $D \to S$ is locally quasi-finite, flat, and locally of finite presentation, and \item $D \to S$ is locally quasi-finite and flat. \end{enumerate} We always have the implications $$(1) \Rightarrow (2) \Leftrightarrow (3) \Rightarrow (4)$$ If $S$ is locally Noetherian, then the last arrow is an if and only if. If $X \to S$ is proper (and $S$ arbitrary), then the first arrow is an if and only if. \end{lemma} \begin{proof} Equivalence of (2) and (3). This follows from Divisors, Lemma \ref{divisors-lemma-fibre-Cartier} if we can show the equivalence of (2) and (3) when $S$ is the spectrum of a field $k$. Let $x \in X$ be a closed point. As $X$ is smooth of relative dimension $1$ over $k$ and we see that $\mathcal{O}_{X, x}$ is a regular local ring of dimension $1$ (see Varieties, Lemma \ref{varieties-lemma-smooth-regular}). Thus $\mathcal{O}_{X, x}$ is a discrete valuation ring (Algebra, Lemma \ref{algebra-lemma-characterize-dvr}) and hence a PID. It follows that every sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ which is nonvanishing at all the generic points of $X$ is invertible (Divisors, Lemma \ref{divisors-lemma-effective-Cartier-in-points}). In other words, every closed subscheme of $X$ which does not contain a generic point is an effective Cartier divisor. It follows that (2) and (3) are equivalent. \medskip\noindent If $S$ is Noetherian, then any locally quasi-finite morphism $D \to S$ is locally of finite presentation (Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation}), whence (3) is equivalent to (4). \medskip\noindent If $X \to S$ is proper (and $S$ is arbitrary), then $D \to S$ is proper as well. Since a proper locally quasi-finite morphism is finite (More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite}) and a finite, flat, and finitely presented morphism is finite locally free (Morphisms, Lemma \ref{morphisms-lemma-finite-flat}), we see that (1) is equivalent to (2). \end{proof} \begin{lemma} \label{lemma-sum-divisors-on-curves} Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. Then $D_1 + D_2$ is finite locally free of degree $d_1 + d_2$ over $S$. \end{lemma} \begin{proof} By Lemma \ref{lemma-divisors-on-curves} we see that $D_1$ and $D_2$ are relative effective Cartier divisors on $X/S$. Thus $D = D_1 + D_2$ is a relative effective Cartier divisor on $X/S$ by Divisors, Lemma \ref{divisors-lemma-sum-relative-effective-Cartier-divisor}. Hence $D \to S$ is locally quasi-finite, flat, and locally of finite presentation by Lemma \ref{lemma-divisors-on-curves}. Applying Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated} the the surjective integral morphism $D_1 \amalg D_2 \to D$ we find that $D \to S$ is separated. Then Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper} implies that $D \to S$ is proper. This implies that $D \to S$ is finite (More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite}) and in turn we see that $D \to S$ is finite locally free (Morphisms, Lemma \ref{morphisms-lemma-finite-flat}). Thus it suffice to show that the degree of $D \to S$ is $d_1 + d_2$. To do this we may base change to a fibre of $X \to S$, hence we may assume that $S = \Spec(k)$ for some field $k$. In this case, there exists a finite set of closed points $x_1, \ldots, x_n \in X$ such that $D_1$ and $D_2$ are supported on $\{x_1, \ldots, x_n\}$. In fact, there are nonzerodivisors $f_{i, j} \in \mathcal{O}_{X, x_i}$ such that $$D_1 = \coprod \Spec(\mathcal{O}_{X, x_i}/(f_{i, 1})) \quad\text{and}\quad D_2 = \coprod \Spec(\mathcal{O}_{X, x_i}/(f_{i, 2}))$$ Then we see that $$D = \coprod \Spec(\mathcal{O}_{X, x_i}/(f_{i, 1}f_{i, 2}))$$ From this one sees easily that $D$ has degree $d_1 + d_2$ over $k$ (if need be, use Algebra, Lemma \ref{algebra-lemma-ord-additive}). \end{proof} \begin{lemma} \label{lemma-difference-divisors-on-curves} Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. If $D_1 \subset D_2$ (as closed subschemes) then there is a closed subscheme $D \subset X$ finite locally free of degree $d_2 - d_1$ over $S$ such that $D_2 = D_1 + D$. \end{lemma} \begin{proof} This proof is almost exactly the same as the proof of Lemma \ref{lemma-sum-divisors-on-curves}. By Lemma \ref{lemma-divisors-on-curves} we see that $D_1$ and $D_2$ are relative effective Cartier divisors on $X/S$. By Divisors, Lemma \ref{divisors-lemma-difference-relative-effective-Cartier-divisor} there is a relative effective Cartier divisor $D \subset X$ such that $D_2 = D_1 + D$. Hence $D \to S$ is locally quasi-finite, flat, and locally of finite presentation by Lemma \ref{lemma-divisors-on-curves}. Since $D$ is a closed subscheme of $D_2$, we see that $D \to S$ is finite. It follows that $D \to S$ is finite locally free (Morphisms, Lemma \ref{morphisms-lemma-finite-flat}). Thus it suffice to show that the degree of $D \to S$ is $d_2 - d_1$. This follows from Lemma \ref{lemma-sum-divisors-on-curves}. \end{proof} \noindent Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. By Lemma \ref{lemma-divisors-on-curves} for a scheme $T$ over $S$ and $D \in \text{Hilb}^d_{X/S}(T)$, we can view $D$ as a relative effective Cartier divisor on $X_T/T$ such that $D \to T$ is finite locally free of degree $d$. Hence, by Lemma \ref{lemma-sum-divisors-on-curves} we obtain a transformation of functors $$\text{Hilb}^{d_1}_{X/S} \times \text{Hilb}^{d_2}_{X/S} \longrightarrow \text{Hilb}^{d_1 + d_2}_{X/S},\quad (D_1, D_2) \longmapsto D_1 + D_2$$ If $\text{Hilb}^d_{X/S}$ is representable for all