Picard Schemes of Curves

Introduction

In this chapter we do just enough work to construct the Picard scheme of a projective nonsingular curve over an algebraically closed field. See [Kleiman-Picard] for a more thorough discussion as well as historical background.

Later in the Stacks project we will discuss Hilbert and Quot functors in much greater generality.

Hilbert scheme of points

Let \(X \to S\) be a morphism of schemes. Let \(d \geq 0\) be an integer. For a scheme \(T\) over \(S\) we let \[\Hilbfunctor^d_{X/S}(T) = \left\{ \begin{matrix} Z \subset X_T\text{ closed subscheme such that }\\ Z \to T\text{ is finite locally free of degree }d \end{matrix} \right\}\] If \(T' \to T\) is a morphism of schemes over \(S\) and if \(Z \in \Hilbfunctor^d_{X/S}(T)\), then the base change \(Z_{T'} \subset X_{T'}\) is an element of \(\Hilbfunctor^d_{X/S}(T')\). In this way we obtain a functor \[\Hilbfunctor^d_{X/S} : (\Sch/S)^{opp} \longrightarrow \textit{Sets},\quad T \longrightarrow \Hilbfunctor^d_{X/S}(T)\] In general \(\Hilbfunctor^d_{X/S}\) is an algebraic space (insert future reference here). In this section we will show that \(\Hilbfunctor^d_{X/S}\) 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 \(\Hilbfunctor^d_{X/S}\) is representable by a scheme, we often denote this scheme by \(\underline{\Hilbfunctor}^d_{X/S}\).

Lemma

Let \(X \to S\) be a morphism of schemes. The functor \(\Hilbfunctor^d_{X/S}\) satisfies the sheaf property for the fpqc topology (Topologies, Definition 022G).

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 022D) and that \(X_{T_i \times_T T_{i'}} = X_i \times_{X_T} X_{i'}\). Suppose that \(Z_i \in \Hilbfunctor^d_{X/S}(T_i)\) is a collection of elements such that \(Z_i\) and \(Z_{i'}\) map to the same element of \(\Hilbfunctor^d_{X/S}(T_i \times_T T_{i'})\). By effective descent for closed immersions (Descent, Lemma 03I0) 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 02VO.

Lemma

Let \(X \to S\) be a morphism of schemes. If \(X \to S\) is of finite presentation, then the functor \(\Hilbfunctor^d_{X/S}\) is limit preserving (Limits, Remark 05LX).

Proof

Let \(T = \lim T_i\) be a limit of affine schemes over \(S\). We have to show that \(\Hilbfunctor^d_{X/S}(T) = \colim \Hilbfunctor^d_{X/S}(T_i)\). Observe that if \(Z \to X_T\) is an element of \(\Hilbfunctor^d_{X/S}(T)\), then \(Z \to T\) is of finite presentation. Hence by Limits, Lemma 01ZM 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 01ZP to see that we may assume \(Z_i \to X_{T_i}\) is a closed immersion after increasing \(i\). We apply Limits, Lemma 06AC to see that \(Z_i \to T_i\) is finite locally free of degree \(d\) after possibly increasing \(i\). Then \(Z_i \in \Hilbfunctor^d_{X/S}(T_i)\) as desired.

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 \[\Hilbfunctor^d_{X/S} \longrightarrow \Hilbfunctor^d_{Y/S}\] which maps an element \(Z \in \Hilbfunctor^d_{X/S}(T)\) to \(i_T(Z) \subset Y_T\) in \(\Hilbfunctor^d_{Y/S}\). Here \(i_T : X_T \to Y_T\) is the base change of \(i\).

Lemma

Let \(S\) be a scheme. Let \(i : X \to Y\) be a closed immersion of schemes. If \(\Hilbfunctor^d_{Y/S}\) is representable by a scheme, so is \(\Hilbfunctor^d_{X/S}\) and the corresponding morphism of schemes \(\underline{\Hilbfunctor}^d_{X/S} \to \underline{\Hilbfunctor}^d_{Y/S}\) is a closed immersion.

Proof

Let \(T\) be a scheme over \(S\) and let \(Z \in \Hilbfunctor^d_{Y/S}(T)\). 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 \(\Hilbfunctor^d_{Y/S}\) and the universal object¹ \(Z_{univ} \in \Hilbfunctor^d_{Y/S}(T_{univ})\) 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 \(\Hilbfunctor^d_{X/S}(T_{univ, X})\). A formal argument then shows that \(T_{univ, X}\) is a scheme representing \(\Hilbfunctor^d_{X/S}\) with universal object \(Z_{univ, X}\).

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 07AI. 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 \(\Spec(A/(g_\beta^j))\).

