(page 1) Chapter 8

Picard functor

effective Cartier divisor divisor, relative Cartier divisor Weil divisor divisor, Cartier divisor, Weil functor of curves complete linear system

Linear Systems

In the previous chapter, we saw that there was a natural transformation

Φ : Curves_{X} → Pic_{X},

where X is a smooth projective surface over a field k. The first functor classifies families of curves on X, and the second functor classifies invertible sheaves. As we’ve said many times by now, the first main goal of these notes is to show that both these functors are actually schemes. If that claim is true, then the fibre of Φ should also be a scheme. In this chapter, we are going to identify the fibres of Φ as schemes directly.

To get an idea of what the answer should look like, we need to step back from the rarified air of functors and recall some facts from the classical theory of curves on surfaces. As a starting point, note that Pic_{X}(k) = Pic(X × k)/Pic(k) = Pic(X). So, the k-points of the Picard functor of X are nothing other than the usual isomorphism classes of invertible sheaves on X. In the same way, Curves_{X}(k) is the set of effective Cartier divisors on X × k = X relative to k. Since k is a field, every module over k is flat. Thus, the relativity condition is irrelevant. Since X is smooth and projective, Cartier divisors and Weil divisors are the same thing. In brief, the k-points on the functor of curves on X are nothing other than the effective Weil divisors on X.

Now let L ∈ Pic(X) be an invertible sheaf. The fibre of Φ over the k-point L should (if there’s any justice) consist of those effective divisors D on X such that L ≈ O_{X}(D). In classical terms, this fibre is a complete linear system on X, and its points correspond in a natural way to the points of a projective space of dimension N − 1, where N = dim_{k}(Γ(X, L)).

This analysis suggests that we should expect the fibres of Φ to be projective spaces. Before we can prove such a result, however, we need to be able to compute the fibre of a natural transformation in a completely functorial manner.

Definition 8.1. Let C be an arbitrary category, and let S ∈ C be any object. Define C/S to be the category in which an object is an arrow T → S in C, and in which a morphism from T → S to U → S is given by an arrow T → U in C making

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2 Chapter 8: Linear Systems

fibre functor fibre functor fibre functor

the following diagram commute:

\begin{matrix}T & ⟶ & U \\ ↓ & & ↓ \\ S & ⟶^{=} & S. \end{matrix}

Definition 8.2. Let F, G : C^{op} → SET be contravariant functors, and let Φ : F → G be a natural transformation. Let S ∈ C be any object, and let α ∈ G(S). The fibre functor of Φ over α is defined to be the contravariant functor

Φ^{α} : (C/S)^{op} → SET

given by

Φ^{α}(f : T → S) = {β ∈ F (T) : Φ(β) = f^{∗}(α) ∈ G(T)}.

Example 8.3. Suppose C = SCH/k is the category of k-schemes. Assume that F = h_{X} and G = h_{Y} are representable functors; i.e., for all k-schemes T, one has F (T) = Hom(T, X) and G(T) = Hom(T, Y). Then the natural transformation Φ : F → G must be given by composition with some morphism of schemes ϕ : X → Y. That is, if β ∈ F (T), we can view β as a morphism β : T → X and then Φ(β) = ϕ ∘ β : T → Y.

Now let α ∈ G(k) be any k-point of G. By definition, the fibre Φ^{α} is another functor on the category of k-schemes. Let f : T → Spec(k) be the structure morphism of a k-scheme T. An element β ∈ F (T) = Hom(T, X) lies in the fibre Φ^{α}(T) if and only if Φ(β) = f^{∗}(α). That is, β is a point of the fibre if and only if there is a commutative diagram

T  ⟶^{β} X
↓^{f}  ↓^{ϕ}
Spec(k) ⟶^{α} Y.

By the universality of fibre products, we can reinterpret this in the following way:

Φ^{α}(T) = Φ^{α}(f : T → Spec(k)) = {β : T → ϕ^{−1}(α)} = Hom(T, ϕ^{−1}(α)).

So, if F = h_{X} and G = h_{Y} are both schemes, then the fibre functor of the natural transformation associated to a morphism ϕ : X → Y is represented by the scheme ϕ^{−1}(α).

The point of this laborious example, of course, is to convince you that the definition of the fibre functor of a natural transformation is not total nonsense. In the case where everything can be interpreted in terms of schemes and morphisms, the fibre functor is just the usual scheme theoretic fibre.