degrees $d$, then this transformation of functors corresponds to a morphism of schemes $$\underline{\text{Hilb}}^{d_1}_{X/S} \times_S \underline{\text{Hilb}}^{d_2}_{X/S} \longrightarrow \underline{\text{Hilb}}^{d_1 + d_2}_{X/S}$$ over $S$. Observe that $\underline{\text{Hilb}}^0_{X/S} = S$ and $\underline{\text{Hilb}}^1_{X/S} = X$. A special case of the morphism above is the morphism $$\underline{\text{Hilb}}^d_{X/S} \times_S X \longrightarrow \underline{\text{Hilb}}^{d + 1}_{X/S},\quad (D, x) \longmapsto D + x$$ \begin{lemma} \label{lemma-universal-object} Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$ such that the functors $\text{Hilb}^d_{X/S}$ are representable. The morphism $\underline{\text{Hilb}}^d_{X/S} \times_S X \to \underline{\text{Hilb}}^{d + 1}_{X/S}$ is finite locally free of degree $d + 1$. \end{lemma} \begin{proof} Let $D_{univ} \subset X \times_S \underline{\text{Hilb}}^{d + 1}_{X/S}$ be the universal object. There is a commutative diagram $$\xymatrix{ \underline{\text{Hilb}}^d_{X/S} \times_S X \ar[rr] \ar[rd] & & D_{univ} \ar[ld] \ar@{^{(}->}[r] & \underline{\text{Hilb}}^{d + 1}_{X/S} \times_S X \\ & \underline{\text{Hilb}}^{d + 1}_{X/S} }$$ where the top horizontal arrow maps $(D', x)$ to $(D' + x, x)$. We claim this morphism is an isomorphism which certainly proves the lemma. Namely, given a scheme $T$ over $S$, a $T$-valued point $\xi$ of $D_{univ}$ is given by a pair $\xi = (D, x)$ where $D \subset X_T$ is a closed subscheme finite locally free of degree $d + 1$ over $T$ and $x : T \to X$ is a morphism whose graph $x : T \to X_T$ factors through $D$. Then by Lemma \ref{lemma-difference-divisors-on-curves} we can write $D = D' + x$ for some $D' \subset X_T$ finite locally free of degree $d$ over $T$. Sending $\xi = (D, x)$ to the pair $(D', x)$ is the desired inverse. \end{proof} \begin{lemma} \label{lemma-hilb-d-smooth} Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$ such that the functors $\text{Hilb}^d_{X/S}$ are representable. The schemes $\underline{\text{Hilb}}^d_{X/S}$ are smooth over $S$ of relative dimension $d$. \end{lemma} \begin{proof} We have $\underline{\text{Hilb}}^d_{X/S} = S$ and $\underline{\text{Hilb}}^1_{X/S} = X$ thus the result is true for $d = 0, 1$. Assuming the result for $d$, we see that $\underline{\text{Hilb}}^d_{X/S} \times_S X$ is smooth over $S$ (Morphisms, Lemma \ref{morphisms-lemma-base-change-smooth} and \ref{morphisms-lemma-composition-smooth}). Since $\underline{\text{Hilb}}^d_{X/S} \times_S X \to \underline{\text{Hilb}}^{d + 1}_{X/S}$ is finite locally free of degree $d + 1$ by Lemma \ref{lemma-universal-object} the result follows from Descent, Lemma \ref{descent-lemma-smooth-permanence}. We omit the verification that the relative dimension is as claimed (you can do this by looking at fibres, or by keeping track of the dimensions in the argument above). \end{proof} \noindent We collect all the information obtained sofar in the case of a proper smooth curve over a field. \begin{proposition} \label{proposition-hilb-d} Let $X$ be a geometrically irreducible smooth proper curve over a field $k$. \begin{enumerate} \item The functors $\text{Hilb}^d_{X/k}$ are representable by smooth proper varieties $\underline{\text{Hilb}}^d_{X/k}$ of dimension $d$ over $k$. \item For a field extension $k'/k$ the $k'$-rational points of $\underline{\text{Hilb}}^d_{X/k}$ are in $1$-to-$1$ bijection with effective Cartier divisors of degree $d$ on $X_{k'}$. \item For $d_1, d_2 \geq 0$ there is a morphism $$\underline{\text{Hilb}}^{d_1}_{X/k} \times_k \underline{\text{Hilb}}^{d_2}_{X/k} \longrightarrow \underline{\text{Hilb}}^{d_1 + d_2}_{X/k}$$ which is finite locally free of degree ${d_1 + d_2 \choose d_1}$. \end{enumerate} \end{proposition} \begin{proof} The functors $\text{Hilb}^d_{X/k}$ are representable by Proposition \ref{proposition-hilb-d-representable} (see also Remark \ref{remark-when-proposition-applies}) and the fact that $X$ is projective (Varieties, Lemma \ref{varieties-lemma-dim-1-proper-projective}). The schemes $\underline{\text{Hilb}}^d_{X/k}$ are separated over $k$ by Lemma \ref{lemma-hilb-d-separated}. The schemes $\underline{\text{Hilb}}^d_{X/k}$ are smooth over $k$ by Lemma \ref{lemma-hilb-d-smooth}. Starting with $X = \underline{\text{Hilb}}^1_{X/k}$, the morphisms of Lemma \ref{lemma-universal-object}, and induction we find a morphism $$X^d = X \times_k X \times_k \ldots \times_k X \longrightarrow \underline{\text{Hilb}}^d_{X/k},\quad (x_1, \ldots, x_d) \longrightarrow x_1 + \ldots + x_d$$ which is finite locally free of degree $d!$. Since $X$ is proper over $k$, so is $X^d$, hence $\underline{\text{Pic}}^d_{X/k}$ is proper over $k$ by Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}. Since $X$ is geometrically irreducible over $k$, the product $X^d$ is irreducible (Varieties, Lemma \ref{varieties-lemma-bijection-irreducible-components}) hence the image is irreducible (in fact geometrically irreducible). This proves (1). Part (2) follows from the definitions. Part (3) follows from the commutative diagram $$\xymatrix{ X^{d_1} \times_k X^{d_2} \ar[d] \ar@{=}[r] & X^{d_1 + d_2} \ar[d] \\ \underline{\text{Hilb}}^{d_1}_{X/k} \times_k \underline{\text{Hilb}}^{d_2}_{X/k} \ar[r] & \underline{\text{Hilb}}^{d_1 + d_2}_{X/k} }$$ and multiplicativity of degrees of finite locally free morphisms. \end{proof} \begin{remark} \label{remark-universal-object-hilb-d} Let $X$ be a geometrically irreducible smooth proper curve over a field $k$ as in