Lemma

Let \(X \to S\) be a morphism of schemes. If \(X \to S\) is separated and \(\Hilbfunctor^d_{X/S}\) is representable, then \(\underline{\Hilbfunctor}^d_{X/S} \to S\) is separated.

Proof

In this proof all unadorned products are over \(S\). Let \(H = \underline{\Hilbfunctor}^d_{X/S}\) and let \(Z \in \Hilbfunctor^d_{X/S}(H)\) be the universal object. Consider the two objects \(Z_1, Z_2 \in \Hilbfunctor^d_{X/S}(H \times H)\) 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 \(\Hilbfunctor^d_{X/S}\), 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 \(\Hilbfunctor^d_{X/S}(T)\). 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 07AI that \(\Delta\) is a closed immersion. In the proof of Lemma 0B97 the reader finds an alternative easier proof of the needed result in our special case.

Lemma

Let \(X \to S\) be a morphism of affine schemes. Let \(d \geq 0\). Then \(\Hilbfunctor^d_{X/S}\) is representable.

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 0B97 we may assume that \(X = \Spec(A)\) where \(A = R[x_i; i \in I]\). We will use Schemes, Lemma 01JJ to prove the lemma in this case.

Condition (1) of the lemma follows from Lemma 0B95.

For every subset \(W \subset A\) of cardinality \(d\) we will construct a subfunctor \(F_W\) of \(\Hilbfunctor^d_{X/S}\). (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 \(Z \in \Hilbfunctor^d_{X/S}(T)\) 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 \(Z \in \Hilbfunctor^d_{X/S}(T)\) we denote \(f|_Z\) the pullback of \(f\) by the morphism \(Z \to X_T \to X\).

Openness, i.e., condition (2)(b) of the lemma. This follows from Algebra, Lemma 00O0.

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 \(Z_V \in F_W(V)\).

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 through 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 \(T_{univ} = \Spec(R_{univ})\) as follows. First, we set \[R_{univ} = R[c_{kl}^m, b^l, b_i^l]/\mathfrak a_{univ}\] for some ideal \(\mathfrak a_{univ}\) which we describe below. Here the indices \(k, l, m, i\) are all ranging through \(\{1, \ldots, d\}\). Then we set \[B_{univ} = R_{univ}e_1 \oplus R_{univ}e_2 \oplus \ldots \oplus R_{univ}e_d\] as an \(R_{univ}\)-module with algebra structure given by the rule \[e_ke_l = \sum c_{kl}^m e_m\] The closed immersion \(Z_{univ} \to X_{T_{univ}}\) is given by the \(R_{univ}\)-algebra map \[\Psi : A \otimes_R R_{univ} \longrightarrow B_{univ}\] mapping \(1 \otimes 1\) to \(\sum b^le_l\) and \(x_i\) to \(\sum b_i^le_l\) (for existence see below). As for the ideal \(\mathfrak a_{univ}\) we have the following prescription:

1. multiplication on \(B_{univ}\) is commutative, i.e., we should have \(c_{lk}^m - c_{kl}^m \in \mathfrak a_{univ}\),

2. multiplication on \(B_{univ}\) is associative, i.e., we should have \(c_{lk}^m c_{m n}^p - c_{lq}^p c_{kn}^q \in \mathfrak a_{univ}\),

3. \(\sum b^le_l\) 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).

Since we’ve guaranteed that \(B_{univ}\) is a commutative \(R_{univ}\)-algebra with \(1\) given by \(\sum b^le_l\) the homomorphism \(\Psi\) exists. The final condition is that

1. \(f_l\) to map to \(e_l\) in \(B_{univ}\). Write \(\Psi(f_l) - e_l \equiv \sum h_l^me_m\) for some \(h_l^m \in R[c_{kl}^m, b^l, b_i^l]\) then we should have \(h_l^m \in \mathfrak a_{univ}\).

With \(\mathfrak a_{univ} \subset R[c_{kl}^m, b^l, b_i^l]\) generated by the elements listed in (1) – (5) it is clear that \(F_W\) is represented by \(\Spec(R_{univ})\).

Proposition

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 \(\Hilbfunctor^d_{X/S}\) is representable by a scheme.

Proof

Either using relative glueing (Constructions, Section 01LG) or using the functorial point of view (Schemes, Lemma 01JJ) we reduce to the case where \(S\) is affine. Details omitted.

Assume \(S\) is affine. For \(U \subset X\) affine open, denote \(F_U \subset \Hilbfunctor^d_{X/S}\) the subfunctor such that for a scheme \(T/S\) an element \(Z \in \Hilbfunctor^d_{X/S}(T)\) is in \(F_U(T)\) if and only if \(Z \subset U_T\). We will use Schemes, Lemma 01JJ and the subfunctors \(F_U\) to conclude.

Condition (1) is Lemma 0B95.

