(page 1) Chapter 12

Picard functor functor of curves

The Picard Scheme

We’d like to start studying the Picard functor Pic_{X}. As with the functor of curves, we need to decompose this functor as a disjoint union of related functors. However, we have to confront an arithmetic problem that did not arise in the case of curves. The natural index for the pieces of the Picard functor are the numerical equivalence classes of divisors or of invertible sheaves. The arithmetic problem arises when we try to decide if we should take numerical equivalence classes over the base field k or over its algebraic closure k̄.

Base Change

Proposition 12.1. (Hilbert’s Theorem 90) Let G_{m} denote the group of units of X, viewed as a sheaf in either the flat or étale topologies. Then the canonical maps

H^{1}(X, O_{X}^{∗}) → H^{1}(X_{et}, G_{m}) → H^{1}(X_{fl}, G_{m})

are isomorphisms. In particular, H^{1}(X_{fl}, G_{m}) = Pic(X).

Proof: Let f : X_{fl} → X_{Zar} be the canonical morphism of sites induced by the identity on X. The Leray spectral sequence of f is

E_{2}^{p,q} = H^{p}(X_{Zar}, R^{q}f_{∗}G_{m}) ⇒ H^{p+q}(X_{fl}, G_{m}).

Looking at the sequence of low degree terms, we see that it suffices to show that R^{1}f_{∗}G_{m} = 0. To do this, we must show that any element of H^{1}(U_{fl}, G_{m}), where U is an arbitrary Zariski open subset of X, becomes zero in H^{1}((U_{i})_{fl}, G_{m}), where the U_{i} are some Zariski open cover of U. This reduces, in turn, to showing that H^{1}(U_{fl}, G_{m}), where U = Spec(A) and A is a local ring. Since Čech cohomology always suffices to compute the first cohomology group. Some other stuff goes here, but I’m getting nauseous.

Corollary 12.2. Let k be a separably closed field and let k̄ be its algebraic closure. Let X be a smooth, projective, geometrically irreducible k-scheme, and let X̄ denote

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2 Chapter 12: The Picard Scheme

its base extension to k̄. Then pulling back along the natural map X̄ → X induces intersection number an isomorphism Pic(X) → Pic(X̄).

Proof: Because the Picard group can be computed in the étale topology, every invertible sheaf on X̄ already comes from a sheaf defined on X.

Corollary 12.3. Let X be a smooth, projective, geometrically irreducible scheme over an arbitrary field k, and let X̄ be the corresponding scheme over the algebraic closure k̄. Pullback along the natural map X̄ → X induces an injection Pic(X) → Pic(X̄).

Proof: By the previous corollary, we can replace k̄ by the separable closure. Let G = Gal(k̄/k), and compute in the étale topology. The sequence of low degree terms in the Lyndon-Hochschild-Serre spectral sequence is

H^{1}(G, H^{0}(X̄, G_{m})) → H^{1}(X, G_{m}) → H^{0}(G, H^{1}(X̄, G_{m})).

The first term vanishes by Hilbert’s Theorem 90, and the remaining terms translate into an injection

Pic(X) → Pic(X̄)^{G}.

Theorem 12.4. Let X be a smooth, projective surface over a field k, and let X̄ be the corresponding surface over the algebraic closure. If the functor PicX̄ is represented by a k̄-scheme, then the functor Pic_{X} is represented by a k-scheme.

Proof: Who knows? Probably descent theory.

From now on, we can assume that the field k is algebraically closed.

Intersection Theory Revisited

Let L and M be two invertible sheaves on X × T. For all points t ∈ T, we get induced sheaves L_{t} and M_{t} on the fibre X_{t}. Since X_{t} is a surface, we get intersection numbers (L_{t}.M_{t}).

Proposition 12.5. The intersection numbers (L_{t}.M_{t}) are constant on connected components of T.

Proof: Let’s quickly review how to compute intersection numbers on a surface X. If C and D are smooth, irreducible curves that meet properly, then the intersection number C.D is just the number of points of intersection. That is, if we write Z = C ∩ D, then

C.D = #Z = deg(Z).

