The Picard Functor

Urs T. Hartl

Definition

Let S be a base scheme and let f : X ⟶ S be a morphism of schemes. Consider the contravariant functor from the category of S-schemes to the category of abelian groups

P_{X/S} : (Sch/S) ⟶ (Ab) , S′ ⟼ Pic(X ×_{S} S′) = H^{1}(X ×_{S} S′, O_{X×_{S}S′}^{∗}) .

The relative Picard functor is the (fppf)-sheaf associated to the functor P_{X/S}.

Pic_{X/S} = R^{1}f_{∗} 𝔾_{m} as (fppf)-sheaves on S .

This means Pic_{X/S} is a contravariant functor from (Sch/S) to (Ab) such that, for each S-scheme T and for each morphism T′ ⟶ T which is either faithfully flat and of finite presentation, i.e. (fppf) or a Zariski-covering, the following sequence is exact:

Pic_{X/S}(T) ⟶ Pic_{X/S}(T′) \overrightarrow{⟶} Pic_{X/S}(T′ ×_{T} T′) .

Every Element of Pic_{X/S}(T) for a quasi-compact S-scheme T can be given by a line bundle L′ on X ×_{S} T′ for some scheme T′ which is (fppf) over T . Furthermore there exists an (fppf)-morphism T̃ ⟶ T′ ×_{T} T′, such that the pullbacks with respect to the two projections T̃ ⟶ T′ are isomorphic.

Now consider the case:

(∗) Let f be quasi-compact and quasi-separated with a section x : S ⟶ X and let f_{∗}O_{X} = O_{S} universally, i.e. still valid after any base change. This holds for example if f is proper and flat with geometrically reduced and irreducible fibers.

Then Pic_{X/S} is the contravariant Functor from (Sch/S) to (Ab) given by

S′ ⟼ { Isomclasses of (L, λ) : L line bundle on X ×_{S} S′
λ : O_{S′} ⟶^{∼} (x × id_{S′})^{∗} L rigidification }
= Pic(X ×_{S} S′)/ Pic(S′) .

The rigidification λ has two effects. It kills all line bundles coming from S′ and secondly it causes the automorphism group of (L, λ) to be trivial.

Representability

The functor Pic_{X/S} is called representable if there exists an S-scheme P such that there is an isomorphism of functors

Pic_{X/S} ≅ Hom_{(Sch/S)}( _{•} , P) =: P ( _{•} ) .

In the case (∗) this means, that there exists a rigidified line bundle (P, ρ) ∈ Pic_{X/S}(P) called the Poincaré-bundle, with the universal property (Yoneda Lemma):

for every S-scheme S′ and for every line bundle (L, λ) ∈ Pic_{X/S}(S′) there exists a unique morphism g : S′ ⟶ P with

(L, λ) ≅ ( id_{X} ×g)^{∗}(P, ρ) .

Concerning the representability there is the following theorem.

Theorem 1. (Grothendieck) Let f : X ⟶ S be projective and finitely presented, flat with geometrically reduced and irreducible fibers. Then Pic_{X/S} is representable by a separated S-scheme which is locally of finite presentation over S.

\underline{Proof:} cf. [FGA, n^{o} 232, Thm. 3.1], [BLR, Thm. 8.2.1] \underline{1.} One introduces effective, relative Cartier divisors D on X over S, i.e. D is a closed subscheme of X, flat over S, which in each fibre is an effective Cartier divisor. One considers the contravariant functor

Div_{X/S} : (Sch/S) ⟶ (Sets)
S′  ⟼ { effective relative Cartier divisors on X ×_{S} S′/S′ } .

There is a morphism of functors Div_{X/S} ⟶ Pic_{X/S} sending D ↦ O_{X}(D) , which is shown to be relatively representable, i.e. for each S-scheme S′ the morphism

Div_{X/S} ×_{Pic_{X/S}}S′ ⟶ S′

is a morphism of schemes. So the divisors inducing a given line bundle in Pic_{X/S}(S′) are parameterized by a scheme.

