(page 1) Chapter 4

algebraic cycle cycle, algebraic degree universal family Hilbert polynomial numerical function numerical polynomial difference function

Hilbert Polynomials

Let’s start this chapter by recalling the motivating problems that were advanced in the introduction. There we saw several examples of surfaces on which we could classify algebraic cycles first by specifying some discrete invariants, such as the degree, and then by producing a continuous universal family of cycles with those fixed discrete invariants. The primary goal of this chapter is to describe a particularly important discrete invariant: the Hilbert polynomial.

Definition 4.1. We call any function f : Z → Z a numerical function. A polynomial p(z) ∈ Q[z] with rational coefficients is called a numerical polynomial if p(m) ∈ Z for all m >> 0, m ∈ Z. In other words, a polynomial p(z) is numerical if it represents a numerical function for sufficiently large integers m.

Proposition 4.2. If p(z) is a numerical polynomial, then there exist integers c_{0}, c_{1}, …, c_{r} such that

p(z) = c_{0}\binom{z}{r} + c_{1}\binom{z}{r − 1} + ⋯ + c_{r},

where

\binom{z}{r} = \frac{z(z − 1) ⋯ (z − r + 1)}{r!} = \frac{z^{r}}{r!} + · · · .

Proof: The case of degree zero being obvious, we shall proceed by induction on the degree of p(z). Since \binom{z}{r} is a polynomial of degree exactly r, we clearly have such an expression with coefficients c_{i} ∈ Q. Define the difference function of p to be Δp(z) = p(z + 1) − p(z). It is easy to check that Δ\binom{z}{r} = \binom{z}{r−1}. So,

Δp(z) = c_{0}\binom{z}{r − 1} + c_{1}\binom{z}{r − 2} + ⋯ + c_{r−1}.

By induction, the coefficients c_{0}, …, c_{r−1} ∈ Z. But then c_{r} ∈ Z, since p(m) ∈ Z for m >> 0.

◯^{c}Coombes 1996  1

(page 2)

2 Chapter 4: Hilbert Polynomials

numerical function numerical polynomial numerical polynomial difference function coherent sheaf Euler characteristic Hilbert function numerical polynomial Hilbert polynomial support coherent sheaf noetherian scheme Hilbert function

Corollary 4.3. If p(z) ∈ Q[z] is a numerical polynomial, then p(m) ∈ Z for all m ∈ Z.

Proof: This follows from the proposition, since it is true for \binom{z}{r}.

Proposition 4.4. Let f be a numerical function and assume that there exists a numerical polynomial q(z) ∈ Q[z] such that Δf(m) = q(m) for all m >> 0, m ∈ Z. Then there exists a numerical polynomial p(z) such that f(m) = p(m) for all m >> 0.

Proof: Using the previous proposition, we can write

q(z) = c_{0}\binom{z}{r} + c_{1}\binom{z}{r − 1} + ⋯ + c_{r},

with c_{0}, c_{1}, …, c_{r} ∈ Z. Start with the polynomial

p(z) = c_{0}\binom{z}{r + 1} + c_{1}\binom{z}{r} + ⋯ + c_{r}\binom{z}{1}.

Then Δp = q, which equals Δf for m >> 0. So, Δ(f − p)(m) = 0 for all m >> 0. Thus, (f − p)(m) is equal to some constant c_{r+1} ∈ Z for all m >> 0, and so

f(m) = p(m) + c_{r+1}.

Remark 4.5. The proof also shows that the degree of the numerical polynomial representing f is always exactly one greater than the degree of the numerical polynomial representing Δf.

Definition 4.6. Let k be a field and let F be a coherent sheaf on the projective space P_{k}^{n}. The numerical function

m ↦ χ(F(m)) = ∑_{i=0}^{n}(−1)^{i} dim(H^{i}(P^{n}, F(m)))

is called the Hilbert function of the sheaf F.

Proposition 4.7. Let F be a coherent sheaf on P_{k}^{n}. There exists a unique numerical polynomial p_{F}(z) ∈ Q[z] such that χ(F(m)) = p_{F}(m) for all m ∈ Z. This polynomial is called the Hilbert polynomial of F.

Proof: We proceed by induction on the support of F, which is defined to be

Supp(F) = {x ∈ P^{n} : F_{x} ≠ 0}.

