Relative divisors and the existence of the Picard scheme

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

A nonzero section of a line bundle on an integral projective variety cuts out a divisor. Vary the section while keeping the bundle fixed, and the resulting divisors form a projective space. Vary the bundle as well, and these projective spaces become the fibres of the Abel map. This gives a way to construct the space of line-bundle classes from the already constructed Hilbert scheme.

The proof has three distinct jobs. We must construct complete linear systems over arbitrary parameter schemes, including nilpotent ones. We must show that linear equivalence defines a proper flat relation on a suitable open divisor scheme. We must then construct the quotient as a scheme. We carry out that last step by parametrizing the equivalence classes themselves as closed subschemes in a Hilbert scheme.

For the main existence theorem, \(S\) is Noetherian and \(f:X\to S\) is projective and flat, with geometrically integral fibres. Fix a relatively very ample bundle \(\mathcal O_X(1)\). The complete-linear-system theorem has the broader hypothesis that \(f\) is proper, flat and finitely presented. Throughout, \(\operatorname{Pic}_{X/S}\) denotes the étale Picard sheaf. It equals the fppf Picard sheaf under these hypotheses, by The Picard functor and the Picard scheme of a curve, Theorem 3.1.

We use the quotient convention

\[ \mathbb P(Q)=\operatorname{Proj}_S(\operatorname{Sym}Q). \tag{0.1} \]

It parametrizes quotient line bundles of \(Q\). If \(V\) is a bundle of sections, their lines are parametrized by \(\mathbb P(V^\vee)\).

1 Relative divisors form an open Hilbert locus

An effective Cartier divisor on \(Y\) is a closed subscheme whose ideal is locally generated by a nonzerodivisor. Equivalently it is the zero scheme of a regular section of an invertible sheaf. The empty divisor is allowed. For a flat finitely presented morphism \(Y\to T\), a relative effective Cartier divisor is an effective Cartier divisor \(D\subset Y\) that is flat over \(T\).

The fibrewise Cartier criterion gives an equivalent condition: locally write the equation as \(a\); its image must be a nonzerodivisor in every fibre where it vanishes. Then \(a\) is a nonzerodivisor on \(Y\), and \(\mathcal O_Y/(a)\) is flat over \(T\) [Stacks, Tag 062Y]. In particular the condition persists after arbitrary base change. This criterion includes points outside \(D\), where the equation is a unit.

There is also a useful formulation starting with a flat family of closed subschemes. If \(Y\) and \(D\) are flat and finitely presented over \(T\), then \(D\) is a relative effective Cartier divisor exactly when every geometric fibre is an effective Cartier divisor in the corresponding fibre of \(Y\). The criterion applies locally, and faithful flatness of a field extension detects the fibre condition.

Define

\[ \operatorname{Div}_{X/S}(T) =\{D\subset X_T:\ D\text{ is a relative effective Cartier divisor}\}. \tag{1.1} \]

Divisors here are closed subschemes, so no choice of an equation or its scalar is part of the datum.

Proposition 1.1. For a flat projective finitely presented family, \(\operatorname{Div}_{X/S}\) is represented by an open subscheme of \(\operatorname{Hilb}_{X/S}\). Over a Noetherian base its fixed-Hilbert-polynomial pieces are quasi-projective. Geometric integrality is not needed for this openness assertion.

Proof. Let \(H\) be the Hilbert scheme and \(Z\subset X_H\) its universal flat closed subscheme. Let \(B\subset Z\) be the locus where \(Z\) fails to be a relative Cartier divisor in \(X_H/H\). The Cartier criterion makes its complement open, so \(B\) is closed in \(Z\). Since \(Z\to H\) is proper, its image is closed. Put

\[ U=H\setminus\operatorname{image}(B). \tag{1.2} \]

The family \(Z_U\) is a relative divisor. Conversely, a relative divisor \(D\subset X_T\) gives a unique Hilbert map \(g:T\to H\). For every \(t\), its fibre is Cartier. Faithfully flat extension from \(\kappa(g(t))\) to \(\kappa(t)\) detects the same condition for \(Z_{g(t)}\). The Cartier criterion for the flat universal family then says the whole fibre over \(g(t)\) avoids \(B\). Thus \(g(t)\in U\). A map with image in an open subscheme factors through that open, for arbitrary \(T\); hence \(g\) factors uniquely through \(U\).

These constructions are inverse and commute with pullback. They prove representability on all test schemes, rather than only a classification of geometric points. The fixed-polynomial Hilbert scheme is projective by Hilbert and Quot schemes. Its divisor open is quasi-projective and, over Noetherian \(S\), of finite type. \(\square\)

The natural Abel map is the map of sheaves

\[ a:\operatorname{Div}_{X/S}\longrightarrow\operatorname{Pic}_{X/S}, \qquad D\longmapsto[\mathcal O_{X_T}(D)]. \tag{1.3} \]

For now its target is a sheaf, not an assumed scheme.

2 The universal module of sections

We need a parameter module that still works when the dimension of sections jumps. Simply writing \((f_*L)^\vee\) would not do that.

Lemma 2.1. Let \(f:X\to S\) be proper and finitely presented, and let \(F\) be a finitely presented quasi-coherent sheaf flat over \(S\). There is a finitely presented quasi-coherent module \(Q_F\) such that, for every \(g:T\to S\) and every quasi-coherent \(N\) on \(T\),

\[ \mathcal Hom_T(g^*Q_F,N) \simeq (f_T)_*(F_T\otimes f_T^*N). \tag{2.1} \]

It is unique with this universal isomorphism, and its formation commutes with every base change. If \(S\) is locally Noetherian, \(Q_F\) is coherent.