Condition (2)(a) follows from the fact that \(F_U = \Hilbfunctor^d_{U/S}\) and that this is representable by Lemma 0B99. Namely, if \(Z \in F_U(T)\), then \(Z\) can be viewed as a closed subscheme of \(U_T\) which is finite locally free of degree \(d\) over \(T\) and hence \(Z \in \Hilbfunctor^d_{U/S}(T)\). Conversely, if \(Z \in \Hilbfunctor^d_{U/S}(T)\) then \(Z \to U_T \to X_T\) is a closed immersion² and we may view \(Z\) as an element of \(F_U(T)\).

Let \(Z \in \Hilbfunctor^d_{X/S}(T)\) 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).

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 \(T \to S\).

Remark

Let \(f : X \to S\) be a morphism of schemes. The assumption of Proposition 0B9A and hence the conclusion holds in each of the following cases:

1. \(X\) is quasi-affine,

2. \(f\) is quasi-affine,

3. \(f\) is quasi-projective,

4. \(f\) is locally projective,

5. there exists an ample invertible sheaf on \(X\),

6. there exists an \(f\)-ample invertible sheaf on \(X\), and

7. there exists an \(f\)-very ample invertible sheaf on \(X\).

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 01ZY.

Moduli of divisors on smooth curves

For a smooth morphism \(X \to S\) of relative dimension \(1\) the functor \(\Hilbfunctor^d_{X/S}\) parametrizes relative effective Cartier divisors as defined in Divisors, Section 056P.

Lemma

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

1. \(D \to S\) is finite locally free,

2. \(D\) is a relative effective Cartier divisor on \(X/S\),

3. \(D \to S\) is locally quasi-finite, flat, and locally of finite presentation, and

4. \(D \to S\) is locally quasi-finite and flat.

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.

Proof

Equivalence of (2) and (3). This follows from Divisors, Lemma 062Y 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 056S). Thus \(\mathcal{O}_{X, x}\) is a discrete valuation ring (Algebra, Lemma 00PD) 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 0AG8). 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.

If \(S\) is Noetherian, then any locally quasi-finite morphism \(D \to S\) is locally of finite presentation (Morphisms, Lemma 01TX), whence (3) is equivalent to (4).

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 02LS) and a finite, flat, and finitely presented morphism is finite locally free (Morphisms, Lemma 02KB), we see that (1) is equivalent to (2).

Lemma

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 \(S\).

Proof

By Lemma 0B9D 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 0B8U. Hence \(D \to S\) is locally quasi-finite, flat, and locally of finite presentation by Lemma 0B9D. Applying Morphisms, Lemma 09MQ the surjective integral morphism \(D_1 \amalg D_2 \to D\) we find that \(D \to S\) is separated. Then Morphisms, Lemma 03GN implies that \(D \to S\) is proper. This implies that \(D \to S\) is finite (More on Morphisms, Lemma 02LS) and in turn we see that \(D \to S\) is finite locally free (Morphisms, Lemma 02KB). 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 02MC).

Lemma

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 \(D_2 = D_1 + D\).

Proof

This proof is almost exactly the same as the proof of Lemma 0B9E. By Lemma 0B9D we see that \(D_1\) and \(D_2\) are relative effective Cartier divisors on \(X/S\). By Divisors, Lemma 0B8V 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 0B9D. 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 02KB). Thus it suffice to show that the degree of \(D \to S\) is \(d_2 - d_1\). This follows from Lemma 0B9E.

Let \(X \to S\) be a smooth morphism of schemes of relative dimension \(1\). By Lemma 0B9D for a scheme \(T\) over \(S\) and \(D \in \Hilbfunctor^d_{X/S}(T)\), 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 0B9E we obtain a transformation of functors \[\Hilbfunctor^{d_1}_{X/S} \times \Hilbfunctor^{d_2}_{X/S} \longrightarrow \Hilbfunctor^{d_1 + d_2}_{X/S},\quad (D_1, D_2) \longmapsto D_1 + D_2\] If \(\Hilbfunctor^d_{X/S}\) is representable for all degrees \(d\), then this transformation of functors corresponds to a morphism of schemes \[\underline{\Hilbfunctor}^{d_1}_{X/S} \times_S \underline{\Hilbfunctor}^{d_2}_{X/S} \longrightarrow \underline{\Hilbfunctor}^{d_1 + d_2}_{X/S}\] over \(S\). Observe that \(\underline{\Hilbfunctor}^0_{X/S} = S\) and \(\underline{\Hilbfunctor}^1_{X/S} = X\). A special case of the morphism above is the morphism \[\underline{\Hilbfunctor}^d_{X/S} \times_S X \longrightarrow \underline{\Hilbfunctor}^{d + 1}_{X/S},\quad (D, x) \longmapsto D + x\]

Lemma

Let \(X \to S\) be a smooth morphism of schemes of relative dimension \(1\) such that the functors \(\Hilbfunctor^d_{X/S}\) are representable. The morphism \(\underline{\Hilbfunctor}^d_{X/S} \times_S X \to \underline{\Hilbfunctor}^{d + 1}_{X/S}\) is finite locally free of degree \(d + 1\).