Proposition \ref{proposition-hilb-d}. Let $d \geq 0$. The universal closed object is a relatively effective divisor $$D_{univ} \subset \underline{\text{Hilb}}^{d + 1}_{X/k} \times_k X$$ over $\underline{\text{Hilb}}^{d + 1}_{X/k}$ by Lemma \ref{lemma-divisors-on-curves}. In fact, $D_{univ}$ is isomorphic as a scheme to $\underline{\text{Hilb}}^d_{X/k} \times_k X$, see proof of Lemma \ref{lemma-universal-object}. In particular, $D_{univ}$ is an effective Cartier divisor and we obtain an invertible module $\mathcal{O}(D_{univ})$. If $[D] \in \underline{\text{Hilb}}^{d + 1}_{X/k}$ denotes the $k$-rational point corresponding to the effective Cartier divisor $D \subset X$ of degree $d$, then the restriction of $\mathcal{O}(D_{univ})$ to to the fibre $[D] \times X$ is $\mathcal{O}_X(D)$. \end{remark} \section{The Picard functor} \label{section-picard-functor} \noindent Given any scheme $X$ we denote $\text{Pic}(X)$ the set of isomorphism classes of invertible $\mathcal{O}_X$-modules. See Modules, Definition \ref{modules-definition-pic}. Given a morphism $f : X \to Y$ of schemes, pullback defines a group homomorphism $\text{Pic}(Y) \to \text{Pic}(X)$. The assignment $X \leadsto \text{Pic}(X)$ is a contravariant functor from the category of schemes to the category of abelian groups. This functor is not representable, but it turns out that a relative variant of this construction sometimes is representable. \medskip\noindent Let us define the Picard functor for a morphism of schemes $f : X \to S$. The idea behind our construction is that we'll take it to be the sheaf $R^1f_*\mathbf{G}_m$ where we use the fppf topology to compute the higher direct image. Unwinding the definitions this leads to the following more direct definition. \begin{definition} \label{definition-picard-functor} Let $\Sch_{fppf}$ be a big site as in Topologies, Definition \ref{topologies-definition-big-small-fppf}. Let $f : X \to S$ be a morphism of this site. The {\it Picard functor} $\text{Pic}_{X/S}$ is the fppf sheafification of the functor $$(\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad T \longmapsto \text{Pic}(X_T)$$ If this functor is representable, then we denote $\underline{\text{Pic}}_{X/S}$ a scheme representing it. \end{definition} \noindent An often used remark is that if $T \in \Ob((\Sch/S)_{fppf})$, then $\text{Pic}_{X_T/T}$ is the restriction of $\text{Pic}_{X/S}$ to $(\Sch/T)_{fppf}$. It turns out to be nontrivial to see what the value of $\text{Pic}_{X/S}$ is on schemes $T$ over $S$. Here is a lemma that helps with this task. \begin{lemma} \label{lemma-flat-geometrically-connected-fibres} Let $f : X \to S$ be as in Definition \ref{definition-picard-functor}. If $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T \in \Ob((\Sch/S)_{fppf})$, then $$0 \to \text{Pic}(T) \to \text{Pic}(X_T) \to \text{Pic}_{X/S}(T)$$ is an exact sequence for all $T$. \end{lemma} \begin{proof} We may replace $S$ by $T$ and $X$ by $X_T$ and assume that $S = T$ to simplify the notation. Let $\mathcal{N}$ be an invertible $\mathcal{O}_S$-module. If $f^*\mathcal{N} \cong \mathcal{O}_X$, then we see that $f_*f^*\mathcal{N} \cong f_*\mathcal{O}_X \cong \mathcal{O}_S$ by assumption. Since $\mathcal{N}$ is locally trivial, we see that the canonical map $\mathcal{N} \to f_*f^*\mathcal{N}$ is locally an isomorphism (because $\mathcal{O}_S \to f_*f^*\mathcal{O}_S$ is an isomorphism by assumption). Hence we conclude that $\mathcal{N} \to f_*f^*\mathcal{N} \to \mathcal{O}_S$ is an isomorphism and we see that $\mathcal{N}$ is trivial. This proves the first arrow is injective. \medskip\noindent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module which is in the kernel of $\text{Pic}(X) \to \text{Pic}_{X/S}(S)$. Then there exists an fppf covering $\{S_i \to S\}$ such that $\mathcal{L}$ pulls back to the trivial invertible sheaf on $X_{S_i}$. Choose a trivializing section $s_i$. Then $\text{pr}_0^*s_i$ and $\text{pr}_1^*s_j$ are both trivialising sections of $\mathcal{L}$ over $X_{S_i \times_S S_j}$ and hence differ by a multiplicative unit $$f_{ij} \in \Gamma(X_{S_i \times_S S_j}, \mathcal{O}_{X_{S_i \times_S S_j}}^*) = \Gamma(S_i \times_S S_j, \mathcal{O}_{S_i \times_S S_j}^*)$$ (equality by our assumption on pushforward of structure sheaves). Of course these elements satisfy the cocycle condition on $S_i \times_S S_j \times_S S_k$, hence they define a descent datum on invertible sheaves for the fppf covering $\{S_i \to S\}$. By Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_S$-module $\mathcal{N}$ with trivializations over $S_i$ whose associated descent datum is $\{f_{ij}\}$. Then $f^*\mathcal{N} \cong \mathcal{L}$ as the functor from descent data to modules is fully faithful (see proposition cited above). \end{proof} \begin{lemma} \label{lemma-flat-geometrically-connected-fibres-with-section} Let $f : X \to S$ be as in Definition \ref{definition-picard-functor}. Assume $f$ has a section $\sigma$ and that $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T \in \Ob((\Sch/S)_{fppf})$. Then $$0 \to \text{Pic}(T) \to \text{Pic}(X_T) \to \text{Pic}_{X/S}(T) \to 0$$ is a split exact sequence with splitting given by $\sigma_T^* : \text{Pic}(X_T) \to \text{Pic}(T)$. \end{lemma} \begin{proof} Denote $K(T) = \Ker(\sigma_T^* : \text{Pic}(X_T) \to \text{Pic}(T))$. Since $\sigma$ is a section of $f$ we see that $\text{Pic}(X_T)$ is the direct sum of $\text{Pic}(T)$ and $K(T)$. Thus by Lemma \ref{lemma-flat-geometrically-connected-fibres} we