\underline{2.} One shows the representability of the functor Div_{X/S} using the existence of the Hilbertscheme. This is the hardest part of the proof.

\underline{3.} For a fixed Φ ∈ ℚ[t] one considers the subfunctor Pic_{X/S}^{Φ} of Pic_{X/S}, which consists of all elements having Hilbert-polynomial Φ with respect to the given projective embedding of X.

For suitable Φ there exists a finite union D(Φ) of connected components of Div_{X/S} such that the functor Pic_{X/S}^{Φ} is the quotient

D(Φ) ⟶ Pic_{X/S}^{Φ}

by a proper and smooth equivalence relation. One now shows that therefore it is representable by a scheme.

For general Φ there exists an n_{Φ} ∈ ℤ such that the translate Pic_{X/S}^{Φ} +O_{X}(n_{Φ}) is of the special case above.

Since Pic_{X/S}^{Φ} is an open and closed subfunctor of Pic_{X/S} one finds that

Pic_{X/S} = ∐_{Φ∈ℚ[t]} Pic_{X/S}^{Φ}

is representable by a scheme over S.

A further theorem on the representability is the following.

Theorem 2. (Murre and Oort) Let X be a proper scheme over a field k. Then Pic_{X/k} is representable by a scheme which is locally of finite type over k.

\underline{Proof:} cf. [Mu], [Oo] One reduces to the projective case which was done by Grothendieck [FGA, n^{o} 232, Sect. 6].

If now X is proper over k, we define Pic_{X/k}^{0} as the connected component which contains the unit element. It is a group scheme of finite type over k.

If X is proper over S, we define the groupfunctor

Pic_{X/S}^{0}(T) := { ξ ∈ Pic_{X/S}(T) : ξ|_{X_{t}} ∈ Pic_{X_{s}/k(s)}^{0}(k(t)) ∀t ∈ T } .

The case of curves

In the case of curves more can be said.

Theorem 3. Let f : X ⟶ S be a proper, flat curve, locally of finite presentation. Then Pic_{X/S} is a formally smooth functor over S.

\underline{Proof:} cf. [BLR, Prop. 8.4.2] That Pic_{X/S} is formally smooth over S means that for each affine S-scheme Z and for each closed subscheme Z_{0} ⊆ Z which is given by a sheaf of ideals N with N^{2} = 0 the canonical map is surjective:

Hom(Z, Pic_{X/S}) ⟶ Hom(Z_{0}, Pic_{X/S}) .

This follows from considering the exact sequence

0 ⟶ N ⊗_{O_{S}} O_{X} ⟶ O_{X×_{S}Z}^{∗} ⟶ O_{X×_{S}Z_{0}}^{∗} ⟶ 0 .
n  ⟼ 1 + n

In fact applying (f × id_{Z})_{∗} yields

R^{1}(f × id_{Z})_{∗} O_{X×_{S}Z}^{∗} ⟶ R^{1}(f × id_{Z_{0}})_{∗} O_{X×_{S}Z_{0}}^{∗} ⟶ R^{2}(f × id_{Z})_{∗} N ⊗_{O_{S}} O_{X} .

Since X is a curve over S and N is quasi-coherent, the last term is 0 . So applying H^{0}(Z, _{•} ) and observing Z affine, one gets

H^{0}(Z, R^{1}(f × id_{Z})_{∗} O_{X×_{S}Z}^{∗}) ⟶ H^{0}(Z_{0}, R^{1}(f × id_{Z_{0}})_{∗} O_{X×_{S}Z_{0}}^{∗}) .
= Pic_{X/S}(Z)  = Pic_{X/S}(Z_{0})

and thus the above morphism is surjective.

If X is a proper curve over a field k, then Pic_{X/k} is a scheme locally of finite type over k by Theorem 2. So Pic_{X/k}^{0} is a scheme of finite type over k. In this case smoothness and formal smoothness are the same. So we see that Pic_{X/k}^{0} is a smooth k-scheme.