Proof

Let \(D_{univ} \subset X \times_S \underline{\Hilbfunctor}^{d + 1}_{X/S}\) be the universal object. There is a commutative diagram \[\xymatrix{ \underline{\Hilbfunctor}^d_{X/S} \times_S X \ar[rr] \ar[rd] & & D_{univ} \ar[ld] \ar@{^{(}->}[r] & \underline{\Hilbfunctor}^{d + 1}_{X/S} \times_S X \\ & \underline{\Hilbfunctor}^{d + 1}_{X/S} }\] 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 0B9F 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.

Lemma

Let \(X \to S\) be a smooth morphism of schemes of relative dimension \(1\) such that the functors \(\Hilbfunctor^d_{X/S}\) are representable. The schemes \(\underline{\Hilbfunctor}^d_{X/S}\) are smooth over \(S\) of relative dimension \(d\).

Proof

We have \(\underline{\Hilbfunctor}^0_{X/S} = S\) and \(\underline{\Hilbfunctor}^1_{X/S} = X\) thus the result is true for \(d = 0, 1\). Assuming the result for \(d\), we see that \(\underline{\Hilbfunctor}^d_{X/S} \times_S X\) is smooth over \(S\) (Morphisms, Lemma 01VB and 01VA). Since \(\underline{\Hilbfunctor}^d_{X/S} \times_S X \to \underline{\Hilbfunctor}^{d + 1}_{X/S}\) is finite locally free of degree \(d + 1\) by Lemma 0B9G the result follows from Descent, Lemma 05B5. 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).

We collect all the information obtained sofar in the case of a proper smooth curve over a field.

Proposition

Let \(X\) be a geometrically irreducible smooth proper curve over a field \(k\).

1. The functors \(\Hilbfunctor^d_{X/k}\) are representable by smooth proper varieties \(\underline{\Hilbfunctor}^d_{X/k}\) of dimension \(d\) over \(k\).

2. For a field extension \(k'/k\) the \(k'\)-rational points of \(\underline{\Hilbfunctor}^d_{X/k}\) are in \(1\)-to-\(1\) bijection with effective Cartier divisors of degree \(d\) on \(X_{k'}\).

3. For \(d_1, d_2 \geq 0\) there is a morphism \[\underline{\Hilbfunctor}^{d_1}_{X/k} \times_k \underline{\Hilbfunctor}^{d_2}_{X/k} \longrightarrow \underline{\Hilbfunctor}^{d_1 + d_2}_{X/k}\] which is finite locally free of degree \({d_1 + d_2 \choose d_1}\).

Proof

The functors \(\Hilbfunctor^d_{X/k}\) are representable by Proposition 0B9A (see also Remark 0B9B) and the fact that \(X\) is projective (Varieties, Lemma 0A26). The schemes \(\underline{\Hilbfunctor}^d_{X/k}\) are separated over \(k\) by Lemma 0B98. The schemes \(\underline{\Hilbfunctor}^d_{X/k}\) are smooth over \(k\) by Lemma 0B9H. Starting with \(X = \underline{\Hilbfunctor}^1_{X/k}\), the morphisms of Lemma 0B9G, and induction we find a morphism \[X^d = X \times_k X \times_k \ldots \times_k X \longrightarrow \underline{\Hilbfunctor}^d_{X/k},\quad (x_1, \ldots, x_d) \longrightarrow x_1 + \ldots + x_d\] which is finite locally free of degree \(d!\). Since \(X\) is proper over \(k\), so is \(X^d\), hence \(\underline{\Hilbfunctor}^d_{X/k}\) is proper over \(k\) by Morphisms, Lemma 03GN. Since \(X\) is geometrically irreducible over \(k\), the product \(X^d\) is irreducible (Varieties, Lemma 038F) 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 \[\xymatrix{ X^{d_1} \times_k X^{d_2} \ar[d] \ar@{=}[r] & X^{d_1 + d_2} \ar[d] \\ \underline{\Hilbfunctor}^{d_1}_{X/k} \times_k \underline{\Hilbfunctor}^{d_2}_{X/k} \ar[r] & \underline{\Hilbfunctor}^{d_1 + d_2}_{X/k} }\] and multiplicativity of degrees of finite locally free morphisms.

