(page 1) Chapter 7

Picard functor flat morphism Weil divisor divisor, Weil principal divisor divisor, principal Chow group

Relative Effective Cartier Divisors

We’re now almost ready to write down, in a precise form, the functors that measure families (of linear equivalence classes) of curves on a smooth projective surface X over a field k. As it turns out, the definitions are much easier for invertible sheaves than for divisors. For any k-scheme T, we define the Picard functor of X at T to be

Pic_{X}(T) = Pic(X ×_{k} T)/p_{T}^{∗} (Pic(T))
= \frac{\{isomorphism classes of invertible sheaves on X × T \}}{L_{1} ∼ L_{2} ⟺ ∃M ∈ Pic(T) : L_{1} ≈ L_{2} ⊗ p_{T}^{∗} (M)}

If f : S → T is any morphism of k-schemes, then pullback of invertible sheaves defines a group homomorphism (1 × f)^{∗} : Pic_{X}(T) → Pic_{X}(S). Thus, we see immediately that Pic_{X} is a contravariant functor from the category of k-schemes to the category of abelian groups. It is much harder to construct an analogous functor for families of actual divisors on X.

Example 7.1. Consider the projection Y = Spec(k[x, y]/(xy)) → X = Spec(k[x]) from the coordinate axes in the plane onto the x-axis. The divisor x = 0 on X pulls back to an entire component of Y. There is no way to fix this example to convert the pullback into a divisor; the source of the difficulty lies in the fact that the morphism is not flat.

A Plethora of Divisors

It’s time for a quick review of several different notions of divisor. For the proofs of the assertions in this section, you should consult @@99@@. For purposes of this discussion, we will say that a scheme X is “nice” if it is integral, noetherian, separated, and regular in codimension one. When X is nice, we can take the most geometrically appealing notion of divisor. A Weil divisor on X is a finite sum of irreducible closed subschemes of codimension one with integral coefficients. Using the notation of Chapter 5, we write Z^{1}(X) for the group of all Weil divisors. The “nice” hypotheses are used to attach a divisor to any nonzero rational function f ∈ k(X)^{∗}. Divisors of functions are called principal, and the quotient of the group of Weil divisors by the principal Weil divisors is the first Chow group, CH^{1}(X).

◯^{c}Coombes 1996  1

(page 2)

2 Chapter 7: Relative Effective Cartier Divisors

Cartier divisor divisor, Cartier principal divisor divisor, principal Cartier class group effective Cartier divisor

An obvious weakness of Weil divisors, for our purposes, is that we have to look at the entire category of k-schemes, and we can’t restrict ourselves to the nice ones. So, let X be any scheme. Let K = K_{X} be the sheaf of total quotient rings of X. Taking units, one obtains an exact sequence of sheaves

0 → O^{∗} → K^{∗} → K^{∗}/O^{∗} → 0.

A Cartier divisor on X is defined to be a section of CD(X) = Γ(X, K^{∗}/O^{∗}). By the definition of a quotient sheaf, each Cartier divisor is represented by an open cover U = {U_{i}} and functions f_{i} ∈ Γ(U_{i}, K^{∗}) such that f_{i}/f_{j} ∈ Γ(U_{i} ∩ U_{j}, O^{∗}).

A Cartier divisor is called a principal divisor if it lies in the image of Γ(X, K^{∗}). The quotient of the Cartier divisors by the principal Cartier divisors will be called the Cartier class group and denoted CaCl(X). The following statements can easily be derived from the definitions.

• If X is a nice scheme, then the Cartier divisors on X can be identified with the locally principal Weil divisors.

• If X is nice and locally factorial, then CD(X) = Z^{1}(X).

• If X is nice, then there is an injection α : CaCl(X) → CH^{1}(X).

• If X is nice and locally factorial, then α is an isomorphism.

• There is an injection β : CaCl(X) → Pic(X) = H^{1}(X, O^{∗}).