◯^{c}Coombes 1996 Riemann-Roch Theorem for Curves Euler characteristic numerical equivalence Picard functor numerical equivalence

Since Z is the restriction of the divisor D to the curve C, we have O_{C}(D) = O_{C}(Z). By the Riemann-Roch Theorem for Curves, we get

χ(O_{C}(−D)) = −C.D + 1 − g(C)

and

χ(O_{C}) = 1 − g(C).

Therefore,

C.D = χ(O_{C}) − χ(O_{C}(−D)).

We can transport this computation back to the surface X as follows. Since C is an effective divisor on X, we have an exact sequence of sheaves

0 → O_{X}(−C) → O_{X} → O_{C} → 0.

Twisting by −D gives the exact sequence

0 → O_{X}(−C − D) → O_{X}(−D) → O_{C}(−D) → 0.

By the additivity of the Euler characteristic, we get

C.D = χ(O_{C}) − χ(O_{C}(−D))
= (χ(O_{X}) − χ(O_{X}(−C))) − (χ(O_{X}(−D)) − χ(O_{X}(−C − D)))
= χ(O_{X}) − χ(O_{X}(−C)) − χ(O_{X}(−D)) − χ(O_{X}(−C − D))

This computation generalizes to all pairs of invertible sheaves, giving

(L.M) = χ(O_{X}) − χ(L^{v}) − χ(M^{v}) + χ(L^{v} ⊗ M^{v}).

The result now follows from the invariance of Euler characteristics (and Hilbert polynomials) in flat families.

Definition 12.6. Let Pic^{τ} (X) denote the subgroup of Pic(X) consisting of all invertible sheaves L such that (L.M) = 0 for all M ∈ Pic(X). The quotient Num(X) = Pic(X)/Pic^{τ} (X) is called the group of numerical equivalence classes of divisors on X.

Remark 12.7. Because the intersection numbers are constant on connected components, we can decompose the Picard functor as a disjoint union of the subfunctors Pic_{X}^{ξ}, where Pic_{X}^{ξ}(T) is the subset of Pic_{X}(T) consisting of all invertible sheaves L on X × T such that each fibre L_{t} at a closed point t ∈ T has numerical equivalence class ξ.

Lemma 12.8. All the functors Pic_{X}^{ξ} are isomorphic.

Proof: Let ξ_{1} and ξ_{2} be numerical equivalence classes represented by invertible sheaves K_{1} and K_{2}. Let L be an invertible sheaf on X × T representing an element of Pic_{X}^{ξ_{1}}(T). Map this to the element L ⊗ p_{X}^{∗}(K_{1} ⊗ K_{2}^{−1}) in Pic_{X}^{ξ_{2}}. This defines an isomorphism between the functors.

◯^{c}Coombes 1996 linear equivalence numerical equivalence Picard functor separated scheme effective Cartier divisor d

So, it is enough to pick one ξ and show that the corresponding functor is represented by a scheme. Everybody’s first choice is probably to take ξ = 0, so that the resulting functor Pic_{X}^{0} = Pic_{X}^{τ} is a group scheme. In fact, however, we will instead take a numerical equivalance class ξ with enough nice properties to ensure that we can pick out good curves representing each linear equivalence class corresponding to ξ. If we can do that, then the following proposition shows us how to complete the proof.

Proposition 12.9. Let ξ ∈ Num(X) be a numerical equivalence class. If the map ϕ : Curves_{X}^{ξ} → Pic_{X}^{ξ} admits a section σ, then Pic_{X}^{ξ} is represented by a closed subscheme of Curves_{X}^{ξ}.