Remark

Let \(X\) be a geometrically irreducible smooth proper curve over a field \(k\) as in Proposition 0B9I. Let \(d \geq 0\). The universal closed object is a relatively effective divisor \[D_{univ} \subset \underline{\Hilbfunctor}^{d + 1}_{X/k} \times_k X\] over \(\underline{\Hilbfunctor}^{d + 1}_{X/k}\) by Lemma 0B9D. In fact, \(D_{univ}\) is isomorphic as a scheme to \(\underline{\Hilbfunctor}^d_{X/k} \times_k X\), see proof of Lemma 0B9G. In particular, \(D_{univ}\) is an effective Cartier divisor and we obtain an invertible module \(\mathcal{O}(D_{univ})\). If \([D] \in \underline{\Hilbfunctor}^{d + 1}_{X/k}\) denotes the \(k\)-rational point corresponding to the effective Cartier divisor \(D \subset X\) of degree \(d + 1\), then the restriction of \(\mathcal{O}(D_{univ})\) to the fibre \([D] \times X\) is \(\mathcal{O}_X(D)\).

The Picard functor

Given any scheme \(X\) we denote \(\Pic(X)\) the set of isomorphism classes of invertible \(\mathcal{O}_X\)-modules. See Modules, Definition 01CX. Given a morphism \(f : X \to Y\) of schemes, pullback defines a group homomorphism \(\Pic(Y) \to \Pic(X)\). The assignment \(X \leadsto \Pic(X)\) 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.

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.

Definition

Let \(\Sch_{fppf}\) be a big site as in Topologies, Definition 021S. Let \(f : X \to S\) be a morphism of this site. The Picard functor \(\Picardfunctor_{X/S}\) is the fppf sheafification of the functor \[(\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad T \longmapsto \Pic(X_T)\] If this functor is representable, then we denote \(\underline{\Picardfunctor}_{X/S}\) a scheme representing it.

Remark

There are three related objects in Definition 0B9L. First, the Picard stack remembers invertible modules and their isomorphisms; see Examples of Stacks, Section 0372. Second, taking isomorphism classes in every fibre gives the presheaf \[P(T) = \Pic(X_T).\] Third, the Picard functor is the associated fppf sheaf \(P^\#\). The second and third objects need not agree. Indeed, equality of two classes over an overlap only records the existence of an isomorphism between the corresponding invertible modules. It does not record a choice of isomorphism satisfying the cocycle condition; the automorphisms retained by the Picard stack have been forgotten. This also explains the relation with the Picard stack given in Quot and Hilbert Spaces, Lemma 0D26.

By Sites, Lemma 00WK, a section of \(\Picardfunctor_{X/S}\) over \(T\) can be represented by an fppf covering \(\{T_i \to T\}\) and classes \[\xi_i \in \Pic(X_{T_i})\] such that, for every \(i,j\), the two pullbacks of \(\xi_i\) and \(\xi_j\) become equal over an fppf covering of \(T_i \times_T T_j\). Two such presentations determine the same section precisely when their pullbacks become equal after passing to a common refinement and covering once more. Thus sheafification imposes local equality of classes; it does not assert that a section is represented by a single invertible module on \(X_T\).

Historically, [FGA, Exposé 232, Section 1] first considered the intermediate presheaf \[P'(T) = H^0(T, R^1f_{T, *}\mathcal{O}_{X_T}^*)\] in the topology used there and called it the restricted relative Picard functor. In that setting the map \(P \to P'\) is asserted to induce an isomorphism on the associated sheaves for faithfully flat quasi-compact coverings. The intermediate presheaf \(P'\) should not be confused with the Picard functor \(P^\#\) of Definition 0B9L. For example, the Brauer–Severi example in the cited section shows that \(P'\) may fail to be representable even when the sheafified functor has a natural Picard scheme.

If an intermediate presheaf such as \(P'\) is representable, then it already satisfies fpqc descent; see Ètale Cohomology, Lemma 03O3. Hence, whenever its associated sheaf agrees with \(P^\#\), it represents the same Picard functor. In general, representability of Definition 0B9L means exactly that there is a scheme \(Q\) over \(S\) and functorial isomorphisms \[\Mor_S(T, Q) = \Picardfunctor_{X/S}(T)\] for all \(T\) over \(S\).

An often used remark is that if \(T \in \Ob((\Sch/S)_{fppf})\), then \(\Picardfunctor_{X_T/T}\) is the restriction of \(\Picardfunctor_{X/S}\) to \((\Sch/T)_{fppf}\). It turns out to be nontrivial to see what the value of \(\Picardfunctor_{X/S}\) is on schemes \(T\) over \(S\). Here is a lemma that helps with this task.

Lemma

Let \(f : X \to S\) be as in Definition 0B9L. If \(\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}\) is an isomorphism for all \(T \in \Ob((\Sch/S)_{fppf})\), then \[0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/S}(T)\] is an exact sequence for all \(T\).

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.

Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module which is in the kernel of \(\Pic(X) \to \Picardfunctor_{X/S}(S)\). 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 023T 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).

Lemma

Let \(f : X \to S\) be as in Definition 0B9L. 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 \[0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/S}(T) \to 0\] is a split exact sequence with splitting given by \(\sigma_T^* : \Pic(X_T) \to \Pic(T)\).

Proof

Denote \(K(T) = \Ker(\sigma_T^* : \Pic(X_T) \to \Pic(T))\). Since \(\sigma\) is a section of \(f\) we see that \(\Pic(X_T)\) is the direct sum of \(\Pic(T)\) and \(K(T)\). Thus by Lemma 0B9M we see that \(K(T) \subset \Picardfunctor_{X/S}(T)\) for all \(T\). Moreover, it is clear from the construction that \(\Picardfunctor_{X/S}\) 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.

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 023T 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.

A representability criterion

To prove the Picard functor is representable we will use the following criterion.

Lemma

Let \(k\) be a field. Let \(G : (\Sch/k)^{opp} \to \textit{Groups}\) be a functor. With terminology as in Schemes, Definition 01JI, assume that

1. \(G\) satisfies the sheaf property for the Zariski topology,

2. there exists a subfunctor \(F \subset G\) such that

   1. \(F\) is representable,

   2. \(F \subset G\) is representable by open immersion,

   3. for every field extension \(K\) of \(k\) and \(g \in G(K)\) there exists a \(g' \in G(k)\) such that \(g'g \in F(K)\).

Then \(G\) is representable by a group scheme over \(k\).

Proof

This follows from Schemes, Lemma 01JJ. 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.

The Picard scheme of a curve

In this section we will apply Lemma 0B9Q to show that \(\Picardfunctor_{X/k}\) 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.

Lemma

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 0B9N are satisfied.

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 04L2 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 02KH) that \(\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}\) is an isomorphism. This finishes the proof.

Let \(X\) be a smooth projective curve over a field \(k\) with a \(k\)-rational point \(\sigma\). Then the functor \[\Picardfunctor_{X/k, \sigma} : (\Sch/k)^{opp} \longrightarrow \textit{Ab},\quad T \longmapsto \Ker(\Pic(X_T) \xrightarrow{\sigma_T^*} \Pic(T))\] is isomorphic to \(\Picardfunctor_{X/k}\) on \((\Sch/k)_{fppf}\) by Lemmas 0B9U and 0B9N. Hence it will suffice to prove that \(\Picardfunctor_{X/k, \sigma}\) is representable. We will use the notation “\(\mathcal{L} \in \Picardfunctor_{X/k, \sigma}(T)\)” 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 \(\mathcal{O}_T\).

Lemma

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 \(F(T) \subset \Picardfunctor_{X/k, \sigma}(T)\) consisting of \(\mathcal{L}\) such that \(Rf_{T, *}\mathcal{L}\) is isomorphic to an invertible \(\mathcal{O}_T\)-module placed in degree \(0\). Then \(F \subset \Picardfunctor_{X/k, \sigma}\) is a subfunctor and the inclusion is representable by open immersions.

Proof

Immediate from Derived Categories of Schemes, Lemma 0B9S applied with \(i = 0\) and \(r = 1\) and Schemes, Definition 01JI.

To continue it is convenient to make the following definition.

Definition

Let \(k\) be a field. Let \(X\) be a smooth projective geometrically irreducible curve over \(k\). The genus of \(X\) is \(g = \dim_k H^1(X, \mathcal{O}_X)\).

Lemma

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 0B9V is representable by an open subscheme of \(\underline{\Hilbfunctor}^g_{X/k}\).

Proof

In this proof unadorned products are over \(\Spec(k)\). By Proposition 0B9I the scheme \(H = \underline{\Hilbfunctor}^g_{X/k}\) exists. Consider the universal divisor \(D_{univ} \subset H \times X\) and the associated invertible sheaf \(\mathcal{O}(D_{univ})\), see Remark 0B9J. We adjust by tensoring with the pullback via \(\sigma_H : H \to H \times X\) to get \[\mathcal{L}_H = \mathcal{O}(D_{univ}) \otimes_{\mathcal{O}_{H \times X}} \text{pr}_H^*\sigma_H^*\mathcal{O}(D_{univ})^{\otimes -1} \in \Picardfunctor_{X/k, \sigma}(H)\] By the Yoneda lemma (Categories, Lemma 001P) the invertible sheaf \(\mathcal{L}_H\) defines a natural transformation \[h_H \longrightarrow \Picardfunctor_{X/k, \sigma}\] 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 0B9S 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 \[\xymatrix{ h_W \ar[d] \ar[r] & F \ar[d] \\ h_H \ar[r] & \Picardfunctor_{X/k, \sigma} }\] To finish the proof we will show that the top horizontal arrow is an isomorphism.