• If X is integral, then β is an isomorphism. It is worth examining the injection β in more detail. Let D = {(U_{i}, f_{i})} be a

Cartier divisor on a scheme X. On U_{i}, the element 1/f_{i} generates a one-dimensional O_{X}-submodule of K_{X}. Since the ratios f_{i}/f_{j} are regular units on U_{i} ∩ U_{j}, these submodules glue together into an invertible sheaf on X. This sheaf is denoted O_{X}(D). It is a subsheaf of K, and the map β : CaCl(X) → Pic(X) is given by β : D ↦ O_{X}(D).

Definition 7.2. A Cartier divisor D = {(U_{i}, f_{i})} on a scheme X is called an effective Cartier divisor if one of the following equivalent conditions holds.

(i) Each of the local equations satisfies f_{i} ∈ Γ(X, O_{X}).

(ii) There are inclusions O_{X} ⊂ O_{X}(D) ⊂ K.

(iii) O_{X}(−D) ⊂ O_{X} is an ideal sheaf.

When X is a nice scheme, effective Cartier divisors are identified with effective locally principle Weil divisors; i.e., Weil divisors that can be written as a finite sum of irreducible closed subschemes of codimension one with positive integer coefficients.

We have seen that Cartier’s notion of a divisor extends Weil’s notion in a natural way from nice schemes to all schemes. Unfortunately, this new notion only solves part of our problem. We still can’t pull Cartier divisors back along arbitrary morphisms. The obvious way to try to define a pullback is as follows. Let g : Y → X be a morphism of schemes, and let D = {(U_{i}, f_{i})} be a Cartier divisor on X. Then g^{∗}(f_{i}) = f_{i} ∘ g is a function on g^{−1}(U_{i}), so we can try to construct a Cartier divisor using the data

(7.3)  g^{∗}(D) = {(g^{−1}(U_{i}), g^{∗}(f_{i}))}.

◯^{c}Coombes 1996 associated point germ annihilator support generic point annihilator generic point generic point embedded point flat

To see that this attempt fails, let’s reconsider our earlier example. The divisor x = 0 on X = Spec(k[x]) is a principal (Weil or Cartier) divisor. When we try to pull it back to Y = Spec(k[x, y]/(xy)), the local equation x becomes a zero divisor. That’s forbidden in the definition of a Cartier divisor, since the local equations must be units in the total quotient ring. In the next section, we will see that this example illustrates the only real obstruction to forming pullbacks, and we’ll find a way to avoid it.

Associated Points

Let F be a coherent sheaf on a noetherian scheme X. We say that x ∈ X is an associated point of F if there exists a germ s ∈ F_{x} whose annihilator is primary to the maximal ideal M_{x} ⊂ O_{X,x}. Equivalently, x ∈ X is an associated point of F if we can find an open neighborhood U of x and a section s ∈ Γ(U, F) such that the support of s is contained in {x}^{−}.

Neither of those descriptions is terribly enlightening, so we should probably look at some examples. First, let η ∈ X be the generic point of an irreducible component. Take U to be an open neighborhood of η that does not meet any other irreducible component. Since the closure of η in U is the whole thing, the support of any nonzero section s ∈ F(U) must lie in the closure. So, η is an associated point of any sheaf F that is not identically zero along that irreducible component.

Let’s write A(F) for the set of associated points of a sheaf F. As a special case, we’ll write A(X) = A(O_{X}). Now let x ∈ A(X) be an associated point. Let f ∈ O_{X,x} be a nonzero section whose annihilator I is primary to the maximal ideal M_{x}. If g ∈ I, then, by the definition of the annihilator, gf = 0. Thus, g is a zero divisor. Since I is primary for M_{x}, for each g ∈ M there is a positive integer n such that g^{n} ∈ I. But then g^{n}f = 0, and g is also a zero divisor. Thus,

A(X) = {x ∈ X : M_{x} consists entirely of zero divisors}.

In particular, if X is reduced, then A(X) is equal to the set of generic points of the irreducible components of X.

Example 7.4. Let X = Spec(k[x, y]/(xy, y^{2})). As a scheme, X is just the affine line together with an extra nilpotent element in the local ring at the origin. Let O be the local ring at the origin. The annihilator of y in O is clearly equal to the maximal ideal M = (x, y). So, the origin is an associated point of X. An associated point that is not the generic point of an irreducible component is often called an embedded point of X.