Next we want to study Pic_{X/k}^{0} in terms of divisors. So let X be a proper curve over a field k and D an effective Cartier divisor on X. Locally at a point x ∈ X the divisor D is given by a regular element l. We define the order of D in x as

ord_{x}(D) := length_{k}(O_{X,x}/(l)) .

If X is regular at x, then O_{X,x} is a discrete valuation ring and ord_{x}(D) is just the order of l in that ring. We now define the degree of D

deg(D) := ∑_{x∈X} ord_{x}(D) · [k(x) : k] .

It has the following properties:

1. deg is additive, so we can also define it for non-effective divisors.

2. deg is not altered by field extensions.

3. If X is reduced, the degree of a divisor on X is the same as the degree of its pullback to the normalization of X.

4. The degree of the divisor of a meromorphic function is zero.

So we can define the degree for line bundles. Let L be a line bundle on X, then there are two effective Cartier divisors D and D′ on X with L ≅ O(D − D′) . We define the degree of L

deg(L) := deg(D) − deg(D′) .

If X is a flat, proper curve of finite presentation over an arbitrary base S and L a line bundle on X, the function s ↦ deg(L|_{X_{s}}) is locally constant on S.

Next if X = X_{1} ∪ … ∪ X_{r} is the decomposition in irreducible components, we define the partial degree of L on X_{i}:

deg_{X_{i}}(L) := deg(L|_{X_{i}}) .

The partial degrees are related to the total degree by the formula

deg(L) = ∑_{i=1}^{n} d_{i} · deg_{X_{i}}(L) ,

where d_{i} is the multiplicity of X_{i} in X, i.e. d_{i} := length(O_{X,η_{i}}) for the generic point η_{i} of X_{i}.

Now let X be a smooth proper geometrically irreducible curve of genus g over a field k. Assume that X has a rational point x_{0} . Then there is a morphism

X^{g} ⟶ X^{(g)} = X^{g}/𝔖_{g} ⟶ Pic_{X/k}^{0} , x_{1} + … + x_{g} ↦ O_{X}(Σ(x_{i} − x_{0})) ,

where 𝔖_{g} is the symmetric group on g letters and X^{(g)} is the symmetric product of X. The latter morphism is an epimorphism and birational. The whole Picard variety decomposes

Pic_{X/k} = ∐_{d∈ℤ} Pic_{X/k}^{d} .

where Pic_{X/k}^{d} is the connected component of Pic_{X/k} representing the line bundles of degree d. It is a Pic_{X/k}^{0}-torsor.

In the following we consider arbitrary proper curves over a field k.

Proposition 4. Let X be a proper curve over a field k. Then Pic_{X/k}^{0} consists of all elements of Pic_{X/k} whose partial degree on each irreducible component of X ⊗_{k} \overline{k} is zero.

\underline{Proof:} cf. [BLR, Cor. 9.3.13] Let k be algebraically closed, X_{red} = ⋃ X_{i} be the irreducible components with normalizations X̃_{i} and g : ∐ X̃_{i} ⟶ X. Then Pic_{X/k} is an extension

1 ⟶ L ⟶ Pic_{X/k} ⟶^{g^{∗}} ∏_{i} Pic_{X̃_{i}/k} ⟶ 1

by a connected linear group L. Therefore we have

Pic_{X/k}^{0} = (g^{∗})^{−1}(∏_{i} Pic_{X̃_{i}/k}^{0}) .

The proposition now follows from the fact, that Pic_{X̃_{i}/k}^{0}(k) are exactly the line bundles having degree zero on X_{i}.

Description of Pic_{X/k}^{0}

Let X be a proper curve over a perfect field k and X̃ the normalization of X_{red} . We want to introduce an intermediate curve lying between X_{red} and X̃.

X ⟵⊃ X_{red} ⟵ X′ ⟵ X̃ .