see that $K(T) \subset \text{Pic}_{X/S}(T)$ for all $T$. Moreover, it is clear from the construction that $\text{Pic}_{X/S}$ is the sheafification of the presheaf $K$. To finish the proof it suffices to show that $K$ satisfies the sheaf condition for fppf coverings which we do in the next paragraph. \medskip\noindent Let $\{T_i \to T\}$ be an fppf covering. Let $\mathcal{L}_i$ be elements of $K(T_i)$ which map to the same elements of $K(T_i \times_T T_j)$ for all $i$ and $j$. Choose an isomorphism $\alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i$ for all $i$. Choose an isomorphism $$\varphi_{ij} : \mathcal{L}_i|_{X_{T_i \times_T T_j}} \longrightarrow \mathcal{L}_j|_{X_{T_i \times_T T_j}}$$ If the map $$\alpha_j|_{T_i \times_T T_j} \circ \sigma_{T_i \times_T T_j}^*\varphi_{ij} \circ \alpha_i|_{T_i \times_T T_j} : \mathcal{O}_{T_i \times_T T_j} \to \mathcal{O}_{T_i \times_T T_j}$$ is not equal to multiplication by $1$ but some $u_{ij}$, then we can scale $\varphi_{ij}$ by $u_{ij}^{-1}$ to correct this. Having done this, consider the self map $$\varphi_{ki}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{jk}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{ij}|_{X_{T_i \times_T T_j \times_T T_k}} \quad\text{on}\quad \mathcal{L}_i|_{X_{T_i \times_T T_j \times_T T_k}}$$ which is given by multiplication by some regular function $f_{ijk}$ on the scheme $X_{T_i \times_T T_j \times_T T_k}$. By our choice of $\varphi_{ij}$ we see that the pullback of this map by $\sigma$ is equal to multiplication by $1$. By our assumption on functions on $X$, we see that $f_{ijk} = 1$. Thus we obtain a descent datum for the fppf covering $\{X_{T_i} \to X\}$. By Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_{X_T}$-module $\mathcal{L}$ and an isomorphism $\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}$ whose pullback to $X_{T_i}$ recovers $(\mathcal{L}_i, \alpha_i)$ (small detail omitted). Thus $\mathcal{L}$ defines an object of $K(T)$ as desired. \end{proof} \section{A representability criterion} \label{section-representability} \noindent To prove the Picard functor is representable we will use the following criterion. \begin{lemma} \label{lemma-criterion} Let $k$ be a field. Let $G : (\Sch/k)^{opp} \to \textit{Groups}$ be a functor. With terminology as in Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}, assume that \begin{enumerate} \item $G$ satisfies the sheaf property for the Zariski topology, \item there exists a subfunctor $F \subset G$ such that \begin{enumerate} \item $F$ is representable, \item $F \subset G$ is representable by open immersion, \item for every field extension $K$ of $k$ and $g \in G(K)$ there exists a $g' \in G(k)$ such that $g'g \in F(K)$. \end{enumerate} \end{enumerate} Then $G$ is representable by a group scheme over $k$. \end{lemma} \begin{proof} This follows from Schemes, Lemma \ref{schemes-lemma-glue-functors}. Namely, take $I = G(k)$ and for $i = g' \in I$ take $F_i \subset G$ the subfunctor which associates to $T$ over $k$ the set of elements $g \in G(T)$ with $g'g \in F(T)$. Then $F_i \cong F$ by multiplication by $g'$. The map $F_i \to G$ is isomorphic to the map $F \to G$ by multiplication by $g'$, hence is representable by open immersions. Finally, the collection $(F_i)_{i \in I}$ covers $G$ by assumption (2)(c). Thus the lemma mentioned above applies and the proof is complete. \end{proof} \section{The Picard scheme of a curve} \label{section-picard-curve} \noindent In this section we will apply Lemma \ref{lemma-criterion} to show that $\text{Pic}_{X/k}$ is representable, when $k$ is an algebraically closed field and $X$ is a smooth projective curve over $k$. To make this work we use a bit of cohomology and base change developed in the chapter on derived categories of schemes. \begin{lemma} \label{lemma-check-conditions} Let $k$ be a field. Let $X$ be a smooth projective curve over $k$ which has a $k$-rational point. Then the hypotheses of Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section} are satisfied. \end{lemma} \begin{proof} The meaning of the phrase has a $k$-rational point'' is exactly that the structure morphism $f : X \to \Spec(k)$ has a section, which verifies the first condition. By Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety} we see that $k' = H^0(X, \mathcal{O}_X)$ is a field extension of $k$. Since $X$ has a $k$-rational point there is a $k$-algebra homomorphism $k' \to k$ and we conclude $k' = k$. Since $k$ is a field, any morphism $T \to \Spec(k)$ is flat. Hence we see by cohomology and base change (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) that $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism. This finishes the proof. \end{proof} \noindent Let $X$ be a a smooth projective curve over a field $k$ with a $k$-rational point $\sigma$. Then the functor $$\text{Pic}_{X/k, \sigma} : (\Sch/S)^{opp} \longrightarrow \textit{Ab},\quad T \longmapsto \Ker(\text{Pic}(X_T) \xrightarrow{\sigma_T^*} \text{Pic}(T))$$ is isomorphic to $\text{Pic}_{X/k}$ on $(\Sch/S)_{fppf}$ by Lemmas \ref{lemma-check-conditions} and \ref{lemma-flat-geometrically-connected-fibres-with-section}. Hence it will suffice to prove that $\text{Pic}_{X/k, \sigma}$ is representable. We will use the notation $\mathcal{L} \in \text{Pic}_{X/k, \sigma}(T)$'' to signify that $T$ is a scheme over $k$ and $\mathcal{L}$ is an invertible $\mathcal{O}_{X_T}$-module whose restriction to $T$ via $\sigma_T$ is isomorphic to $\mathcal{O}_T$. \begin{lemma} \label{lemma-define-open} Let $k$ be a field. Let $X$ be a smooth projective curve over $k$ with a $k$-rational point $\sigma$. For a scheme $T$ over $k$, consider the subset $F(T) \subset \text{Pic}_{X/k, \sigma}(T)$ consisting of $\mathcal{L}$ such that $Rf_{T, *}\mathcal{L}$ is isomorphic to an invertible $\mathcal{O}_T$-module placed in degree $0$. Then $F \subset \text{Pic}_{X/k, \sigma}$ is a subfunctor and the inclusion is representable by open immersions. \end{lemma} \begin{proof} Immediate from Derived Categories of Schemes, Lemma \ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric} applied with $i = 0$ and $r = 1$ and Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}. \end{proof} \noindent To continue it is convenient to make the following definition. \begin{definition} \label{definition-genus} Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. The {\it genus} of $X$ is $g = \dim_k H^1(X, \mathcal{O}_X)$. \end{definition} \begin{lemma} \label{lemma-open-representable} Let $k$ be a field. Let $X$ be a smooth projective curve of genus $g$ over $k$ with a $k$-rational point $\sigma$. The open subfunctor $F$ defined in Lemma \ref{lemma-define-open} is representable by an open subscheme of $\underline{\text{Hilb}}^g_{X/k}$. \end{lemma} \begin{proof} In this proof unadorned products are over $\Spec(k)$. By Proposition \ref{proposition-hilb-d} the scheme $H = \underline{\text{Hilb}}^g_{X/k}$ exists. Consider the universal divisor $D_{univ} \subset H \times X$ and the associated invertible sheaf $\mathcal{O}(D_{univ})$, see Remark \ref{remark-universal-object-hilb-d}. We adjust by tensoring with the pullback via $\sigma_H : H \to H \times X$ to get $$\mathcal{L}_H = \mathcal{O}(D_{univ}) \otimes_{\mathcal{O}_{H \times X}} \text{pr}_H^*\sigma_H^*\mathcal{O}(D_{univ})^{\otimes -1} \in \text{Pic}_{X/k, \sigma}(H)$$ By the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}) the invertible sheaf $\mathcal{L}_H$ defines a natural transformation $$h_H \longrightarrow \text{Pic}_{X/k, \sigma}$$ Because $F$ is an open subfuctor, there exists a maximal open $W \subset H$ such that $\mathcal{L}_H|_{W \times X}$ is in $F(W)$. Of course, this open is nothing else than the open subscheme constructed in Derived Categories of Schemes, Lemma \ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric} with $i = 0$ and $r = 1$ for the morphism $H \times X \to H$ and the sheaf $\mathcal{F} = \mathcal{O}(D_{univ})$. Applying the Yoneda lemma again we obtain a commutative diagram $$\xymatrix{ h_W \ar[d] \ar[r] & F \ar[d] \\ h_H \ar[r] & \text{Pic}_{X/k, \sigma} }$$ To finish the proof we will show that the top horizontal arrow is an isomorphism. \medskip\noindent Let $\mathcal{L} \in F(T) \subset \text{Pic}_{X/k, \sigma}(T)$. Let $\mathcal{N}$ be the invertible $\mathcal{O}_T$-module such that $Rf_{T, *}\mathcal{L} \cong \mathcal{N}[0]$. The adjunction map $$f_T^*\mathcal{N} \longrightarrow \mathcal{L} \quad\text{corresponds to a section }s\text{ of}\quad \mathcal{L} \otimes f_T^*\mathcal{N}^{\otimes -1}$$ on $X_T$. Claim: The zero scheme of $s$ is a relative effective Cartier divisor $D$ on $(T \times X)/T$ finite locally free of degree $g$ over $T$. \medskip\noindent Let us finish the proof of the lemma admitting the claim. Namely, $D$ defines a morphism $m : T \to H$ such that $D$ is the pullback of $D_{univ}$. Then $$(m \times \text{id}_X)^*\mathcal{O}(D_{univ}) \cong \mathcal{O}_{T \times X}(D)$$ Hence $(m \times \text{id}_X)^*\mathcal{L}_H$ and $\mathcal{O}(D)$ differ by the pullback of an invertible sheaf on $H$. This in particular shows that $m : T \to H$ factors through the open $W \subset H$ above. Moreover, it follows that these invertible modules define, after adjusting by pullback via $\sigma_T$ as above, the same element of $\text{Pic}_{X/k, \sigma}(T)$. Chasing diagrams using Yoneda's lemma we see that $m \in h_W(T)$ maps to $\mathcal{L} \in F(T)$. We omit the verification that the rule $F(T) \to h_W(T)$, $\mathcal{L} \mapsto m$ defines an inverse of the transformation of functors above. \medskip\noindent Proof of the claim. Since $D$ is a locally principal closed subscheme of $T \times X$, it suffices to show that the fibres of $D$ over $T$ are effective Cartier divisors, see Lemma \ref{lemma-divisors-on-curves} and Divisors, Lemma \ref{divisors-lemma-fibre-Cartier}. Because taking cohomology of $\mathcal{L}$ commutes with base change (Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}) we reduce to $T = \Spec(K)$ where $K/k$ is a field extension. Then $\mathcal{L}$ is an invertible sheaf on $X_K$ with $H^0(X_K, \mathcal{L}) = K$ and $H^1(X_K, \mathcal{L}) = 0$. Thus $$\deg(\mathcal{L}) = \chi(X_K, \mathcal{L}) - \chi(X_K, \mathcal{O}_{X_K}) = 1 - (1 - g) = g$$ See Varieties, Definition \ref{varieties-definition-degree-invertible-sheaf}. To finish the proof we have to show a nonzero section of $\mathcal{L}$ defines an effective Cartier divisor on $X_K$. This is clear. \end{proof} \begin{lemma} \label{lemma-twist-with-general-divisor} Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. Let $K/k$ be a field extension and let $\mathcal{L}$ be an invertible sheaf on $X_K$. Then there exists an invertible sheaf $\mathcal{L}_0$ on $X$ such that $\dim_K H^0(X_K, \mathcal{L} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}_0|_{X_K}) = 1$ and $\dim_K H^1(X_K, \mathcal{L} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}_0|_{X_K}) = 0$. \end{lemma} \begin{proof} This proof is