Proof. Perfect cohomology with arbitrary base change [Stacks, Tag 0A1H] supplies a perfect complex \(K=Rf_*F\). Locally on an affine open in \(S\), it can be represented by finite free modules in nonnegative degrees,

\[ K^0\xrightarrow{d^0}K^1\longrightarrow\cdots\longrightarrow K^b. \tag{2.2} \]

Here is why negative terms can be removed. On any residue field the derived base change is the cohomology complex of a sheaf, so its negative cohomology vanishes. Start at the lowest degree of a finite free representative. If this degree is negative, the differential is injective on the fibre and has an invertible maximal minor near the point. Split off the resulting contractible pair. Repeating removes all negative terms near that point. This is a local assertion, which is all we need.

For any base module \(N\), derived base change and the projection formula identify the right side of (2.1) with the degree-zero kernel

\[ \ker(K^0\otimes N\longrightarrow K^1\otimes N). \tag{2.3} \]

There is no derived flatness issue in this expression: the modules in (2.2) are free, and \(F\) is flat over the base. Define locally

\[ Q_F=\operatorname{coker}\bigl((K^1)^\vee \xrightarrow{(d^0)^\vee}(K^0)^\vee\bigr). \tag{2.4} \]

Applying \(\operatorname{Hom}(-,N)\) to this presentation gives exactly (2.3). Thus it proves (2.1). Intrinsically the module is

\[ Q_F=H^0\bigl(R\mathcal Hom(K,\mathcal O_S)\bigr). \tag{2.5} \]

The intrinsic formula or the universal property identifies the constructions on overlaps, so they glue. The presentation (2.4) is finite, and taking its cokernel commutes with arbitrary tensor pullback. The base-changed complex computes the base-changed cohomology by Tag 0A1H, proving the asserted base-change property. Uniqueness follows from the module version of Yoneda applied to (2.1). On a locally Noetherian scheme, finite presentation implies coherence. \(\square\)

On a fibre,

\[ Q_F\otimes\kappa(s) \simeq H^0(X_s,F_s)^\vee. \tag{2.6} \]

If the higher cohomology of \(F_s\) vanishes throughout a neighbourhood, the same cancellation argument reduces (2.2) to a finite locally free module \(V\) in degree zero. Then

\[ V=f_*F,\qquad Q_F=V^\vee, \tag{2.7} \]

and \(V\) commutes with every base change. The more familiar sufficient condition \(H^1(X_s,F_s)=0\) also gives local freeness of the section module near \(s\). To see this directly, cancel all contractible pairs whose differential has a nonzero entry on that residue field. The remaining differentials vanish on the fibre. Its degree-one term must then have rank zero, since its fibre cohomology in degree one is zero. After shrinking, the complex therefore has no degree-one term, and its degree-zero term splits off as a free summand. Formula (2.4) identifies the section module with its dual; all further base changes preserve this splitting. We will use the stronger all-higher-cohomology vanishing condition in the existence proof.

The distinction between finite presentation and coherence matters. The theorem is valid over arbitrary schemes \(S\). Calling \(Q_F\) coherent in that generality would impose a property that finite presentations over noncoherent rings need not have.

3 Complete linear systems on every test scheme

For proper flat finitely presented \(f\) with geometrically integral fibres, we have

\[ \mathcal O_T\simeq(f_T)_*\mathcal O_{X_T} \quad\text{for all }T\to S. \tag{3.1} \]

We use the precise proper-flat cohomology theorem for geometrically reduced connected fibres [Stacks, Tag 0E0L]. It applies because an integral geometric fibre is reduced and connected. Consequently pullback of base line bundles is fully faithful: homomorphisms between them are detected by pushing forward, locally trivializing them and using (3.1).

Let \(L\) be invertible on \(X\). Its complete relative linear system is

\[ \operatorname{LinSys}_L(T)= \{D\in\operatorname{Div}_{X/S}(T): \mathcal O_{X_T}(D)\simeq L_T\otimes f_T^*N \text{ for some invertible }N\text{ on }T\}. \tag{3.2} \]

Theorem 3.1 (complete linear systems). With these hypotheses, \(\operatorname{LinSys}_L\) is represented by \(\mathbb P(Q_L)\), where \(Q_L\) is the finitely presented module of Lemma 2.1. This is also the fibre product of the Abel map with the Picard class defined by \(L\). If \(S\) is locally Noetherian the module is coherent. If \(f_*L=V\) is locally free and commutes with every base change, the linear system is \(\mathbb P(V^\vee)\).

Proof. A \(T\)-point of \(\mathbb P(Q_L)\) is a surjection

\[ g^*Q_L\twoheadrightarrow N \tag{3.3} \]

onto an invertible sheaf, considered up to isomorphism of the target. By (2.1), it gives a section

\[ s\in\Gamma(X_T,L_T\otimes f_T^*N). \tag{3.4} \]

The quotient is surjective precisely when its residue-field map is nonzero at every point of \(T\), by Nakayama applied to its cokernel. By (2.6), this means \(s_t\ne0\) in \(H^0(X_t,L_t)\) at every point. On an integral fibre, a nonzero global section of a line bundle is regular: in a trivialization it is nonzero at the generic point, hence is a nonzerodivisor in every domain where it is considered. The fibrewise Cartier criterion therefore makes the zero scheme \(D\) of \(s\) a relative divisor. Its divisor bundle is \(L_T\otimes f_T^*N\).

Conversely, start with \(D\) in (3.2). Choose the displayed isomorphism. The canonical section of \(\mathcal O(D)\) becomes (3.4), and the fibre criterion makes it nonzero on every fibre. Equation (2.1) gives a map (3.3), which is surjective by Nakayama.