Since F is a coherent sheaf on a noetherian scheme, its support is a closed subscheme X = Supp(F) ⊂ P^{n}. If X is zero-dimensional, then F(m) is isomorphic to F for all m, and so the Hilbert function is a constant. Thus, the result is true for sheaves

support support difference function Hilbert polynomial Serre Euler characteristic Hilbert polynomial support Hilbert polynomial Hilbert polynomial projection formula

with zero-dimensional support. If F is any coherent sheaf, choose a linear form x ∈ Γ(P^{n}, O(1)) such that the hyperplane x = 0 does not contain any of the components of the support of F. Define sheaves R and Q by the exact sequence

0 → R → F(−1)⟶^{x}F → Q → 0.

Then these new sheaves have support of strictly smaller dimension than the support of F. Since twisting by m preserves exactness, we have

χ(F(m)) − χ(F(m − 1)) = χ(Q(m)) − χ(R(m)).

By induction, the right-hand side of this equation is represented by a numerical polynomial. Since the left-hand side is the difference function of the Hilbert polynomial of F, it follows from Prop. 4.4 that there is a unique numerical polynomial p(z) such that p(m) = χ(F(m)) for all m >> 0. Now use the additivity of the Euler characteristic in exact sequences to induct downward and show that the equality holds for all m ∈ Z.

Remark 4.8. By a theorem of Serre, the higher cohomology of F(m) vanishes if we take m sufficiently large. So, we could have replaced the Euler characteristic with the numerical function m ↦ dim(H^{0}(P^{n}, F(m)) in the definition of the Hilbert polynomial. Had we done so, then we would only know that the Hilbert polynomial agreed with the numerical function for m >> 0, since the downward induction in the last step of the proof would have failed.

Corollary 4.9. Let F be a coherent sheaf on P^{n}. Then

dim(Supp(F)) = deg(p_{F}(z)).

Proof: This follows immediately from the proof of the previous proposition and Remark 4.5.

Remark 4.10. Let X ⊂ P^{n} be a closed subscheme. Then the structure sheaf of X is a coherent sheaf on P^{n}, so we can use the proceeding machinery to associate to X a Hilbert polynomial. By abuse of notation, we will write p_{X}(z) = p_{O_{X}} (z). This notation is particularly infuriating since it seems to imply that the Hilbert polynomial only depends on X. As we shall see, however, it actually depends on the way X is embedded in projective space.

The following result is extremely useful. Out first application will be to show that the Hilbert polynomial of X can be computed using only the cohomology of sheaves on X.

Lemma 4.11. (Projection Formula) Let f : X → Y be any morphism of schemes, let F be an arbitrary sheaf on X, and let E be a locally free sheaf on Y. Then there is a natural isomorphism

f_{∗}(F) ⊗_{Y} E⟶^{≈}f_{∗}(F ⊗_{X} f^{∗}(E)).

◯^{c}Coombes 1996 closed immersion very ample Euler characteristic projection formula Euler characteristic Hilbert polynomial very ample Hilbert polynomial d-uple embedding

Proof: Since f_{∗} and f^{∗} are adjoint functors, there are natural maps

f^{∗}f_{∗}H → H for all H ∈ MOD(X)

and

G → f_{∗}f^{∗}G for all G ∈ MOD(Y).

Using the fact that pullback commutes with tensor products, we get the morphism

(f_{∗}F ⊗ E) → f_{∗}f^{∗}(f_{∗}F ⊗ E) = f_{∗}(f^{∗}f_{∗}F ⊗ f^{∗}E) → f_{∗}(F ⊗ f^{∗}E).

Producing the map is the hard part. Now to check that this map is an isomorphism of sheaves on Y, we can work locally. So, we can assume that E is free, when the result becomes obvious.

Proposition 4.12. Let f : X → P^{n} be a closed immersion, and let O_{X}(H) = f^{∗}(O(1)) be the very ample invertible sheaf that determines the embedding. Then

p_{X}(m) = χ(O_{X}(mH))

for all m ∈ Z.

Proof: Since f^{∗} commutes with tensor product on the category of locally free sheaves, we have f^{∗}(O(m)) = O_{X}(mH) for all m. Now the projection formula gives

f_{∗}(O_{X}(mH)) = f_{∗}(f^{∗}(O(m))) = f_{∗}(O_{X} ⊗ f^{∗}(O(m))) = (f_{∗}O_{X}) ⊗ O(m).