Proposition 7.5. Let f : X → Y be a morphism of noetherian schemes. Let F be a coherent sheaf on X. If F is flat over Y, then f(A(F)) ⊂ A(Y).

Proof: Let x ∈ X and suppose y = f(x) ∉ A(Y). Then there exists a nonunit a ∈ M_{y} ⊂ O_{Y,y} such that multiplication by a defines an injection

0 → O_{Y,y}⟶^{a}O_{Y,y}.

◯^{c}Coombes 1996 flat generic point associated point closed point point, closed Cartier divisor divisor, Cartier

Since f^{∗} : O_{Y,y} → O_{X,x} is a local homomorphism, a pulls back to an element f^{∗}a ∈ M_{x} ⊂ O_{X,x}. Since F is flat over Y, we can tensor the exact sequence with F to get an injection

0 → F_{x}⟶^{f^{∗}a}F_{x}.

So, multiplication by (f^{∗}a)^{n} is an injection on F_{x} for every positive integer n. In other words, no section s ∈ F_{x} can be killed by an ideal primary to the maximal ideal M_{x}. Therefore, x ∉ A(F), and the result follows.

Remark 7.6. Let f : X → Y be a morphism of noetherian schemes, F coherent on X and flat over Y. Let x ∈ X and write y = f(x). Although we will not need this fact, it is true that x ∈ A(F) if and only if y ∈ A(Y) and x ∈ A(F ⊗ k(y)).

Proposition 7.7. Let f : X → Y be a morphism of schemes. Assume that X is noetherian and that Y is a regular, integral curve. Let F be a coherent sheaf on X. Then F is flat over Y if and only if for every point x ∈ A(F), the image f(x) is equal to the generic point of Y.

Proof: (⇒) This direction was proved in the previous proposition. (⇐) Assume that F is not flat over Y. In particular, we can find a point x ∈ X such that F_{x} is not a flat module over A = O_{Y,y}, where y = f(x). If A were a ring of dimension zero, then, by regularity, A would be a field. So, every module over A would be free and hence flat. Thus, A must be a regular local ring of dimension one. Let π ∈ A generate the maximal ideal. Now F_{x} is a flat A-module if and only if f^{∗}π : F_{x} → F_{x} is injective. Since we assumed that it’s not flat, there must be a nonzero section s ∈ F_{x} such that f^{∗}π.s = 0. Let I ⊂ O_{X,x} be the annihilator of s. Choose a minimal prime ideal P among those which contain I. This prime ideal corresponds to a point z ∈ {x}^{−} ⊂ X. Now z is an associated point of F that maps to the closed point y ∈ Y.

Corollary 7.8. Let Y be an integral, regular curve, y ∈ Y a closed point, X ⊂ P_{Y −y}^{N} a closed subscheme that is flat over Y − y. Then there exists a unique closed subscheme X̄ ⊂ P_{Y}^{n} such that X̄ → Y is flat and X̄ ∩ P_{Y −y}^{n} = X.

Proof: Take X̄ to be the scheme theoretic closure of X. The result follows from the previous proposition after observing that A(X) = A(X̄).

Let’s return to the problem of pulling a Cartier divisor back along a morphism g : X → Y. The only conceivable definition is given by equation (7.3). As we have seen, there are circumstances where this formula fails to yield a Cartier divisor. So, our real task is to find a useful criterion that will ensure that this definition works as desired.

Lemma 7.9. Let g : X → Y be a morphism of schemes, and let D be a Cartier divisor on Y. Assume for all x ∈ A(X) that g(x) ∉ Supp(D). Then equation (7.3) describes a well-defined Cartier divisor g^{∗}(D) on X.

◯^{c}Coombes 1996 support Cartier divisor divisor, Cartier associated point divisor, relative flat morphism effective Cartier divisor

Proof: The support of a Cartier divisor is defined to be

Supp(D) = {x ∈ X : 1 is not a local equation of D at x}.

Thus, by avoiding the associated points, the hypotheses ensure that the local equations of D do not become zero divisors when pulled back to X. Therefore, equation (7.3) defines a Cartier divisor on X. It is easy to check that this definition is independent of the choice of open cover and local equations.