This point of projective space is independent of the choices. Two choices of \(N\) are isomorphic, since cancelling \(L_T\) gives isomorphic pullbacks of them and (3.1) detects that isomorphism on \(T\). Two isomorphisms with fixed \(N\) differ by a unit of \(X_T\), which is a unique base unit by (3.1). It changes the quotient by an automorphism of its target and therefore leaves the projective point unchanged. In the other direction, isomorphic quotient pairs have identical zero schemes.

The two constructions are inverse: the canonical divisor section recovers the original section up to the allowed scalar, and the zero scheme recovers the original divisor ideal. Each operation commutes with pullback. This proves the statement for arbitrary test schemes, including nonreduced ones.

Finally, equality between the Abel class of \(D\) and the class of \(L_T\) in the Picard sheaf implies that their bundles differ by a base line bundle: this is the injection of the raw relative Picard quotient into its sheaf proved in the previous lesson, Proposition 1.1, under (3.1). Hence the sheaf fibre is exactly (3.2). If sections form a base-change-compatible bundle \(V\), the universal identity (2.1) identifies \(Q_L\) with \(V^\vee\). \(\square\)

On a field-valued class actually represented by \(L\), the fibre is

\[ \mathbb P\bigl(H^0(X,L)^\vee\bigr). \tag{3.5} \]

The general relative \(\mathbb P(Q_L)\) need not be a projective-space bundle: its fibre dimensions can jump. There is a separate descent issue if a Picard class has no global line-bundle representative. It becomes a complete linear system after an étale cover, but the projective spaces can descend to a Brauer–Severi scheme instead of the projectivization of a global vector bundle. We will use local representatives and effective descent explicitly.

4 Separatedness before representability

We prove separatedness at the sheaf level. This avoids assuming the Picard scheme while constructing it.

Proposition 4.1. Under the main projective-flat geometrically integral hypotheses, the diagonal of \(\operatorname{Pic}_{X/S}\) is represented by closed immersions.

Proof. First take two line bundles \(L,M\) on \(X_T\). We construct the closed subscheme of \(T\) where their relative classes are equal, in the functorial sense that must hold after every further base change.

The fibre Hilbert polynomials of \(L\) and \(M\) are locally constant. Indeed the proper-flat perfect cohomology complexes of finitely many twists have locally constant Euler characteristic [Stacks, Tags 0A1H and 0B9T]; finitely many values determine a polynomial of degree bounded by the projective ambient dimension. The fibre dimension \(n\) is locally constant as well. We may work on one such open and closed piece.

If \(n=0\), the proper flat family is an isomorphism. It is finite because its fibres are finite [Stacks, Tag 02LS], and finite locally free of rank one because each geometric fibre is the single reduced point. The unit map \(\mathcal O_T\to(f_T)_*\mathcal O_{X_T}\) is an isomorphism, so its finite algebra is \(\mathcal O_T\). The relative Picard sheaf is zero there.

Suppose \(n\geq1\). Let \(T_0\subset T\) be the open and closed locus on which the coefficient of degree \(n-1\) in \(P_M-P_L\) is zero. Equal classes must lie in this locus. On an integral projective fibre, a nonzero homomorphism between line bundles is injective. If it is not an isomorphism, its cokernel is a twist of the structure sheaf of a nonempty effective Cartier divisor. That sheaf has dimension \(n-1\) and positive leading Hilbert coefficient. The Hilbert-polynomial dimension and positivity statement was established in Castelnuovo–Mumford regularity and boundedness, Section 3. Thus, on \(T_0\), every nonzero fibre homomorphism \(L_t\to M_t\) is an isomorphism.

Set \(A=M\otimes L^{-1}\) on \(X_{T_0}\). The scheme

\[ E=\mathbb P(Q_A)\longrightarrow T_0 \tag{4.1} \]

parametrizes fibrewise nonzero sections of \(A\) up to a base line-bundle factor. By the preceding paragraph these sections are fibrewise isomorphisms. A homomorphism of line bundles which is an isomorphism on every fibre is an isomorphism everywhere: in a trivialization its coefficient is a unit at every point, as detected in that point's fibre local ring. Therefore (4.1) parametrizes exactly equality of the two relative classes on every test scheme over \(T_0\).

There is at most one such projective point on any test scheme. Two isomorphisms with their base factors differ by a unique isomorphism of the base factors and then by a base unit, using (3.1). Both changes are identified in the quotient-line parametrization. Hence \(E\to T_0\) is a monomorphism. It is projective; a proper morphism with these finite fibres is finite [Stacks, Tag 02LS]. A finite monomorphism is a closed immersion [Stacks, Tag 03BB]. Thus \(E\) is a closed subscheme of \(T_0\), and also of \(T\). It represents equality of the relative classes; maps factoring through a closed subscheme include their nilpotent conditions, not just their point sets.

For two arbitrary Picard classes over \(T\), take an étale cover on which they are represented by bundles. The closed equality subschemes just constructed have canonical identical pullbacks on overlaps, since they represent the same equality functor. Their ideal sheaves descend effectively [Stacks, Tag 023R]. The descended closed subscheme represents equality over \(T\) by the sheaf property. This proves the diagonal assertion. \(\square\)

The argument isolates the role of integral fibres: a nonzero section has a regular divisor, and a nonempty such divisor has a strictly positive top Hilbert coefficient. Both assertions can fail in the form needed here on a reducible fibre.