Proof: By hypothesis, ϕ∘σ is the identity. On the other hand, σ ∘ϕ is a morphism from Curves_{X}^{ξ} to itself that projects the entire functor onto a subfunctor isomorphic to the Picard functor. Write C(ξ) for the scheme that represents Curves_{X}^{ξ}. Let f : C(ξ) → C(ξ) be the homomorpism that corresponds to the composite natural transformation σ ∘ ϕ. Define P (ξ) by the pullback diagram

P (ξ) ⟶^{g}  C(ξ)
↓  ↓^{(1,f)}
C(ξ) ⟶^{Δ} C(ξ) × C(ξ),

where Δ is the diagonal map.

By the definition of the fibre product, Hom(T, P (ξ)) can be identified with the set of pairs α, β ∈ Hom(T, C(ξ)) such that Δ(α) = (1, f)(β). In other words, we have an equality α × α = β × f(β) ∈ Hom(T, C(ξ) × C(ξ)). Thus, Hom(T, P (ξ)) is isomorphic to the subset of Hom(T, C(ξ)) left fixed by f; i.e., to the subset of Curves_{X}^{ξ}(T) left fixed by σ∘ϕ. But we’ve already seen that this description identifies Pic_{X}^{ξ}(T). Finally, we know that C(ξ), being a projective scheme, is separated. So, Δ is a closed immersion, and thus its pullback g is also a closed immersion.

Now we need to solve the following problem: Given an invertible sheaf L on X × T that has numerical class ξ along the fibres, construct a relative effective Cartier divisor ivisor, relative D ⊂ X × T such that O_{X×T} (D) = L ⊗ p_{T}^{∗} (K) for some invertible sheaf K ∈ Pic(T). This construction must have two additional nice properties

(i) If we replace L by L ⊗ P_{T}^{∗}(K_{1}) for any invertible sheaf K_{1} on T, then we must get the same divisor D; and

(ii) It must commute with base extensions S → T.

So, let L be an invertible sheaf on X × T and let x ∈ X be a closed point. Write ι_{x} : T → X × T for the section of p_{T} that maps T onto {x} × T. Consider the sheaves

M_{x} = ι_{x}^{∗}(L)

◯^{c}Coombes 1996 exterior power effective Cartier divisor divisor, relative

and

E = (p_{T})_{∗}(L)

on the scheme T. There is a canonical homomorphism

h_{x} : E → M_{x},

which can be described as follows. If s ∈ E(U) is a section of E over the open set U ⊂ T, then the equality E(U) = L(X × U) allows us to view s as a section of L over X × U. Now can pull this section back along ι_{x} to get a section of M_{x} over U = ι_{x}^{−1}(X × U). Condition I. Suppose we know that any sheaf of type ξ is very ample, and that its higher cohomology vanishes.

Since the restriction of L to any fibre of p_{T} is of type ξ, then we would know that E is a locally free sheaf of rank r determined completely by the class ξ. Now let x_{1}, …, x_{r−1} be a collection of closed points in X. Putting together the h_{x} maps for all these points defines a morphism

h : E → ⨁_{i=1}^{r−1} M_{x_{i}}.

Taking exterior powers of this map defines a homomorphism

∧h : ⋀^{r−1}(E) → ⨂_{i=1}^{r−1} M_{x_{i}}.

Taking the dual of this map, one gets a homomorphism

(∧h)^{∗} : ⨂_{i=1}^{r−1} M_{x_{i}}^{−1} → Hom( ⋀^{r−1}(E), O_{T}) = E ⊗ (⋀^{r}(E))^{−1}.

Twisting by all the invertible sheaves in sight, we thus get a homomorphism

h̃ : O_{T} → E ⊗ {(⋀^{r}(E))^{−1} ⊗ [⨂_{i=1}^{r−1} M_{x_{i}}]} .

By the projection formula, we can transport this element back up to X × T, where it represents a global section σ of the invertible sheaf

(12.1)  L ⊗ p_{T}^{∗} {(⋀^{r}(E))^{−1} ⊗ [⨂_{i=1}^{r−1} M_{x_{i}}]} .

Condition II. Suppose in addition that σ does not vanish entirely on any of the fibres of p_{T}.