a variant of the proof of Varieties, Lemma \ref{varieties-lemma-general-degree-g-line-bundle}. We encourage the reader to read that proof first. \medskip\noindent First we pick an ample invertible sheaf $\mathcal{L}_0$ and we replace $\mathcal{L}$ by $\mathcal{L} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}_0^{\otimes n}|_{X_K}$ for some $n \gg 0$. The result will be that we may assume that $H^0(X_K, \mathcal{L}) \not = 0$ and $H^1(X_K, \mathcal{L}) = 0$. Namely, we will get the vanishing by Cohomology of Schemes, Lemma \ref{coherent-lemma-vanshing-gives-ample} and the nonvanishing because the degree of the tensor product is $\gg 0$. We will finish the proof by descending induction on $t = \dim_K H^0(X_K, \mathcal{L})$. The base case $t = 1$ is trivial. Assume $t > 1$. \medskip\noindent Observe that for a closed and hence $k$-rational point $x$ of $X$, the inverse image $x_K$ is a $K$-rational point of $X_K$. Moreover, there are infinitely many $k$-rational points. Therefore the points $x_K$ form a Zariski dense collection of points of $X_K$. \medskip\noindent Let $s \in H^0(X_K, \mathcal{L})$ be nonzero. There exists an $x$ as above such that $s$ does not vanish in $x_K$. Let $\mathcal{I}$ be the ideal sheaf of $i : x_K \to X_K$ as in Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve}. Look at the short exact sequence $$0 \to \mathcal{I} \otimes_{\mathcal{O}_{X_K}} \mathcal{L} \to \mathcal{L} \to i_*i^*\mathcal{L} \to 0$$ Observe that $H^0(X_K, i_*i^*\mathcal{L}) = H^0(x_K, i^*\mathcal{L})$ has dimension $1$ over $K$. Since $s$ does not vanish at $x$ we conclude that $$H^0(X_K, \mathcal{L}) \longrightarrow H^0(X, i_*i^*\mathcal{L})$$ is surjective. Hence $\dim_K H^0(X_K, \mathcal{I} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}) = t - 1$. Finally, the long exact sequence of cohomology also shows that $H^1(X_K, \mathcal{I} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}) = 0$ thereby finishing the proof of the induction step. \end{proof} \begin{proposition} \label{proposition-pic-curve} Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. The Picard functor $\text{Pic}_{X/k}$ is representable. \end{proposition} \begin{proof} Since $k$ is algebraically closed there exists a rational point $\sigma$ of $X$. As discussed above, it suffices to show that the functor $\text{Pic}_{X/k, \sigma}$ classifying invertible modules trivial along $\sigma$ is representable. To do this we will check conditions (1), (2)(a), (2)(b), and (2)(c) of Lemma \ref{lemma-criterion}. \medskip\noindent The functor $\text{Pic}_{X/k, \sigma}$ satisfies the sheaf condition for the fppf topology because it is isomorphic to $\text{Pic}_{X/k}$. It would be more correct to say that we've shown the sheaf condition for $\text{Pic}_{X/k, \sigma}$ in the proof of Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section} which applies by Lemma \ref{lemma-check-conditions}. This proves condition (1) \medskip\noindent As our subfunctor we use $F$ as defined in Lemma \ref{lemma-define-open}. Condition (2)(b) follows. Condition (2)(a) is Lemma \ref{lemma-open-representable}. Condition (2)(c) is Lemma \ref{lemma-twist-with-general-divisor}. \end{proof} \noindent In fact, the proof given above produces more information which we collect here. \begin{lemma} \label{lemma-picard-pieces} Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. \begin{enumerate} \item $\underline{\text{Pic}}_{X/k}$ is a disjoint union of $g$-dimensional smooth proper varieties $\underline{\text{Pic}}^d_{X/k}$, \item $k$-points of $\underline{\text{Pic}}^d_{X/k}$ correspond to invertible $\mathcal{O}_X$-modules of degree $d$, \item $\underline{\text{Pic}}^0_{X/k}$ is an open and closed subgroup scheme, \item for $d \geq 0$ there is a canonical morphism $\gamma_d : \underline{\text{Hilb}}^d_{X/k} \to \underline{\text{Pic}}^d_{X/k}$ \item the morphisms $\gamma_d$ are surjective for $d \geq g$ and smooth for $d \geq 2g - 1$, \item the morphism $\underline{\text{Hilb}}^g_{X/k} \to \underline{\text{Pic}}^g_{X/k}$ is birational. \end{enumerate} \end{lemma} \begin{proof} Pick a $k$-rational point $\sigma$ of $X$. Recall that $\text{Pic}_{X/k}$ is isomorphic to the functor $\text{Pic}_{X/k, \sigma}$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-chi-locally-constant-geometric} for every $d \in \mathbf{Z}$ there is an open subfunctor $$\text{Pic}^d_{X/k, \sigma} \subset \text{Pic}_{X/k, \sigma}$$ whose value on a scheme $T$ over $k$ consists of those $\mathcal{L} \in \text{Pic}_{X/k, \sigma}(T)$ such that $\chi(X_t, \mathcal{L}_t) = d + 1 - g$ and moreover we have $$\text{Pic}_{X/k, \sigma} = \coprod\nolimits_{d \in \mathbf{Z}} \text{Pic}^d_{X/k, \sigma}$$ as fppf sheaves. It follows that the scheme $\underline{\text{Pic}}_{X/k}$ (which exists by Proposition \ref{proposition-pic-curve}) has a corresponding decomposition $$\underline{\text{Pic}}_{X/k, \sigma} = \coprod\nolimits_{d \in \mathbf{Z}} \underline{\text{Pic}}^d_{X/k, \sigma}$$ where the points of $\underline{\text{Pic}}^d_{X/k, \sigma}$ correspond to isomorphism classes of invertible modules of degree $d$ on $X$. \medskip\noindent Fix $d \geq 0$. There is a morphism $$\gamma_d : \underline{\text{Hilb}}^d_{X/k} \longrightarrow \underline{\text{Pic}}^d_{X/k}$$ coming from the invertible sheaf $\mathcal{O}(D_{univ})$ on $\underline{\text{Hilb}}^d_{X/k} \times_k X$ (Remark \ref{remark-universal-object-hilb-d}) by the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}). Our proof of the