There are only finitely many non-smooth points of X_{red} . We define X′ by identifying all points of X̃ lying above such a non-smooth point of X_{red} . (This can be formalized with the amalgamated sum, cf. [BLR, p. 247].) So the singularities of X′ are just ordinary multiple points.

The above maps induce morphisms on the Picard-schemes.

Pic_{X/k}^{0} ⟶ Pic_{X_{red}/k}^{0} ⟶ Pic_{X′/k}^{0} ⟶ Pic_{X̃/k}^{0} .

The next theorem tells more about the structure of these morphisms.

Theorem 5.

a) the map Pic_{X/k}^{0} ⟶ Pic_{X_{red}/k}^{0} is an epimorphism with a smooth connected unipotent group as kernel.

b) the map Pic_{X_{red}/k}^{0} ⟶ Pic_{X′/k}^{0} is an epimorphism with a smooth connected unipotent group as kernel, which is trivial if and only if X′ = X_{red} .

c) the map Pic_{X′/k}^{0} ⟶ Pic_{X̃/k}^{0} is an epimorphism with a torus as kernel, which is trivial if and only if each irreducible component is homeomorphic to its normalization and the configuration of the irreducible components of X ⊗_{k} \overline{k} is tree-like, i.e. if and only if H_{ét}^{1}(X ⊗_{k} \overline{k}, ℤ) = 0 .

d) Pic_{X̃/k}^{0} is an abelian variety.

\underline{Proof:} cf. [BLR, Sect. 9.2]

\underline{Ad c)} Let X′ = X = ⋃_{i=1}^{n} X_{i} be the irreducible components and g : X̃ = ∐_{i=1}^{n} X̃_{i} ⟶ X

be the normalization. Let further x_{1}, …, x_{N} be the singular points of X and x̃_{ν 1}, …, x̃_{ν m_{ν}} the points of X̃ lying above x_{ν}. Then we have the exact sequence

1 ⟶ O_{X}^{∗} ⟶ g_{∗} O∗ ; X̃ ⟶ g_{∗} O∗ ; X̃/O_{X}^{∗} ⟶ 1 .

The sheaf F := g_{∗} O∗ ; X̃/O_{X}^{∗} is concentrated at the points x_{ν}, so we get

\begin{matrix}1 ⟶ f_{∗} O_{X}^{∗} & ⟶ f_{∗}g_{∗} O∗ ; X̃ ⟶ & f_{∗} F ⟶ R^{1}f_{∗} O_{X}^{∗} ⟶ (R^{1}f_{∗})g_{∗} O∗ ; X̃ ⟶ 1 . \\ || & || & \\ \\ H^{0}(X, O_{X})^{∗} & ∏_{i=1}^{n} H^{0}(X̃_{i}, OX̃)^{∗} & ∏_{ν=1}^{N}(∏_{μ=1}^{m_{ν}} k(x̃_{νμ})^{∗})/k(x_{ν})^{∗} \end{matrix}

So the kernel is a quotient of a torus, thus a torus. The remaining assertion follows by combinatorial arguments.

Remark: One can describe the extension c) explicitly (cf. [Zh]).

For the remaining part let us work with curves over a discrete valuation ring. Then in the following case the Picard functor is representable.

Theorem 6. (Raynaud) Let S be the spectrum of a discrete valuation ring. Let f : X ⟶ S be a proper, flat, normal curve with f_{∗}O_{X} = O_{S} and geometrically reduced special fiber. Then Pic_{X/S}^{0} is representable by a separated S-scheme.

For the proof see [Ra, Thm. 8.2.1] or [BLR, Thm. 9.4.2].

Let now R be a complete discrete valuation ring and X a semi-stable curve over R (i.e. a proper, flat scheme whose geometric fibers are reduced and connected curves with only ordinary double points as singularities). Let further the generic fiber X_{K} be smooth over K and the irreducible components of the special fiber X_{k} be smooth over k.

