(page 1) Chapter 9

effective Cartier divisor divisor, relative Hilbert polynomial

Vanishing Theorems

We’re ready to start investigating Curves_{X} more carefully. Let D ∈ Curves_{X}(T). Then D is a flat family of curves parametrized by the scheme T. In our alternative terminology, D is an effective Cartier divisor on X × T relative to T. So, for each point t ∈ T, we get an induced Cartier divisor D_{t} on the fibre X_{t} = X ×_{k} k(t). (At the k-points, therefore, we get a Cartier divisor on X itself.) Associated to the Cartier divisor D and its specializations D_{t} are several coherent sheaves, including the associated invertible sheaf O_{X×T} (D), the ideal sheaf O_{X×T} (−D), and the structure sheaf O_{D}. Fixing an embedding of X into some projective space P^{n}, we can think of each of these sheaves as a coherent sheaf on P_{T}^{N}, with specializations to coherent sheaves on P_{k(t)}^{N}. So, we can associate to each sheaf along a fibre its Hilbert polynomial for this embedding in projective space. By Thm. 6.19, these Hilbert polynomials are constant on connected components of flat families. So, we could use any of them to decompose the functor of curves into disjoint pieces.

For technical reasons, we choose to use the ideal sheaf. So, we define the Hilbert polynomial associated to a curve C on the surface X to be the numerical polynomial determined by the values

P_{C}(m) = χ(O_{X}(−C + m)).

Given a flat family D of curves on X × T, we thus get Hilbert polynomials

P_{t}(m) = χ(O_{X_{t}}(−D_{t} + m)),

which are constant on connected components.

Now fix one of these Hilbert polynomials P. Define a functor Curves_{X}^{P} whose value on a k-scheme T is

Curves_{X}^{P}(T) = {D ∈ Curves_{X}(T) : P_{t} = P for all t}.

Actually, that definition only works if T is connected. We really need to extend it to arbitrary schemes by writing T = ∐_{α} T_{α} as a disjoint union of its connected components, and then taking

Curves_{X}^{P}(T) = ∏_{α} Curves_{X}^{P}(T_{α}).

(page 2)

2 Chapter 9: Vanishing Theorems

m-regular

Castelnuovo’s Regularity Theorem associated point

In any event, we’re supposed to think of this functor as picking out the families of curves that have a fixed Hilbert polynomial P along each fibre. Because the Hilbert polynomials are constant, we have a natural decomposition

Curves_{X}(T) = ∐_{P} Curves_{X}^{P}(T).

So, in order to show that Curves_{X} is represented by some scheme, we can work instead with the subfunctors Curves_{X}^{P}. Here we can take advantage of the extra information that the Hilbert polynomial makes available to us to describe properties of the sheaves O_{X_{t}}(−D_{t} + m).

Definition 9.1. Let F be a coherent sheaf on P^{n}. We say that F is m-regular if H^{i}(P^{n}, F(m − i) = 0 for all i > 0.

Proposition 9.2. (Castelnuovo) Let F be m-regular on P^{n}. Then

(i) For all k > m, the group of global sections H^{0}(P^{n}, F(k)) is spanned by H^{0}(P^{n}, F(k − 1)) ⊗ H^{0}(P^{n}, O(1)).

(ii) For all i > 0 and all k + i > m, we have H^{i}(P^{n}, F(k)) = 0. In particular, F(k) is generated by global sections for all k ≥ m.

Proof: We will induct on the dimension n, since both parts are obvious if n = 0. When n > 0, we can choose a hyperplane H that does not contain any of the associated points of F. Write F_{H} = F ⊗O_{H} for the restriction of F to H. Twisting the standard exact sequence

0 → O(−H) → O → O_{H} → 0

with F(k) gives an exact sequence

0 → F(k − 1) → F(k) → F_{H}(k) → 0.

(Here we have used the hypothesis about H missing the associated points to ensure that multiplication by its local equation remains an injection on F.) Taking cohomology, we find

H^{i}(F(m − i)) → H^{i}(F_{H}(m − i)) → H^{i+1}(F(m − i − 1)).

Thus, if F is m-regular, then the restriction F_{H} is also m-regular. Now we can apply the inductive hypothesis to F_{H}. Continuing the long exact sequence of cohomology gives

H^{i+1}(F(m − i − 1)) → H^{i+1}(F(m − i)) → H^{i+1}(F_{H}(m − i)).

The first term is zero by the m-regularity of F. The final term is zero by property (ii) for F_{H}. Thus, the middle term also vanishes. In particular, we have shown that if F is m-regular, then F is m + 1-regular. Therefore, part (ii) follows in general.

◯^{c}Coombes 1996

Now consider the following commutative diagram. Mumford’s Regularity Theorem

H^{0}(F(k − 1)) ⊗H^{0}(O(1))⟶H^{0}(F_{H}(k − 1)) ⊗H^{0}(O_{H}(1))
↓^{μ}  ↓
H^{0}(F(k − 1))⟶H^{0} (F(k))  ⟶^{ν}  H^{0} (F(k))

The top arrow is surjective for k > m because, by property (ii), H^{1}(F(k − 2)) = 0. The rightmost vertical arrow is surjective by the inductive hypothesis, which gives us property (i) on F_{H}. Now we know that the composite ν ∘ μ is also surjective. Since the bottom sequence is exact, it follows that H^{0}(F(k)) is spanned by Im(μ) and H^{0}(F(k − 1)). But the latter piece comes from multiplication by the class of H in H^{0}(O(1)), so the theorem follows.