representability of the Picard functor of $X/k$ in Proposition \ref{proposition-pic-curve} and Lemma \ref{lemma-open-representable} shows that $\gamma_g$ induces an open immersion on a nonempty open of $\underline{\text{Hilb}}^g_{X/k}$. Moreover, the proof shows that the translates of this open by $k$-rational points of the group scheme $\underline{\text{Pic}}_{X/k}$ define an open covering. Since $\underline{\text{Hilb}}^g_{X/K}$ is smooth of dimension $g$ (Proposition \ref{proposition-hilb-d}) over $k$, we conclude that the group scheme $\underline{\text{Pic}}_{X/k}$ is smooth of dimension $g$ over $k$. \medskip\noindent By Groupoids, Lemma \ref{groupoids-lemma-group-scheme-over-field-separated} we see that $\underline{\text{Pic}}_{X/k}$ is separated. Hence, for every $d \geq 0$, the image of $\gamma_d$ is a proper variety over $k$ (Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-is-proper}). \medskip\noindent Let $d \geq g$. Then for any field extension $K/k$ and any invertible $\mathcal{O}_{X_K}$-module $\mathcal{L}$ of degree $d$, we see that $\chi(X_K, \mathcal{L}) = d + 1 - g > 0$. Hence $\mathcal{L}$ has a nonzero section and we conclude that $\mathcal{L} = \mathcal{O}_{X_K}(D)$ for some divisor $D \subset X_K$ of degree $d$. It follows that $\gamma_d$ is surjective. \medskip\noindent Combining the facts mentioned above we see that $\underline{\text{Pic}}^d_{X/k}$ is proper for $d \geq g$. This finishes the proof of (2) because now we see that $\underline{\text{Pic}}^d_{X/k}$ is proper for $d \geq g$ but then all $\underline{\text{Pic}}^d_{X/k}$ are proper by translation. \medskip\noindent It remains to prove that $\gamma_d$ is smooth for $d \geq 2g - 1$. Consider an invertible $\mathcal{O}_X$-module $\mathcal{L}$ of degree $d$. Then the fibre of the point corresponding to $\mathcal{L}$ is $$Z = \{D \subset X \mid \mathcal{O}_X(D) \cong \mathcal{L}\} \subset \underline{\text{Hilb}}^d_{X/k}$$ with its natural scheme structure. Since any isomorphism $\mathcal{O}_X(D) \to \mathcal{L}$ is well defined up to multiplying by a nonzero scalar, we see that the canonical section $1 \in \mathcal{O}_X(D)$ is mapped to a section $s \in \Gamma(X, \mathcal{L})$ well defined up to multiplication by a nonzero scalar. In this way we obtain a morphism $$Z \longrightarrow \text{Proj}(\text{Sym}(\Gamma(X, \mathcal{L})^*))$$ (dual because of our conventions). This morphism is an isomorphism, because given an section of $\mathcal{L}$ we can take the associated effective Cartier divisor, in other words we can construct an inverse of the displayed morphism; we omit the precise formulation and proof. Since $\dim H^0(X, \mathcal{L}) = d + 1 - g$ for every $\mathcal{L}$ of degree $d \geq 2g - 1$ by Varieties, Lemma \ref{varieties-lemma-vanishing-degree-2g-and-1-line-bundle} we see that $\text{Proj}(\text{Sym}(\Gamma(X, \mathcal{L})^*)) \cong \mathbf{P}^{d - g}_k$. We conclude that $\dim(Z) = \dim(\mathbf{P}^{d - g}_k) = d - g$. We conclude that the fibres of the morphism $\gamma_d$ all have dimension equal to the difference of the dimensions of $\underline{\text{Hilb}}^d_{X/k}$ and $\underline{\text{Pic}}^d_{X/k}$. It follows that $\gamma_d$ is flat, see Algebra, Lemma \ref{algebra-lemma-CM-over-regular-flat}. As moreover the fibres are smooth, we conclude that $\gamma_d$ is smooth by Morphisms, Lemma \ref{morphisms-lemma-smooth-flat-smooth-fibres}. \end{proof} \section{Some remarks on Picard groups} \label{section-remarks-picard} \noindent This section continues the discussion in Varieties, Section \ref{varieties-section-picard-groups} and will be continued in Algebraic Curves, Section \ref{curves-section-torsion-in-pic}. \begin{lemma} \label{lemma-pic-descends} Let $k$ be a field. Let $X$ be a quasi-compact and quasi-separated scheme over $k$ with $H^0(X, \mathcal{O}_X) = k$. If $X$ has a $k$-rational point, then for any Galois extension $k'/k$ we have $$\text{Pic}(X) = \text{Pic}(X_{k'})^{\text{Gal}(k'/k)}$$ Moreover the action of $\text{Gal}(k'/k)$ on $\text{Pic}(X_{k'})$ is continuous. \end{lemma} \begin{proof} Since $\text{Gal}(k'/k) = \text{Aut}(k'/k)$ it acts (from the right) on $\Spec(k')$, hence it acts (from the right) on $X_{k'} = X \times_{\Spec(k)} \Spec(k')$, and since $\text{Pic}(-)$ is a contravariant functor, it acts (from the left) on $\text{Pic}(X_{k'})$. If $k'/k$ is an infinite Galois extension, then we write $k' = \colim k'_\lambda$ as a filtered colimit of finite Galois extensions, see Fields, Lemma \ref{fields-lemma-infinite-galois-limit}. Then $X_{k'} = \lim X_{k_\lambda}$ (as in Limits, Section \ref{limits-section-limits}) and we obtain $$\text{Pic}(X_{k'}) = \colim \text{Pic}(X_{k_\lambda})$$ by Limits, Lemma \ref{limits-lemma-descend-invertible-modules}. Moreover, the transition maps in this system of abelian groups are injective by Varieties, Lemma \ref{varieties-lemma-change-fields-pic}. It follows that every element of $\text{Pic}(X_{k'})$ is fixed by one of the open subgroups $\text{Gal}(k'/k_\lambda)$, which exactly means that the action is continuous. Injectivity of the transition maps implies that it suffices to prove the statement on fixed points in the case that $k'/k$ is finite Galois. \medskip\noindent Assume $k'/k$ is finite Galois with Galois group $G = \text{Gal}(k'/k)$. Let $\mathcal{L}$ be an element of $\text{Pic}(X_{k'})$ fixed by $G$. We will use Galois descent (Descent, Lemma \ref{descent-lemma-galois-descent}) to prove that $\mathcal{L}$ is the pullback of an