(page 1) Chapter 15

independent

Independent Zero Cycles

Now we return our attention to the second condition that we needed to impose on our numeriocal equivalence class ξ. We basically needed to be able to choose points on X that were in sufficiently general position with respect to every invertible sheaf of type ξ.

Definition 15.1. Let n be an integer. Let Z be a zero cycle of degree N on X. We write Z = ∑ i = 1^{N}P_{i} where the points P_{i} need not be distinct. We say that Z is n-independent if for all curves C on X one has

Number of P_{i} in Supp(C) ≤ n(deg(C))^{2}.

Example 15.2. A zero cycle in the plane is 2-independent if no three points lie on a line, no nine points lie on a conic, etc. This is, of course, a very weak condition, since there is no reason why even six points should lie on a single conic.

Proposition 15.3. For all integers N, there exists a zero cycle Z = ∑_{i=1}^{N} P_{i} of degree N on P^{2} such that for all subsets S ⊂ {1, …, N} and all integers d, if

L_{d,S} = {homogeneous forms F of degree d with F (P_{i}) = 0 if i ∈ S}

then

dim(L_{n,S}) = \{ \begin{matrix}0 & if |S| ≥ (n + 1)(n + 2)/2. \\ (n + 1)(n + 2)/2 − |S| & otherwise. \end{matrix}

Proof: By induction on N. When N = 1, any closed point P works. Now suppose that we have constructed a zero cycle Z = ∑_{i=1}^{N−1} that works. We need to add another point P_{N}.

Here there be dragons.

Putting It All Together

Recall that we planned to prove that the Picard functor Pic_{X} is actually a scheme by choosing a nice numerical equivalence class ξ and defining a natural section of the natural transformation Curves_{X}^{ξ} → Pic_{X}^{ξ}. In order to complete this construction,

◯^{c}Coombes 1996  1

(page 2)

2 Chapter 15: Independent Zero Cycles

we needed to satisfy two conditions. We understand one of those conditions, since it reduced to the numerical criteria

• deg(ξ) ≥ c_{2}, and

• χ(ξ) ≥ (1 − ϵ)\frac{deg(ξ)^{2}}{2(O_{X}(1).O_{X}(1))}.

If ξ satisfies these criteria, then every invertible sheaf of type ξ on X is 0-regular and very ample. Before proceeding, let’s recall the construction that produced the second condition.

Let L be a line bundle on X × T that has numerical class ξ on every fibre. We define

E = (p_{T})_{∗}(L),

which is a locally free sheaf of rank r = χ(ξ). For each closed point x ∈ X, we let ι_{x} : T → X × T be the map t ↦ (x, t), and define

M_{x} = ι_{x}^{∗}(L).

There are canonical maps h_{x} : E → M_{x}. Now choose a set of r points x_{1}, …, x_{r} ∈ X. Adding the h_{x} maps produces

h = ∑_{i=1}^{r−1} h_{x_{i}} : E → ⨁_{i=1}^{r−1} M_{x_{i}}  and  h_{0} = ∑_{i=1}^{r} h_{x_{i}} : E → ⨁_{i=1}^{r} M_{x_{i}}

Taking the maximal exterior power that makes any sense, we get maps

∧h : ⋀^{r−1}(E) → ⨂_{i=1}^{r−1} M_{x_{i}}  and  ∧ h_{0} : ⋀^{r}(E) → ⨂_{i=1}^{r} M_{x_{i}}.

Now dualize (using the fact that Hom(⋀^{r−1}(E), O_{T}) = E ⊗ (⋀^{r}(E))^{−1} to get maps

(∧h)^{∗} : ⨂_{i=1}^{r−1} M_{x_{i}}^{−1} → E ⊗ (⋀^{r}(E))^{−1}  and  (∧h_{0})^{∗} : ⨂_{i=1}^{r} M_{x_{i}}^{−1} → (⋀^{r}(E))^{−1}.

Twist to remove all invertible sheaves from the left-hand side of the map to obtain maps

h′ : O_{T} → E ⊗(⋀^{r}(E))^{−1}⊗(⨂_{i=1}^{r−1} M_{x_{1}})  and  h′_{0} : O_{T} → (⋀^{r}(E))^{−1}⊗(⨂_{i=1}^{r} M_{x_{1}}).

Finally, we can pull these sections back along p_{T} and use the projection formula to produce sections

σ ∈ H^{0}(X × T, L ⊗ p_{T}^{∗} ((⋀^{r}(E))^{−1} ⊗ ⨂_{i=1}^{r−1} M_{x_{i}}))

and

σ_{0} ∈ H^{0}(X × T, p_{T}^{∗} ((⋀^{r}(E))^{−1} ⊗ ⨂_{i=1}^{r} M_{x_{i}})).

We tie these two constructions together as follows. Suppose we have an effective zero cycle Z of degree Nr − 1 on X, and choose a decomposition (γ) that allows us to write the zero cycle in the form

Z = ∑_{i=1}^{N−1} b_{i} + b_{N}^{∗},

where each cycle b_{i} for 1 ≤ i ≤ N − 1 contains r points and the final cycle b_{N}^{∗} contains r − 1 points. Then we get sections

σ_{i} ∈ H^{0}(X × T, L ⊗ p_{T}^{∗} ((⋀^{r}(E))^{−1} ⊗ ⨂_{Q∈b_{i}} M_{Q})) if 1 ≤ i ≤ N − 1

and

σ_{N} ∈ H^{0}(X × T, p_{T}^{∗} ((⋀^{r}(E))^{−1} ⊗ ⨂_{Q∈b_{N}^{∗}} M_{Q})).

Tensoring these sections all together produces a section

σ_{γ} = σ_{1} ⊗ ⋯ ⊗ σ_{N} ∈ H^{0}(X × T, L ⊗ (⨂_{Q∈Z} M_{Q})).

It is important to notice that, even though the section produced by this construction depends on the choice of the decomposition (γ), the place where it lives only depends on the zero cycle Z.

Choose a numerical equivalence class ξ ∈ Num(X) satisfying the criteria: By our earlier analysis, every line bundle of type ξ on X is 0-regular and very ample. In particular, the highr cohomology of any such sheaf L vanishes, so we have

r = χ(ξ) = χ(L) = dim(H^{0}(X, L)).

Now let d be an ionteger such that X contains d-independent zero cycles of very possible degree. Choose an integer N such that

N > d(deg(ξ))^{2}.

Next, choose a d-independent zero cycle Z on X of degree

Nr − 1.

Let L be an arbitrary invertible sheaf of type ξ on X. Let ϕ : X → P^{r−1} be the closed immersion defined by the global sections of L. By an earlier proposition, if x ∈ X is an arbitrary closed point, then the zero cycle ϕ_{∗}(Z + x) is strongly stable on P^{r−1}. Moreover, if P ∈ P^{r−1} is an arbitraryp point, then there exists a decomposition (γ) that allows us to write our zero cycle in the form

Z = ∑_{i=1}^{N−1} b_{i} + b_{N}^{∗},

where ϕ_{∗}(b_{i}) consists of r projectively independent points for 1 ≤ i ≤ N − 1 and ϕ_{∗}(b_{N}^{∗}) spans a hyperplane not containing P.