5 An open locus with enough sections

Call a fibre line bundle \(L_t\) positive here if it is globally generated, has a nonzero section, and

\[ H^i(X_t,L_t(n))=0 \quad\text{for every }i>0\text{ and }n\geq0. \tag{5.1} \]

This is a sufficient positivity condition for the construction, not a definition of ampleness. Its all-twists condition makes tensoring with \(\mathcal O_X(1)\) preserve positivity. Global generation also persists because \(\mathcal O_X(1)\) is globally generated.

Let \(P_+\subset\operatorname{Pic}_{X/S}\) be the subfunctor of classes with positive geometric fibres. These conditions do not depend on the representative or its base line-bundle factor, and are invariant under extending a residue field.

Lemma 5.1. The inclusion \(P_+\to\operatorname{Pic}_{X/S}\) is represented by open immersions. Write \(\xi=[\mathcal O_X(1)]\). The open subfunctors

\[ P_+-m\xi=\{\lambda:\lambda+m\xi\in P_+\},\qquad m\geq0, \tag{5.2} \]

cover the Picard sheaf on every test scheme.

Proof. Start with an actual bundle \(L\) over a Noetherian affine test scheme. Relative Serre vanishing and generation give an integer \(N\) such that all twists \(L(n)\), \(n\geq N\), have zero higher fibre cohomology and are generated by their sections. These are the usual projective coherent-sheaf Serre theorem and its flat base-change form [Stacks, Tags 01XO, 0B5T and 02O1]. One may obtain the latter form by applying the proper-flat cohomology complex to the vanishing direct images. It is then concentrated in degree zero. Its derived tensor with any base module has no negative cohomology, because the original bundle is base-flat and direct image of a sheaf has no negative cohomology. Thus its degree-zero module has no higher Tor with any module: it is flat, hence locally free by finite presentation. Its derived base changes have no higher cohomology, and generation remains surjective after pullback.

Consequently only finitely many vanishing tests in (5.1) remain. Each is open by the finite cohomology complex and semicontinuity. On their intersection, \(f_*L\) is a base-change-compatible locally free module. The locus where its fibre has positive rank is open and closed. Its evaluation map

\[ f^*f_*L\longrightarrow L \tag{5.3} \]

has a coherent cokernel. The image in the base of its support is closed by properness. Removing that image is precisely the global-generation condition on fibres. This proves openness.

For an arbitrary affine test scheme \(\operatorname{Spec}B\) over a Noetherian affine open \(\operatorname{Spec}A\subset S\), write \(B\) as the filtered union of its finitely generated \(A\)-subalgebras. The finitely presented bundle descends to \(X_{B_0}\) at a finite stage. To retain invertibility, descend its inverse, their tensor-product isomorphisms and the two inverse identities as well; the finite-presentation descent theorem [Stacks, Tag 01ZR] ensures that a sufficiently large stage retains these data. The stage \(B_0\) is Noetherian. The preceding open locus pulls back to exactly the required locus on \(\operatorname{Spec}B\), because fibre cohomology and generation are invariant under field extension. This proves openness for all affine tests and then for all tests by gluing.

A Picard class is locally represented by a bundle in the étale topology. The just constructed opens descend: membership is invariant on overlaps, and an étale-local invariant open descends to an open of the base. Its universal factorization property follows from the sheaf property. Thus \(P_+\) is an open subfunctor.

Finally take any \(\lambda\in\operatorname{Pic}_{X/S}(T)\) and \(t\in T\). After a field extension, its fibre has a line-bundle representative. Serre's theorem on that projective fibre makes \(L(m)\) positive for sufficiently large \(m\). Hence \(t\) lies in the inverse image of (5.2) for that \(m\). Those inverse images are opens and cover \(T\). Tensoring by \(\mathcal O(1)\) preserves positivity, so the cover is increasing if desired. \(\square\)

For a positive bundle \(L\) over a test scheme, Lemma 2.1 now gives

\[ V=f_*L\text{ locally free of positive rank}, \qquad \operatorname{LinSys}_L=\mathbb P(V^\vee). \tag{5.4} \]

In particular the latter is smooth, proper and surjective over the test scheme.

It is useful to index the positive locus by the Hilbert polynomial of the inverse bundle:

\[ \phi(n)=\chi(X_t,L_t^{-1}(n)). \tag{5.5} \]

This is locally constant by the same finite-cohomology argument as in Section 4. Thus \(P_+\) is a disjoint union of open and closed subfunctors \(P_+^\phi\). Work also on a piece of \(S\) on which \(\psi=P_{\mathcal O_{X_t}}\) is constant. An effective divisor of such a class has Hilbert polynomial

\[ P_D=\psi-\phi, \tag{5.6} \]

by the ideal sequence \(0\to L_t^{-1}\to\mathcal O_{X_t}\to\mathcal O_{D_t}\to0\).

6 The Abel relation on positive divisors

Let \(Y^\phi\) be the inverse image of \(P_+^\phi\) under the Abel map. Proposition 1.1 and Lemma 5.1 make it an open subscheme of the projective scheme \(\operatorname{Hilb}^{\psi-\phi}_{X/S}\). In particular \(Y^\phi\) is Noetherian and quasi-projective over \(S\).

Proposition 6.1. The map \(Y^\phi\to P_+^\phi\) is a surjection of étale sheaves, represented by smooth proper morphisms. Its relation

\[ R^\phi=Y^\phi\times_{P_+^\phi}Y^\phi \tag{6.1} \]

is a closed subscheme of \(Y^\phi\times_S Y^\phi\), and both its projections to \(Y^\phi\) are smooth and projective.

Proof. Given \(T\to P_+^\phi\), represent its class by a positive bundle \(L\) after an étale cover \(T'\to T\). Theorem 3.1 identifies its divisor fibre with \(\mathbb P(((f_{T'})_*L)^\vee)\). This is a positive-rank projective-space bundle. It is smooth and proper and has sections Zariski locally on \(T'\), by trivializing its section bundle and choosing a coordinate line. Thus the Abel map is an epimorphism of étale sheaves.