Then σ = 0 defines a relative effective Cartier divisor D on X × T whose associated sheaf O_{X×T} is the sheaf (12.1), which is exactly what we are trying to construct. It should be clear that all our steps commute with base extension.

◯^{c}Coombes 1996

Lemma 12.2. If we replace L with L ⊗ p_{T}^{∗} (M), then we get the same Cartier divisor D.

Proof: We see first that E is replaced by

(p_{T})_{∗}(L ⊗ p_{T}^{∗} (M)) = (p_{T})_{∗}(L) ⊗ M = E ⊗ M,

and each M_{x} is replaced by

ι_{x}^{∗}(L ⊗ p_{T}^{∗} M) = ι_{x}^{∗}(L) ⊗ M = M_{x} ⊗ M.

The extra copies of M cancel out at the point when we build the map h̃, and the rest of the construction then gives the same divisor D.

So, we see that we can complete the proof if we can verify conditions I and II. Let’s look at the latter of these conditions first. What does it mean to say that σ does not vanish identically on a fibre p_{T}^{−1}(t)? Write L_{t} for the invertible sheaf induced by L on the fibre, and let ϕ_{t} : X → P^{r−1} be the morphism to projective space defined by choosing a basis of H^{0}(X, L_{t}).

Lemma 12.3. The map σ is not identically zero on the closed fibre p_{T}^{−1}(t) if and only if the points ϕ_{t}(x_{1}), …, ϕ_{t}(x_{r−1}) span a hyperplane in P^{r−1}.

Proof: Since the construction of σ is functorial, we can make the base extension to t : Spec(k) → T, replace L by L_{t}, and see when σ turns out to be zero. In this case, we have E = H^{0}(X, L_{t}) and M_{x} = L_{t} ⊗ k(x). So, σ is nonzero if and only if the linear functionals h_{x_{i}} : H^{0}(X, L_{t}) → L_{t} ⊗ k(x_{i}) are linearly independent; i.e., the intersection of the kernels is one-dimensional. However, ϕ_{t} identifies H^{0}(X, L_{t}) with H^{0}(P^{r−1}, O(1)). The condition that the intersection of a collection of linear functionals here is one-dimensional reduces precisely to the statement that there is a unique hyperplane containing the points ϕ_{t}(x_{1}), …, ϕ_{t}(x_{r−1}).

Remark 12.4. We know that the image ϕ_{t}(X) does not lie in any proper linear subspace. So, for each t ∈ T, a general collection of r − 1 points of X must span a hyperplane. The problem is that the quantifiers are stacked in the worng order: we need to find a single collection of points that works for all t ∈ T. Unfortunately, that problem is much too hard to solve (since it might not even be true). Instead, we need to generalize the method we are using to build a section of the canonical map from curves to invertible sheaves.

We start by choosing Nr − 1 points of X, and group them into N − 1 sets of r points and one final set of r − 1 points. Refer to this entire grouping of points as γ. We apply the construction that was just described to the final set of r − 1 points. For each of the sets of r-points, however, we mimic the construction with the map

h : E → ⨂_{i=1}^{r} M_{x_{i}}.

Take the r^{th} exterior power, dualize, and twist away the invertible sheaves to get a map

h̃ : O_{T} → (⋀^{r}(E))^{−1} ⊗ [⨂_{i=1}^{r} M_{x_{i}}] .

Again, we pull these back up to X × T. Combining everything into one huge package, we get a section

σ_{γ} ∈ H^{0}(X × T, L ⊗ p_{T}^{∗} (K)),

where

K = (⋀^{r}(E))^{−N} ⊗ [ ⨂_{i=1}^{Nr−1} M_{x_{i}}] .

Now we are also going to allow ourselves the freedom to assign a scalar a_{γ} ∈ k to each collection γ. Then we can replace Condition II by the following.

Condition II’. There exists a suitable choice of ξ, N, of Nr − 1 points on X and of scalars a_{γ} such that for all invertible sheaves L on X of type ξ, the canonical section ∑_{γ} a_{γ}σγ is never zero.

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