Then by Theorem 5 after a base ring extension J_{k} := Pic_{X_{k}/k}^{0} is an extension

1 ⟶ 𝔾_{m,k}^{r} ⟶^{α} Pic_{X_{k}/k}^{0} ⟶ Pic_{X̃_{k}/k}^{0} ⟶ 1 ,

of an abelian variety by a torus . The rank of the torus is r = rk_{ℤ} H^{1}(X_{k}, ℤ) and the map α can be described as follows. Let t_{i} be the coordinates of the torus, then α is given by the line bundle t_{1}^{a_{1}} ⊗ … ⊗ t_{r}^{a_{r}} on X_{k} ×_{k} 𝔾_{m,k}^{r} for some basis a_{1}, …, a_{r} of H^{1}(X_{k}, ℤ) .

We now want to investigate this situation with formal and rigid geometric methods and consider the formal completion \overline{J} := (Pic_{X/R}^{0})^{∧} = Pic_{X̂/R}^{0} of Pic_{X/R}^{0} along its special fiber. Then the torus lifts over R to a smooth formal torus

1 ⟶ \overline{𝔾}_{m,R}^{r} ⟶ \overline{J} ⟶ B ⟶ 1 .

There \overline{𝔾}_{m,R} is the formal completion of 𝔾_{m,R} along its special fiber. The quotient B is a formal abelian scheme. Actually it is the formal completion of an abelian scheme over R. On the rigid fibers we obtain

1 ⟶ \overline{𝔾}_{m,K}^{r} ⟶ \overline{J}_{rig} ⟶ B_{rig} ⟶ 1 .

\overline{J}_{rig} parameterizes the formally smooth deformations of the trivial line bundle on X_{K} . Or phrased differently it is given by the divisors on X_{K} whose reductions are divisors with all partial degrees equal to zero.

As push-forward of \overline{J}_{rig} via the open immersion of the formal torus into the affine torus we get

[figure: 1; 1; \overline{𝔾}_{m,K}^{r}; 𝔾_{m,K}^{r}; \overline{J}_{rig}; J̃_{rig}; B_{rig}; B_{rig}; 1; 1]

On X\begin{matrix}an \\ K \end{matrix} ×_{K} J̃_{rig} there is a universal line bundle (P̃, ρ̃) , which induces a canonical morphism J̃_{rig} ⟶ Pic_{X_{K}^{an}/K}^{0} = (Pic_{X_{K} /K}^{0})^{an} . The kernel

M = { p ∈ J̃_{rig} : (P̃, ρ̃)|_{X_{K}^{an}×{p}} trivial }

of this morphism is a lattice in J̃_{rig} of full rank. The intersection M ∩ \overline{J}_{rig} contains only the unit element of \overline{J}_{rig} . The quotient

J̃_{rig}/M = (Pic_{X_{K} /K}^{0})^{an}

exists and makes J̃_{rig} into the universal covering of the rigid space (Pic_{X_{K} /K}^{0})^{an} .

For more details see [BL].

References

[BL] Bosch, S., Lütkebohmert, W.: Stable Reduction and Uniformization of Abelian Varieties II, Invent. math. 78, 257-287 (1984).

[BLR] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models, Springer-Verlag, Berlin-Heidelberg 1990.

[FGA] Grothendieck, A.: Fondements de la Géométrie Algébrique, Séminaire Bourbaki 1957-62, Secrétariat Math., Paris 1962.

[Mu] Murre, J.P.: On contravariant functors from the category of preschemes over a field into the category of abelian groups, Publ. Math. IHES 23, 5-43 (1964).

[Oo] Oort, F.: Sur le schéma de Picard, Bul. Soc. Math. Fr. 90, 1-14 (1962).

[Ra] Raynaud, M.: Spécialisation du foncteur de Picard, Publ. Math. IHES 38, 27-76 (1970).

[Zh] Zhang, B.: Sur les jacobiennes de courbes à singularités ordinaires, Manuscripta Math. 92, 1-12 (1997).

Urs T. Hartl University of Ulm Abt. Reine Mathematik D – 89069 Ulm GERMANY

e-mail: hartl@mathematik.uni-ulm.de