We must also check that the fibre over \(T\) is a scheme. The fibre functor is the equality locus of the given class and the divisor bundle on \(X_{T\times Y^\phi}\). Proposition 4.1 makes it a closed subscheme of \(T\times_S Y^\phi\). Smoothness and properness descend from its just described étale pullback. This proves relative representability and the first assertion without assuming representability of the target.

The same diagonal argument gives the closed subscheme (6.1). To see projectivity of either projection, take \(T=Y^\phi\). Its fibre scheme is proper over \(T\), and maps by a locally closed immersion into

\[ T\times_S\operatorname{Hilb}^{\psi-\phi}_{X/S}. \tag{6.2} \]

This map is proper: the target is separated over \(T\), so the graph argument for a map from a proper \(T\)-scheme applies. A proper monomorphism is a closed immersion, by the finite-fibre and finite-monomorphism results used in Section 4. Thus the fibre scheme is closed in (6.2), which is projective over \(T\). Both projections are therefore projective, as well as smooth. \(\square\)

One should not infer projectivity merely from the fact that the fibres become projective spaces after an étale cover. The argument uses the closed embedding into a fixed projective Hilbert scheme. It supplies the global ample bundle that such a conclusion needs.

7 Constructing the quotient by parametrizing its classes

We now prove the required quotient theorem. A schematic equivalence relation \(R\subset Y\times_S Y\) means that for every test scheme the subset \(R(T)\) is the graph of an equivalence relation on \(Y(T)\). This includes its scheme structure, not just a relation on geometric points.

Theorem 7.1 (proper flat quotient). Let \(S\) be Noetherian, let \(Y\) be quasi-projective over \(S\), and let \(R\subset Y\times_S Y\) be a closed schematic equivalence relation whose projections are proper, flat and finitely presented. There is a quasi-projective \(S\)-scheme \(Q\) and a faithfully flat projective finitely presented morphism \(q:Y\to Q\) such that

\[ R=Y\times_QY. \tag{7.1} \]

The scheme \(Q\) represents the quotient sheaf in the fppf topology. If the projections are smooth, \(q\) is smooth and represents the étale quotient sheaf as well.

Proof. Regard the first projection as a family of closed subschemes of \(Y\), by using the second coordinate. These subschemes are the equivalence classes. The family is proper and flat over \(Y\), and is projective there: embed \(Y\) into a projective \(S\)-scheme; the induced immersion of \(R\) in its product with \(Y\) is proper and therefore closed. The Hilbert polynomial of its fibres is locally constant. Since \(Y\) is Noetherian and quasi-compact, only finitely many polynomials \(p_1,\ldots,p_r\) occur. None is zero, since reflexivity supplies a point in every class.

The Hilbert construction in Hilbert and Quot schemes also applies to proper subschemes of a quasi-projective scheme: take the open in the Hilbert scheme of a projective closure where the universal support avoids the boundary. Let

\[ H=\coprod_{j=1}^r\operatorname{Hilb}^{p_j}_{Y/S}, \qquad \pi:D\to H \tag{7.2} \]

be the resulting finite union of quasi-projective Hilbert schemes and its universal family. The morphism \(\pi\) is projective, flat, finitely presented and surjective. Surjectivity holds because a subscheme with nonzero polynomial is nonempty on every fibre.

The relation family defines a morphism

\[ u:Y\longrightarrow H. \tag{7.3} \]

For every test scheme \(T\) and \(x,y\in Y(T)\), we have

\[ (x,y)\in R(T)\quad\Longleftrightarrow\quad u(x)=u(y). \tag{7.4} \]

Indeed related points have identical class subschemes: after every further base change, transitivity and symmetry identify membership in the two classes, so Yoneda identifies their closed subschemes. Conversely if those subschemes are identical, the point \(x\), which lies in its own class by reflexivity, lies in the class of \(y\). This proves (7.4) on all tests, including their nilpotent structure.

Since \(H\) is separated over \(S\), the graph \(\Gamma_u\subset Y\times_S H\) is closed. Reflexivity makes it a closed subscheme of \(D\). We claim it is invariant under the two projections

\[ D\times_H D\rightrightarrows D. \tag{7.5} \]

To check equality of the two pulled-back closed subschemes, take any test point \((x,y,V)\), where \(V\) is a flat proper class-sized subscheme of \(Y_T\) and \(x,y\) lie in \(V\). Membership in the first pullback is \(V=u(x)\); in the second it is \(V=u(y)\). If the first holds, then \(y\) lies in the class of \(x\), so (7.4) gives \(u(y)=u(x)\). The reverse implication is identical with \(x,y\) exchanged. Since this equivalence holds on every test scheme, the two closed subschemes in (7.5) are equal.

Now \(\pi:D\to H\) is faithfully flat and quasi-compact. Effective descent for quasi-coherent ideals [Stacks, Tag 023R] descends the invariant ideal of \(\Gamma_u\) to an ideal on \(H\). The canonical identification on the two pullbacks obeys the cocycle condition, since both are the same ideal inside the structure sheaf. Thus there is a closed subscheme

\[ Q\subset H\quad\text{with}\quad D\times_H Q=\Gamma_u. \tag{7.6} \]

The graph is isomorphic to \(Y\). Base-changing \(\pi\) in (7.6) therefore gives a faithfully flat projective finitely presented map \(q:Y\to Q\). Since \(Q\) is closed in the finite quasi-projective union \(H\), it is quasi-projective. Equation (7.4), together with its all-test-scheme interpretation, identifies its kernel pair with \(R\), proving (7.1).