Theorem 9.3. (Mumford) For all positive integers n, there exists a polynomial F_{n}(X_{0}, …, X_{n}) such that if I is any coherent ideal sheaf on P^{n} and if the integers a_{i} are defined to be the coefficients

χ(I(m)) = ∑_{i=1}^{n} a_{i}\binom{m}{i}

in the Hilbert polynomial of I, then I is F_{n}(a_{0}, …, a_{n})-regular.

Proof: We induct on the dimension n, since the case n = 0 is obvious. Now the ideal sheaf I defines a closed subscheme Z with structure sheaf O_{Z} = O/I. Choose a hyperplane H disjoint from A(Z). It is important to note that I_{H} = I ⊗ O_{H} is an ideal sheaf on H. To see this, tensor the exact sequence

0 → I → O → O_{Z} → 0

with O_{H} to get

T or_{1}(O_{Z}, O_{H}) → I_{H} → O_{H}.

The T or group vanishes because we assumed that H avoids the associated points of Z. In particular, a local equation of H at a point x never becomes a zero-divisor in O_{Z}.

Because O(−1) is locally free, we also have exact sequences

(9.4)  0 → I(m) → I(m + 1) → I_{H}(m + 1) → 0.

Now we are ready to apply the inductive hypothesis to I_{H}. We have

χ(I_{H}(m + 1)) = χ(I(m + 1)) − χ(I(m))

= ∑_{i=0}^{n} a_{i} [\binom{m + 1}{i} − \binom{m}{i}]

= ∑_{i=0}^{n−1} a_{i+1}\binom{m}{i}.

So, there exists a polynomial G(X_{0}, …, X_{n−1}), depending only on n, such that Serre’s Finiteness Theorem every ideal sheaf I_{H} is G(a_{1}, …, a_{n})-regular.

Write m_{1} = G(a_{1}, …, a_{n}). Taking cohomology in (9.4) yields isomorphisms

H^{i}(I(m)) ≈ H^{i}(I(m + 1))

for all i ≥ 2 and all m ≥ m_{1} − i. By Serre’s Finiteness Theorem, these cohomology groups vanish for m >> 0. Thus, we immediately get that

H^{i}(I(m)) = 0 for all i ≥ 2, and all m ≥ m_{1} − i.

So, I is also m_{1}-regular except possibly for the behavior of the first cohomology groups. Now, the low degree terms in the cohomology sequence are

\begin{matrix}0 → & H^{0}(I(m)) → H^{0}(I(m + 1)) ⟶^{ρ_{m+1}}H^{0}(I_{H}(m + 1)) \\ → & H^{1}(I(m)) → H^{1}(I(m + 1)) → 0. \end{matrix}

From this sequence, we conclude that either ρ_{m+1} is surjective, or dim H^{1}(I(m)) > dim H^{1}(I(m + 1)). However, if ρ_{m} is surjective for a particular value of m, then Castelnuovo’s result shows that it remains surjective for all larger values of m. Thus, for all m ≥ m_{1}, the function m ↦ dim H^{1}(I(m)) is a strictly decreasing function of m until it reaches 0, after which it remains constant. Therefore, I is (m_{1} + dim H^{1}(I(m_{1} − 1)))-regular.

Now, at last, we can use the fact that I is an ideal sheaf.

dim H^{1}(I(m_{1} − 1)) = dim H^{0}(I(m_{1} − 1)) − χ(I(m_{1} − 1))
≤ dim H^{0}(O(m_{1} − 1)) − χ(I(m_{1} − 1))
= H(a_{0}, a_{1}, …, a_{n}; m_{1}),

for some polynomial H. So, we can take

F_{n}(a_{0}, …, a_{n}) = G(a_{1}, …, a_{n}) + H(a_{0}, …, a_{n}; G(a_{1}, …, a_{n})).

Remark 9.5. The result is false if you do not insist on using ideal sheaves. For example, take n = 1 and look at the sheaves

S_{k} = O(k) ⊕ O(−k).

Then

χ(S_{k}(m)) = 2(m + 1)

is independent of k, but the smallest m for which S_{k} is m-regular is |k| − 1.

Corollary 9.6. let X ⊂ P^{n} be a projective algebraic scheme of dimension r. Then there exists a polynomial F (X_{0}, …, X_{r}) such that for all ideal sheaves I ⊂ O_{X}, if

χ(I(m)) = ∑_{i=0}^{r} a_{i}\binom{m}{i},

Hilbert polynomial effective Cartier divisor

then I is F (a_{0}, …, a_{r})-regular.

Proof: Let I define a closed subscheme Z ⊂ X ⊂ P^{n} ,and let J be its ideal sheaf in P^{n}. Let K be the ideal sheaf of X. Then there is an exact sequence

0 → K → J → I → 0.

If J is m_{0}-regular and if H^{i}(K(m)) = 0 for i + m > m_{0} + 1, then the long exact sequence of cohmology shows that I is m_{0}-regular. Now the result follows from the additivity of the Euler characteristic.

Corollary 9.7. Let X ⊂ P^{n} be a smooth projective surface. Fix a Hilbert polynomial P, and let D be an effective Cartier divisor on X such that P is the Hilbert polynomial of O_{X}(−D). In other words, for all integers m, we have

P (m) = χ(O_{X}(−D + m)).

then there exists an integer m_{0} depending only on P such that O_{X}(−D) is m_{0}- regular.

Proof: Since O_{X}(−D) is an ideal sheaf, this follows immediately from the theorem. After all, the polynomial in that theorem is evaluated at the coefficients of the Hilbert polynomial of the ideal sheaf.

(page 6)