Let’s reinterpret the fibre functor in our case. As we have seen, a k-point of Pic_{X} is just an invertible sheaf L on the smooth projective surface X. The fibre of Φ over L is the functor that assigns to a k-scheme T the set of all effective Cartier

◯^{c}Coombes 1996 effective Cartier divisor divisor, relative generic point Künneth Formula

divisors D on X × T relative to T such that O_{X×T} (D) ≈ p_{X}^{∗}(L) ⊗ p_{T}^{∗} (K) for some invertible sheaf K ∈ Pic(T). We’ll denote this functor by LIN SYS_{L}.

Proposition 8.4. Let L ∈ Pic(X) be an invertible sheaf on a smooth projective surface X over a field k. Let N = dim_{k}(H^{0}(X, L)). Then

LIN SYS_{L} ≈ h_{P^{N−1}}

Proof: To give a divisor D ∈ LIN SYS_{L}(T), one needs to specify an invertible sheaf K on T and a section s ∈ H^{0}(X × T, p_{X}^{∗}(L) ⊗ p_{T}^{∗} (K)). Moreover, since s = 0 defines a relative effective Cartier divisor, we must have

s(x) ≠ 0 if x ∈ A(p_{T}^{−1}(p_{T} (x))).

Let t ∈ T and write K = k(t), so that p_{T}^{−1}(t) = X ×_{k} K. By definition, the surface X is geometrically irreducible, so it has a unique generic point. This, the condition that s defines a relative effective Cartier divisor is simply that s is not identically zero along any fibre of p_{T}.

Now suppose that two pairs (K_{1}, s_{1}) and (K_{2}, s_{2}) define the same Cartier divisor D on X × T. I claim that there is an isomorphism K_{1} ≈ K_{2} under which s_{1} corresponds to s_{2}. To see this, note first that we have an isomorphism

p_{X}^{∗}(L) ⊗ p_{T}^{∗} (K_{1}) ≈ p_{X}^{∗}(L) ⊗ p_{T}^{∗} (K_{2}).

Twisting away the invertible sheaf p_{X}^{∗}(L), we get an isomorphism

p_{T}^{∗} (K_{1}) ≈ p_{T}^{∗} (K_{2}).

So, we can get an isomorphism between K_{1} and K_{2} if we know that p_{T}^{∗} is an injection. Since base extension to the algebraic closure induces injections Pic(T) → Pic(T̄), it is enough to show the injectivity when k is algebraically closed. Now choosing a k-point x ∈ X, one gets a section ϵ : T → X × T of the projection map. Pulling the above isomorphism back along ϵ gives us an isomorphism between K_{1} and K_{2}. So, we can assume that K_{1} = K_{2}.

Now suppose s_{1} ≠ s_{2}. Since they define the same divisor, their difference is measured by an element α ∈ H^{0}(X × T, O_{X×T}^{∗}). However,

H^{0}(X × T, O_{X×T}^{∗}) = H^{0}(T, p_{T}^{∗} (O_{X×T}^{∗})) = H^{0}(T, O_{T}^{∗}).

So, we can modify the isomorphism to make s_{1} = s_{2}.

We have identified LIN SYS_{L}(T) with the set of pairs (K, s) where K is an invertible sheaf on T and s is a global section of p_{X}^{∗}(L) ⊗ p_{T}^{∗} (K) that does not vanish on any entire fibre of p_{T}, up to isomorphism. By the Künneth Formula, we can write

H^{0}(X × T, p_{X}^{∗}(L) ⊗ p_{T}^{∗} (K)) = H^{0}(X, L) ⊗ H^{0}(T, K).

Choosing a basis e_{1}, …, e_{N} of H^{0}(X, L), we can write s = ∑_{i=1}^{N} e_{i} ⊗ s_{i} where s_{i} ∈ H^{0}(T, K). The condition that s is not identically zero on the fibre over t is

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simply that the s_{i}(t) do not all vanish simultaneously. So, functor of points

LIN SYS_{L}(T) = {(K, s_{1}, …, s_{N})}

where K is an invertible sheaf on T and s_{1}, …, s_{N} are global section that generate K as a sheaf. But that is exactly the description we worked out earlier for the functor of points associated to projective space.

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