Finally any map \(T\to Q\) lifts to \(Y\) after the fppf cover \(T\times_QY\to T\). Two local lifts have identical images precisely when they are related by (7.1). Thus \(h_Q\) is the sheafification of the presheaf quotient. More explicitly a map from \(Y\) to an fppf sheaf which agrees on \(R\) descends uniquely: choose local lifts of a \(Q\)-point, apply that map, and use the equality on the kernel pair to glue. This is the universal property of the quotient sheaf.

If \(R\to Y\) is smooth, the faithfully flat base change of \(q\) by itself is smooth, so \(q\) is smooth by descent. A smooth surjective map has étale-local sections. The same lifting and gluing argument then identifies \(Q\) with the étale quotient as well. \(\square\)

The Hilbert scheme in this proof parametrizes an entire equivalence class, rather than a selected divisor. The closed graph marks the classes that actually arise from the relation. Flat descent turns that marked locus into the quotient scheme. A reference for the classical result and this method is Nitsure, Construction of Hilbert and Quot Schemes, Theorem 6.8; the argument above supplies every quotient step used here.

8 Gluing the Picard scheme

Theorem 8.1 (Grothendieck's existence theorem). Let \(S\) be Noetherian and \(f:X\to S\) projective and flat with geometrically integral fibres. The étale Picard functor is represented by a separated group scheme locally of finite type over \(S\). Formation of this scheme commutes with arbitrary base change. Its positive fixed-inverse-polynomial charts are quasi-projective over \(S\).

Proof. On an open and closed base piece with fixed \(\psi\), apply Theorem 7.1 to \(Y^\phi\) and \(R^\phi\) from Proposition 6.1. It gives a quasi-projective quotient \(Q^\phi\), with a smooth surjective map from \(Y^\phi\). The Abel map is also an epimorphism of étale sheaves and has exactly the same kernel pair. The local-lift universal property in Theorem 7.1 therefore identifies

\[ h_{Q^\phi}=P_+^\phi. \tag{8.1} \]

This identification uses neither an assumed Picard algebraic space nor an existing Picard variety. Taking the disjoint union over \(\phi\), and then over the open and closed pieces for \(\psi\), represents \(P_+\) by a scheme \(Q_+\). Each piece is of finite type over the Noetherian base.

Translation by \(m\xi\) identifies every open subfunctor in (5.2) with \(P_+\). Their pairwise intersections are represented by opens in both charts, because \(P_+\) is relatively open. The identifications represent equality of the same elements of the Picard sheaf and consequently satisfy the cocycle condition. Glue these schemes along the intersections.

To verify the resulting scheme \(P\) represents the whole sheaf, take any \(\lambda\in\operatorname{Pic}_{X/S}(T)\). Lemma 5.1 makes its inverse images of the charts an open cover of \(T\). On each open it determines a unique map to the corresponding chart. Equality of the class on overlaps makes these maps agree, so they glue uniquely to \(T\to P\). Conversely maps to \(P\) give Picard classes on their inverse-image charts and glue by the sheaf property. Both constructions are inverse on every test scheme.

The scheme is locally of finite type because its charts are. Proposition 4.1 says its diagonal is closed, so it is separated over \(S\). Yoneda turns tensor product, dual and the trivial bundle into its group-law, inverse and identity morphisms. Their identities are exactly the line-bundle identities, giving a commutative group scheme.

For any \(S'\to S\), evaluation of the relative Picard sheaf of \(X_{S'}/S'\) on an \(S'\)-scheme \(T\) is the same evaluation as before: the family is \(X_T\), with the same étale covers and base-bundle quotient. Thus \(P\times_S S'\) represents that sheaf, proving arbitrary base-change compatibility. This assertion does not require \(S'\) to be Noetherian. \(\square\)

The result is local finite type, rather than finite type for the whole Picard scheme. Already \(\operatorname{Pic}_{\mathbb P^n_S/S}\) for \(n\geq1\) has one copy of \(S\) for every integer. Finiteness statements for appropriate parts belong to The structure of the Picard scheme.

9 Examples and the boundary of the theorem

Hypersurfaces as divisors in projective space

The base variety in this example is \(\mathbb P^n_S\), with \(n\geq1\). Its relative Picard scheme is the constant integer scheme, proved in the previous lesson. Let \(V_d\) be the free module of homogeneous degree-\(d\) forms in its \(n+1\) coordinates. For \(d\geq0\), the component of its divisor scheme of bundle class \(\mathcal O(d)\) is

\[ \mathbb P(V_d^\vee), \qquad \operatorname{rank}V_d=\binom{n+d}{n}. \tag{9.1} \]

Thus it is the projective space of degree-\(d\) homogeneous forms, whose dimension over a field is \(\binom{n+d}{n}-1\). A quotient of the dual section module gives a form with coefficients generating the unit ideal locally on the base. Exactly this condition keeps its fibre polynomial nonzero. Each nonzero polynomial is a nonzerodivisor on the integral projective-space fibre, so Theorem 3.1 gives a flat relative hypersurface. When \(d=0\), the sole divisor is empty; negative degrees have no effective divisors.

This example says nothing like \(\operatorname{Pic}(Z)=\mathbb Z\) for every hypersurface \(Z\subset\mathbb P^n\). That assertion would be false: a smooth elliptic plane cubic already has a positive-dimensional degree-zero Picard scheme. It is the Picard scheme of the ambient projective space that indexes (9.1).