Finally, since f is a closed immersion, it is finite, and hence affine. Thus, we have

H^{i}(P^{n}, (f_{∗}O_{X})(m)) = H^{i}(P^{n}, f_{∗}(O_{X}(mH))) = H^{i}(X, O_{X}(mH)).

The result follows since the Euler characteristic is the alternating sum of the dimensions of the chomology groups, and the Hilbert polynomial is determined by the Euler characteristic of the twists.

Example 4.13. Take X = P^{n} with its standard very ample sheaf O(1). Then the Hilbert polynomial p_{n} = p_{P^{n}} is determined by the values

p_{n}(m) = χ(O(m)) = dim H^{0}(P^{n}, O(m)),

for m ≥ 0. In turn, we can identify this value with the number of monomials of degree m in n + 1 variables, which is just \binom{m+n}{n}. So,

p_{n}(z) = \binom{z + n}{n} = \frac{(z + n)(z + n − 1) ⋯ (z + 1)}{n!}.

Example 4.14. Let ν_{d} : P^{1} → P^{d} be the d-uple embedding via all monomials of degree d, and let X_{d} = ν_{d}(P^{1}) be its image. Since ν_{d}^{∗}(O(1)) = O(d) on the projective line, we have

p_{X_{d}}(m) = χ(O(md)) = \binom{md + 1}{1} = md + 1.

very ample very ample Hilbert polynomial hypersurface degree Euler characteristic Hilbert polynomial degree Hilbert polynomial genus of curve degree Riemann-Roch Theorem for Curves Hilbert polynomial

Therefore,

p_{X_{d}}(z) = dz + 1.

Recall that the only invertible sheaves on the projective line are the O(d) for d ∈ Z. Moreover, O(d) is very ample if and only if d > 0. Thus, in this example, we have looked at all the very ample invertible sheaves on the projective line, and have discovered that they have distinct Hilbert polynomials. This computation illustrates our earlier claim that the Hilbert polynomial depends on the embedding in projective space, and not just on the variety.

Example 4.15. Let H be a hypersurface of degree d in P = P^{n}. Then we have an exact sequence

0 → O_{P} (−d) → O_{P} → O_{H} → 0

of coherent sheaves on P. Twisting by m preserves exactness, and the Euler characteristic is additive on exact sequences. Consequently, the Hilbert polynomial is also additive on exact sequences. Therefore,

p_{H}(z) = p_{P} (z) − p_{P} (z − d)

= \binom{z + n}{n} − \binom{z − d + n}{n}

= \frac{(z + n)(z + n − 1) ⋯ (z + 1)}{n!} − \frac{(z − d + n) ⋯ (z − d + 1)}{n!}

= (\frac{1}{n!}z^{n} + \frac{∑_{i=1}^{n} i}{n!}z^{n−1} + ⋯) − (\frac{1}{n!}z^{n} + \frac{∑_{i=1}^{n}(i − d)}{n!}z^{n−1} + ⋯)

= \frac{d}{(n − 1)!}z^{n−1} + ⋯

Remark 4.16. The previous example shows that we can recover not only the dimension of H but also its degree from the Hilbert polynomial. This result is true in general: If X is a closed subscheme of degree d and dimension r in P^{n}, then its Hilbert polynomial looks like

p_{X}(z) = \frac{d}{r!}z^{r} + lower degree terms.

Example 4.17. Let X be a smooth projective curve of genus g, and let f : X → P^{n} be an embedding in projective space defined by a divisor D. In other words, O_{X}(D) = f^{∗}(O(1)). Let d = deg(D) be the degree of the divisor. By the Riemann-Roch Theorem for Curves, we have

χ(O_{X}(mD)) = deg(mD) + 1 − g = md + 1 − g.

Therefore, the Hilbert polynomial of X is

p_{X}(z) = mz + 1 − g.

◯^{c}Coombes 1996 arithmetic genus genus, arithmetic birational

Notice in particular that we can recover the genus of X from the constant term of its Hilbert polynomial, since

g = 1 − p_{X}(0).

This computation is particularly interesting because it shows that the constant term of the Hilbert polynomial of a curve is an isomorphism invariant—a behavior that is drastically different from that of the leading coefficient. For a general closed subscheme X in projective space, we define its arithmetic genus to be

p_{a}(X) = (−1)^{dim(X)}(p_{X}(0) − 1).

The arithmetic genus is a birational invariant for smooth projective varieties.