Lemma 7.10. If g : X → Y is flat, then the pullback of Cartier divisors on Y is always well-defined.

Proof: As we have shown earlier, g(A(X)) ⊂ A(Y), and the support of a Cartier divisor cannot contain an associated point.

Definition 7.11. Let f : X → Y be a flat morphism of finite type between noetherian schemes. An effective Cartier divisor D on X is said to be a relative divisor over Y if D is flat over Y.

Proposition 7.12. Let f : X → Y be a flat morphism of finite type between noetherian schemes, and let D be an effective Cartier divisor on X. Then the following are equivalent:

(i) D is flat over Y.

(ii) For all x ∈ X, the local equation f of D at x is a non-zero-divisor in

O_{X,x} ⊗ k(y) where y = f(x). (iii) For all y ∈ Y, one has A(f^{−1}y) ∩ Supp(D) = ∅.

Proof: The equivalence of (ii) and (iii) should be clear. To see their equivalence with (i), notice that the result is local. So, we can assume that Y = Spec(A), X = Spec(B), B is a flat, finitely generated A-algebra, and the local equation of D is a non-zero-divisor G ∈ B.

We would like to translate the geometric condition that a point x ∈ X lies in A(f^{−1}(y)) into algebraic terms. We let x correspond to a prime ideal Q ⊂ B and y ∈ Y correspond to a prime ideal P ⊂ A. By flatness, B/P B is flat over A/P. If Q̄ ⊂ B/P B is associated to 0, then it must contract to 0 in the domain A/P. In other words, if Q ⊂ B is associated to P B, then Q ∩ A = P. Now condition (iii) is equivalent to the statement: for all primes P ⊂ A, G is not contained in any associated prime of P B. Also, (i) is equivalent to the statement: for all primes P ⊂ A, T or_{1}^{A}(B/GB, A/P) = 0. To see that these algebraic conditions are equivalent, we start with the exact sequence

0 → B⟶^{G}B → B/GB → 0.

Tensoring with A/P yields an exact sequence

T or_{1}^{A}(B, A/P) → T or_{1}^{A}(B/GB, A/P) → B/P B⟶^{G}B/P B.

◯^{c}Coombes 1996 effective Cartier divisor divisor, relative associated point support functor of curves

Since B is flat, the first term vanishes. Now the vanishing of the second term is equivalent to the injectivity of multiplication by G on B/P B, and the result follows.

Proposition 7.13. Let u : X → Y be a flat morphism of finite type of noetherian schemes. Let f : Y′ → Y be any morphism from another noetherian scheme, and form the fibre product diagram

\begin{matrix}X′ & ⟶^{g} & X \\ ↓^{v} & & ↓^{u} \\ Y′ & ⟶^{f} & Y. \end{matrix}

If D is an effective Cartier divisor on X relative to Y, then the pullback g^{∗}(D) is defined, and it is an effective Cartier divisor on X′ relative to Y′.

Proof: We need to show that the image of any associated point x′ ∈ A(X′) misses the support of D in X. Write y′ = v(x′) and y = f(y′). Consider the fibre v^{−1}(y′). Since x′ is an associated point of X′ that lies in the fibre, we must have x′ ∈ A(v^{−1}(y′)). Since the pullback of a pullback is a pullback, one has

v^{−1}(y′) = u^{−1}(y) ×_{k(y)} k(y′).

So, v^{−1}(y′) is flat over u^{−1}(y). Hence, g(x′) ∈ A(u^{−1}(y)). By condition (iii) in the definition of a relative effective Cartier divisor, we see that g(x′) ∉ Supp(D). But that was exactly what we needed to show.

Now we can (finally!) explain what we mean by a continuous family of curves on a smooth projective surface X over a field k. For each k-scheme T, we define

Curves_{X}(T) = {effective Cartier divisors D on X × T relative to T }.

The main point, of course, is that the previous proposition makes Curves_{X} into a functor on the category of k-schemes. Moreover, the assignment D ↦ O_{X×T} (D) defines a natural transformation of functors Φ : Curves_{X} → Pic_{X}.

◯^{c}Coombes 1996