Curves and a Brauer–Severi fibre

For a smooth projective geometrically connected curve, the relative divisor scheme of degree \(d\) is its symmetric power, as proved in Hilbert and Quot schemes. A pointed curve has a universal normalized bundle on its Picard scheme. For \(d\geq0\) with \(d>2g-2\), Serre duality and Riemann–Roch give

\[ \operatorname{Sym}^d C \simeq\mathbb P_{\operatorname{Pic}^d_C}(V^\vee), \qquad \operatorname{rank}V=d+1-g. \tag{9.2} \]

For a curve without a point, the same statement holds after an étale cover supplying a representative of the universal class. The global conclusion can be a Brauer–Severi fibration.

For example let \(C/\mathbb R\) be the conic \(x^2+y^2+z^2=0\). Its degree-one Picard component is \(\operatorname{Spec}\mathbb R\), while

\[ \operatorname{Div}^1_C=C\longrightarrow\operatorname{Pic}^1_C \tag{9.3} \]

has no real section. After extending to \(\mathbb C\) it is \(\mathbb P^1\). It cannot be the projectivization of a two-dimensional real vector space, which would have real points. The class in its target exists as an étale Picard class although there is no real degree-one line bundle on \(C\). This is the same descent obstruction analysed in the previous lesson.

A family of elliptic curves

Let \(E\to S\) be a smooth proper projective geometrically connected genus-one family over a Noetherian base, with section \(e\). Degree-one line bundles have \(H^1=0\) and one section on every fibre. Their sections therefore form a line bundle on every test scheme. Evaluation cuts out a unique relative divisor of degree one. Such a divisor is finite locally free of rank one and therefore is the graph of a unique section of \(E_T\to T\). The construction and its inverse commute with all base changes, exactly as in the degree-one argument of the previous lesson. Hence

\[ E\xrightarrow{\sim}\operatorname{Pic}^1_{E/S}, \qquad x\longmapsto[\mathcal O(x)], \tag{9.4} \]

and translation by \(-[\mathcal O(e)]\) gives

\[ E\xrightarrow{\sim}\operatorname{Pic}^0_{E/S}, \qquad x\longmapsto[\mathcal O(x-e)]. \tag{9.5} \]

The tensor group law on the target transports to the elliptic group law with origin \(e\). Here the degree-zero piece is the fibrewise identity component: it is isomorphic to the given smooth connected proper family. No assertion that all higher-dimensional Picard schemes are smooth is used.

Why the historical erratum matters

The existence theorem uses geometric integrality, not just properness, flatness and (3.1). The broader scheme-existence conjecture in FGA 232, Remark 5.2, was explicitly withdrawn in its erratum, p. 303. That erratum also adds the algebraically closed residue-field qualification to the complete-local-base remark. Modern algebraic-space representability under proper flat finite presentation and (3.1) [Stacks, Tag 0D2C] is a different conclusion and is consistent with these corrections.

One mechanism is visible already in our proof. On a reducible fibre a nonzero section need not be regular. Regular sections form an open inside the projective parameter scheme, and that open need not be proper. The relation projections then lose the properness needed in Theorem 7.1. The historical nonexistence of a Picard scheme in some such cases means failure of representation by a scheme; it does not assert failure of the applicable modern algebraic-space theorem. The general reductions and finiteness results of FGA 232 §6 and FGA 236 require their own additional arguments.

10 Exercises

  1. Fibrewise flatness. Let \(X\to S\) be flat and finitely presented, and let an invertible ideal locally generated by \(a\) have fibrewise nonzerodivisor equations. Show its divisor is flat over \(S\), and show every base change remains a Cartier divisor. Explain why an absolute Cartier divisor in a flat family need not suffice.

  2. The open divisor locus. Give the all-test-scheme proof that \(\operatorname{Div}_{X/S}\) is open in the Hilbert scheme for a flat projective family. Identify the proper morphism used to remove the bad locus. Is geometric integrality required in this exercise?

  3. A section bundle. Assume (3.1), integral geometric fibres, and that \(f_*L=V\) is locally free with arbitrary base change. Prove that the complete linear system is \(\mathbb P(V^\vee)\), including the inverse construction on a nonreduced test scheme. Explain what changes if the Picard class has no global representative.

  4. Separatedness. Prove the diagonal assertion for the étale Picard sheaf of the main theorem. Do not assume in advance that the target is a scheme. Explain why merely proving uniqueness of geometric specializations would miss a required part of the argument.

  5. The curve quotient. Let \(C/k\) be smooth projective geometrically connected, with a point, and let \(d\geq0\), \(d\geq2g-1\). Construct \(\operatorname{Pic}^d_C\) completely as \(\operatorname{Sym}^dC\) modulo linear equivalence. Show the equivalence relation is proper and flat, construct its Hilbert-scheme quotient, identify its sheaf of points and compute the relative dimension of the quotient map. Include \(g=0,d=0\).

11 Solutions

1. At a point of the divisor, the fibrewise Cartier criterion [Stacks, Tag 062Y] applies to the flat finitely presented ambient family: multiplication by \(a\) is injective and its cokernel is flat over the base. The latter is the divisor's structure sheaf. At points outside the divisor, \(a\) is a unit and the quotient is zero, hence flat. Therefore the divisor is flat everywhere. Tensor the exact sequence \(0\to\mathcal O_X\xrightarrow{a}\mathcal O_X\to\mathcal O_D\to0\) with any base algebra. Flatness of the quotient keeps the first arrow injective. Thus the pulled-back equation remains a nonzerodivisor, and its quotient remains flat over the new base. For a failure of the absolute condition, take \(X=S=\operatorname{Spec}k[t]\) with its identity map and \(D=(t)\). The equation is an absolute nonzerodivisor, but on the fibre over zero it is zero; \(k[t]/(t)\) is not flat over \(k[t]\).