Let \(\mathcal{L} \in F(T) \subset \Picardfunctor_{X/k, \sigma}(T)\). 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 \(T\).

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 \[(m \times \text{id}_X)^*\mathcal{O}(D_{univ}) \cong \mathcal{O}_{T \times X}(D)\] Since \(\mathcal{O}_{T \times X}(D) \cong \mathcal{L} \otimes f_T^*\mathcal{N}^{\otimes -1}\) we conclude that \((m \times \text{id}_X)^*\mathcal{L}_H\) and \(\mathcal{L}\) differ by the pullback of an invertible module on \(T\). This 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 \(\Picardfunctor_{X/k, \sigma}(T)\). 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.

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 0B9D and Divisors, Lemma 062Y. Because taking cohomology of \(\mathcal{L}\) commutes with base change (Derived Categories of Schemes, Lemma 0B91) 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 0AYR. 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.

Lemma

Let \(k\) be a separably 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 \(\dim_K H^1(X_K, \mathcal{L} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}_0|_{X_K}) = 0\).

Proof

This proof is a variant of the proof of Varieties, Lemma 0B8Z. We encourage the reader to read that proof first.

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 0B5U 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 \(t > 1\).

Observe that for a \(k\)-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 by Varieties, Lemma 056U. Therefore the points \(x_K\) form a Zariski dense collection of points of \(X_K\).

Let \(s \in H^0(X_K, \mathcal{L})\) be nonzero. From the previous paragraph we deduce there exists a \(k\)-rational point \(x\) 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 0B8Y. 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.

Proposition

Let \(k\) be a separably closed field. Let \(X\) be a smooth projective curve over \(k\). The Picard functor \(\Picardfunctor_{X/k}\) is representable.

Proof

Since \(k\) is separably closed there exists a \(k\)-rational point \(\sigma\) of \(X\), see Varieties, Lemma 056U. As discussed above, it suffices to show that the functor \(\Picardfunctor_{X/k, \sigma}\) 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 0B9Q.

The functor \(\Picardfunctor_{X/k, \sigma}\) satisfies the sheaf condition for the fppf topology because it is isomorphic to \(\Picardfunctor_{X/k}\). It would be more correct to say that we’ve shown the sheaf condition for \(\Picardfunctor_{X/k, \sigma}\) in the proof of Lemma 0B9N which applies by Lemma 0B9U. This proves condition (1)

As our subfunctor we use \(F\) as defined in Lemma 0B9V. Condition (2)(b) follows. Condition (2)(a) is Lemma 0B9X. Condition (2)(c) is Lemma 0B9Y.

In fact, the proof given above produces more information which we collect here.

Lemma

Let \(k\) be a separably closed field. Let \(X\) be a smooth projective curve of genus \(g\) over \(k\).

1. \(\underline{\Picardfunctor}_{X/k}\) is a disjoint union of \(g\)-dimensional smooth proper varieties \(\underline{\Picardfunctor}^d_{X/k}\),

2. \(k\)-points of \(\underline{\Picardfunctor}^d_{X/k}\) correspond to invertible \(\mathcal{O}_X\)-modules of degree \(d\),

3. \(\underline{\Picardfunctor}^0_{X/k}\) is an open and closed subgroup scheme,

4. for \(d \geq 0\) there is a canonical morphism \(\gamma_d : \underline{\Hilbfunctor}^d_{X/k} \to \underline{\Picardfunctor}^d_{X/k}\)

5. the morphisms \(\gamma_d\) are surjective for \(d \geq g\) and smooth for \(d \geq 2g - 1\),

6. the morphism \(\underline{\Hilbfunctor}^g_{X/k} \to \underline{\Picardfunctor}^g_{X/k}\) is birational.

Proof

Pick a \(k\)-rational point \(\sigma\) of \(X\). Recall that \(\Picardfunctor_{X/k}\) is isomorphic to the functor \(\Picardfunctor_{X/k, \sigma}\). By Derived Categories of Schemes, Lemma 0B9T for every \(d \in \mathbf{Z}\) there is an open subfunctor \[\Picardfunctor^d_{X/k, \sigma} \subset \Picardfunctor_{X/k, \sigma}\] whose value on a scheme \(T\) over \(k\) consists of those \(\mathcal{L} \in \Picardfunctor_{X/k, \sigma}(T)\) such that \(\chi(X_t, \mathcal{L}_t) = d + 1 - g\) and moreover we have \[\Picardfunctor_{X/k, \sigma} = \coprod\nolimits_{d \in \mathbf{Z}} \Picardfunctor^d_{X/k, \sigma}\] as fppf sheaves. It follows that the scheme \(\underline{\Picardfunctor}_{X/k}\) (which exists by Proposition 0B9Z) has a corresponding decomposition \[\underline{\Picardfunctor}_{X/k, \sigma} = \coprod\nolimits_{d \in \mathbf{Z}} \underline{\Picardfunctor}^d_{X/k, \sigma}\] where the points of \(\underline{\Picardfunctor}^d_{X/k, \sigma}\) correspond to isomorphism classes of invertible modules of degree \(d\) on \(X\).