invertible sheaf on $X$. Recall that $f_\sigma = \text{id}_X \times \Spec(\sigma) : X_{k'} \to X_{k'}$ and that $\sigma$ acts on $\text{Pic}(X_{k'})$ by pulling back by $f_\sigma$. Hence for each $\sigma \in G$ we can choose an isomorphism $\varphi_\sigma : \mathcal{L} \to f_\sigma^*\mathcal{L}$ because $\mathcal{L}$ is a fixed by the $G$-action. The trouble is that we don't know if we can choose $\varphi_\sigma$ such that the cocycle condition $\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma$ holds. To see that this is possible we use that $X$ has a $k$-rational point $x \in X(k)$. Of course, $x$ similarly determines a $k'$-rational point $x' \in X_{k'}$ which is fixed by $f_\sigma$ for all $\sigma$. Pick a nonzero element $s$ in the fibre of $\mathcal{L}$ at $x'$; the fibre is the $1$-dimensional $k' = \kappa(x')$-vector space $$\mathcal{L}_{x'} \otimes_{\mathcal{O}_{X_{k'}, x'}} \kappa(x').$$ Then $f_\sigma^*s$ is a nonzero element of the fibre of $f_\sigma^*\mathcal{L}$ at $x'$. Since we can multiply $\varphi_\sigma$ by an element of $(k')^*$ we may assume that $\varphi_\sigma$ sends $s$ to $f_\sigma^*s$. Then we see that both $\varphi_{\sigma\tau}$ and $f_\sigma^*\varphi_\tau \circ \varphi_\sigma$ send $s$ to $f_{\sigma\tau}^*s = f_\tau^*f_\sigma^*s$. Since $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$ these two isomorphisms have to be the same (as one is a global unit times the other and they agree in $x'$) and the proof is complete. \end{proof} \begin{lemma} \label{lemma-torsion-descends} Let $k$ be a field of characteristic $p > 0$. Let $X$ be a quasi-compact and quasi-separated scheme over $k$ with $H^0(X, \mathcal{O}_X) = k$. Let $n$ be an integer prime to $p$. Then the map $$\text{Pic}(X)[n] \longrightarrow \text{Pic}(X_{k'})[n]$$ is bijective for any purely inseparable extension $k'/k$. \end{lemma} \begin{proof} First we observe that the map $\text{Pic}(X) \to \text{Pic}(X_{k'})$ is injective by Varieties, Lemma \ref{varieties-lemma-change-fields-pic}. Hence we have to show the map in the lemma is surjective. Let $\mathcal{L}$ be an invertible $\mathcal{O}_{X_{k'}}$-module which has order dividing $n$ in $\text{Pic}(X_{k'})$. Choose an isomorphism $\alpha : \mathcal{L}^{\otimes n} \to \mathcal{O}_{X_{k'}}$ of invertible modules. We will prove that we can descend the pair $(\mathcal{L}, \alpha)$ to $X$. \medskip\noindent Set $A = k' \otimes_k k'$. Since $k'/k$ is purely inseparable, the kernel of the multiplication map $A \to k'$ is a locally nilpotent ideal $I$ of $A$. Observe that $$X_A = X \times_{\Spec(k)} \Spec(A) = X_{k'} \times_X X_{k'}$$ comes with two projections $\text{pr}_i : X_A \to X_{k'}$, $i = 0, 1$ which agree over $A/I$. Hence the invertible modules $\mathcal{L}_i = \text{pr}_i^*\mathcal{L}$ agree over the closed subscheme $X_{A/I} = X_{k'}$. Since $X_{A/I} \to X_A$ is a thickening and since $\mathcal{L}_i$ are $n$-torsion, we see that there exists an isomorphism $\varphi : \mathcal{L}_0 \to \mathcal{L}_1$ by More on Morphisms, Lemma \ref{more-morphisms-lemma-torsion-pic-thickening}. We may pick $\varphi$ to reduce to the identity modulo $I$. Namely, $H^0(X, \mathcal{O}_X) = k$ implies $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} and $A \to k'$ is surjective hence we can adjust $\varphi$ by multiplying by a suitable element of $A$. Consider the map $$\lambda : \mathcal{O}_{X_A} \xrightarrow{\text{pr}_0^*\alpha^{-1}} \mathcal{L}_0^{\otimes n} \xrightarrow{\varphi^{\otimes n}} \mathcal{L}_1^{\otimes n} \xrightarrow{\text{pr}_0^*\alpha} \mathcal{O}_{X_A}$$ We can view $\lambda$ as an element of $A$ because $H^0(X_A, \mathcal{O}_{X_A}) = A$ (same reference as above). Since $\varphi$ reduces to the identity modulo $I$ we see that $\lambda = 1 \bmod I$. Then there is a unique $n$th root of $\lambda$ in $1 + I$ (Algebra, Lemma \ref{algebra-lemma-lift-nth-roots}) and after multiplying $\varphi$ by its inverse we get $\lambda = 1$. We claim that $(\mathcal{L}, \varphi)$ is a descent datum for the fpqc covering $\{X_{k'} \to X\}$ (Descent, Definition \ref{descent-definition-descent-datum-quasi-coherent}). If true, then $\mathcal{L}$ is the pullback of an invertible $\mathcal{O}_X$-module $\mathcal{N}$ by Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. Injectivity of the map on Picard groups shows that $\mathcal{N}$ is a torsion element of $\text{Pic}(X)$ of the same order as $\mathcal{L}$. \medskip\noindent Proof of the claim. To see this we have to verify that $$\text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi = \text{pr}_{02}^*\varphi \quad\text{on}\quad X_{k'} \times_X X_{k'} \times_X X_{k'} = X_{k' \otimes_k k' \otimes_k k'}$$ As before the diagonal morphism $\Delta : X_{k'} \to X_{k' \otimes_k k' \otimes_k k'}$ is a thickening. The left and right hand sides of the equality signs are maps $a, b : p_0^*\mathcal{L} \to p_2^*\mathcal{L}$ compatible with $p_0^*\alpha$ and $p_2^*\alpha$ where $p_i : X_{k' \otimes_k k' \otimes_k k'} \to X_{k'}$ are the projection morphisms. Finally, $a, b$ pull back to the same map under $\Delta$. Affine locally (in local trivializations) this means that $a, b$ are given by multiplication by invertible functions which reduce to the same function modulo a locally nilpotent ideal and which have the same $n$th powers. Then it follows from Algebra, Lemma \ref{algebra-lemma-lift-nth-roots} that these functions are the same. \end{proof} \input{chapters} \bibliography{my} \bibliographystyle{amsalpha} \end{document}