2. Use the universal flat closed subscheme \(Z\subset X_H\). Its Cartier-good locus is open by the fibrewise criterion; let \(B\) be its closed complement in \(Z\). The morphism \(Z\to H\), not a morphism from an affine chart of \(Z\), is proper, so its image of \(B\) is closed. On the complementary open \(U\), the universal family is a relative divisor. A divisor over any \(T\) has Cartier geometric fibres. For its Hilbert map \(T\to H\), faithful flatness of residue-field extensions and the criterion for the flat universal family show that each image point avoids \(B\). Hence the map factors through \(U\). The universal divisor pulls back to the original divisor, and uniqueness is Hilbert uniqueness. This works on nonreduced \(T\), because factorization through an open subscheme is determined by its image. Geometric integrality is unnecessary here; it becomes necessary for turning every nonzero section into a regular section later.

3. A quotient \(V_T^\vee\twoheadrightarrow N\) corresponds to a section of \(L_T\otimes f_T^*N\), by the assumed base-change isomorphism and duality for the vector bundle \(V_T\). Surjectivity means its fibre section is nonzero at every point. Integral fibres make those sections regular; the Cartier criterion supplies a flat divisor \(D\), with \(\mathcal O(D)=L_T\otimes f_T^*N\). Conversely the canonical section of such a divisor corresponds to a fibrewise nonzero map \(V_T^\vee\to N\), which is surjective by Nakayama. Two choices of its displayed bundle isomorphism differ by a base unit under (3.1), so give the same quotient-line point. Taking zero schemes and canonical sections are inverse, including the nilpotent structure of their ideals, and commute with pullback. If the class lacks a global representative, perform this construction on an étale cover. The resulting projective spaces glue as the divisor fibre, but may give a Brauer–Severi scheme. The real conic in (9.3) demonstrates the distinction.

4. For actual bundles \(L,M\) over \(T\), use the locally constant Hilbert polynomials to take the open and closed piece where the degree-\((n-1)\) coefficient of their difference is zero. On an integral \(n\)-dimensional fibre, a nonzero nonisomorphism of line bundles has a nonempty Cartier-divisor cokernel with positive leading coefficient. Thus every nonzero homomorphism on this piece is an isomorphism. The complete linear system of \(M\otimes L^{-1}\) parametrizes precisely equality of their relative classes. It is projective and a monomorphism: any two representing isomorphisms differ only by a base factor isomorphism and a scalar. Properness and finite fibres make it finite, and a finite monomorphism is closed. Off that polynomial piece equality is impossible. For zero-dimensional fibres the family is an isomorphism and the relative functor is zero. For arbitrary classes, the closed equality subschemes from an étale cover descend by their ideals and identical universal property. This proves representability of the diagonal by closed immersions before representability of Picard itself. A statement only about geometric points would not construct this equality subscheme or verify its nilpotent conditions on test schemes.

5. Put \(Y=\operatorname{Sym}^dC\), with its universal divisor. For every degree-\(d\) bundle \(L\), Serre duality gives \(H^1(L)=0\), since \(\deg(\omega_C\otimes L^{-1})=2g-2-d<0\). Riemann–Roch gives \(h^0(L)=d+1-g>0\). The point on the curve identifies Picard classes with normalized actual bundles on all test schemes. Their section modules are base-change-compatible bundles of that rank. Thus the Abel map's pullback by any class is the projective-space bundle of section lines, smooth proper and surjective of relative dimension \(d-g\).

Define \(R\subset Y\times Y\) by equality of the two divisor-bundle classes. Proposition 4.1 makes it closed. Its projections are the just described projective-space bundles, hence smooth, projective and flat. Apply the construction of Theorem 7.1: the fibres of \(R\to Y\) define a map \(u:Y\to H\) to a finite union of Hilbert schemes of \(Y\); the closed graph lies in the flat universal family \(D\to H\); equivalence of the relation makes that graph invariant on \(D\times_HD\); its ideal descends to a closed \(Q\subset H\). The descended map \(Y\to Q\) is faithfully flat projective and smooth, and its kernel pair is \(R\).

Every Picard class acquires an effective divisor étale locally, because its positive-rank section bundle locally has a quotient line of its dual. The Abel map is therefore an epimorphism of étale sheaves with kernel pair \(R\). The universal local-lift argument identifies its target with \(h_Q\). Thus \(Q\) is exactly \(\operatorname{Pic}^d_C\), constructed without presupposing its existence. The quotient map has relative dimension \(d-g\); as \(Y\) has dimension \(d\), the quotient has dimension \(g\). In the boundary case \(g=d=0\), the section rank is one, \(Y=\operatorname{Sym}^0C=\operatorname{Spec}k\), the relation is the identity and the quotient map has relative dimension zero. Properness of \(Q\) can also be read from the projective \(Y\): for any base change, the inverse image of a closed subset of \(Q\) is closed in \(Y\), and surjectivity makes their images in the base equal. This proves universal closedness; finite type and separatedness were already established.

What this lesson does not prove

The divisor locus, module of sections, complete-linear-system correspondence, Picard diagonal, positive-locus construction, effective quotient and gluing proof are established here. In particular the proper-flat quotient is not an unproved input.

The foundational results used are these precise ones:

The general algebraic-space theorem [Stacks, Tag 0D2C] is stated only to delineate the boundary of the scheme theorem; it is not used in the proof. The reductions of FGA 232 §6, component finiteness and the theorem of the base are not proved here. Nor do we assert that a universal line bundle exists without a section, or that the Picard scheme is smooth in arbitrary dimension.

References