Fix \(d \geq 0\). There is a morphism \[\gamma_d : \underline{\Hilbfunctor}^d_{X/k} \longrightarrow \underline{\Picardfunctor}^d_{X/k}\] coming from the invertible sheaf \(\mathcal{O}(D_{univ})\) on \(\underline{\Hilbfunctor}^d_{X/k} \times_k X\) (Remark 0B9J) by the Yoneda lemma (Categories, Lemma 001P). Our proof of the representability of the Picard functor of \(X/k\) in Proposition 0B9Z and Lemma 0B9X shows that \(\gamma_g\) induces an open immersion on a nonempty open of \(\underline{\Hilbfunctor}^g_{X/k}\). Moreover, the proof shows that the translates of this open by \(k\)-rational points of the group scheme \(\underline{\Picardfunctor}_{X/k}\) define an open covering. Since \(\underline{\Hilbfunctor}^g_{X/K}\) is smooth of dimension \(g\) (Proposition 0B9I) over \(k\), we conclude that the group scheme \(\underline{\Picardfunctor}_{X/k}\) is smooth of dimension \(g\) over \(k\).

By Groupoids, Lemma 047L we see that \(\underline{\Picardfunctor}_{X/k}\) is separated. Hence, for every \(d \geq 0\), the image of \(\gamma_d\) is a proper variety over \(k\) (Morphisms, Lemma 0AH6).

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.

Combining the facts mentioned above we see that \(\underline{\Picardfunctor}^d_{X/k}\) is proper for \(d \geq g\). This finishes the proof of (2) because now we see that \(\underline{\Picardfunctor}^d_{X/k}\) is proper for \(d \geq g\) but then all \(\underline{\Picardfunctor}^d_{X/k}\) are proper by translation.

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 \[Z = \{D \subset X \mid \mathcal{O}_X(D) \cong \mathcal{L}\} \subset \underline{\Hilbfunctor}^d_{X/k}\] 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 0B90 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 \(\underline{\Hilbfunctor}^d_{X/k}\) and \(\underline{\Picardfunctor}^d_{X/k}\). It follows that \(\gamma_d\) is flat, see Algebra, Lemma 00R4. As moreover the fibres are smooth, we conclude that \(\gamma_d\) is smooth by Morphisms, Lemma 01V8.

Some remarks on Picard groups

This section continues the discussion in Varieties, Section 0BEG and will be continued in Algebraic Curves, Section 0C1Y.

Lemma

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 \[\Pic(X) = \Pic(X_{k'})^{\text{Gal}(k'/k)}\] Moreover the action of \(\text{Gal}(k'/k)\) on \(\Pic(X_{k'})\) is continuous.

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 \(\Pic(-)\) is a contravariant functor, it acts (from the left) on \(\Pic(X_{k'})\). 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 0BU2. Then \(X_{k'} = \lim X_{k_\lambda}\) (as in Limits, Section 01YV) and we obtain \[\Pic(X_{k'}) = \colim \Pic(X_{k_\lambda})\] by Limits, Lemma 0B8W. Moreover, the transition maps in this system of abelian groups are injective by Varieties, Lemma 0CC5. It follows that every element of \(\Pic(X_{k'})\) 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.

Assume \(k'/k\) is finite Galois with Galois group \(G = \text{Gal}(k'/k)\). Let \(\mathcal{L}\) be an element of \(\Pic(X_{k'})\) fixed by \(G\). We will use Galois descent (Descent, Lemma 0CDR) 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 \(\Pic(X_{k'})\) 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.

Lemma

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 \[\Pic(X)[n] \longrightarrow \Pic(X_{k'})[n]\] is bijective for any purely inseparable extension \(k'/k\).

Proof

First we observe that the map \(\Pic(X) \to \Pic(X_{k'})\) is injective by Varieties, Lemma 0CC5. 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 \(\Pic(X_{k'})\). 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 \(X\).

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 0C6S. 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 02KH 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 0CAP) 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 023B). If true, then \(\mathcal{L}\) is the pullback of an invertible \(\mathcal{O}_X\)-module \(\mathcal{N}\) by Descent, Proposition 023T. Injectivity of the map on Picard groups shows that \(\mathcal{N}\) is a torsion element of \(\Pic(X)\) of the same order as \(\mathcal{L}\).

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 0CAP that these functions are the same.