SMOOTH HYPERSURFACE SECTIONS CONTAINING A GIVEN SUBSCHEME OVER A FINITE FIELD

BJORN POONEN

1. Introduction

Let 𝔽_{q} be a finite field of q = p^{a} elements. Let X be a smooth quasi-projective subscheme of ℙ^{n} of dimension m ≥ 0 over 𝔽_{q}. N. Katz asked for a finite field analogue of the Bertini smoothness theorem, and in particular asked whether one could always find a hypersurface H in ℙ^{n} such that H ∩ X is smooth of dimension m − 1. A positive answer was proved in [Gab01] and [Poo04] independently. The latter paper proved also that in a precise sense, a positive fraction of hypersurfaces have the required property.

The classical Bertini theorem was extended in [Blo70,KA79] to show that the hypersurface can be chosen so as to contain a prescribed closed smooth subscheme Z, provided that the condition dim X > 2 dim Z is satisfied. (The condition arises naturally from a dimensioncounting argument.) The goal of the current paper is to prove an analogous result over finite fields. In fact, our result is stronger than that of [KA79] in that we do not require Z ⊆ X, but weaker in that we assume that Z ∩ X be smooth. (With a little more work and complexity, we could prove a version for a non-smooth intersection as well, but we restrict to the smooth case for simplicity.) One reason for proving our result is that it is used by [SS07].

Let S = 𝔽_{q}[x_{0}, …, x_{n}] be the homogeneous coordinate ring of ℙ^{n}. Let S_{d} ⊆ S be the 𝔽_{q}- subspace of homogeneous polynomials of degree d. For each f ∈ S_{d}, let H_{f} be the subscheme Proj(S/(f)) ⊆ ℙ^{n}. For the rest of this paper, we fix a closed subscheme Z ⊆ ℙ^{n}. For d ∈ ℤ_{≥0}, let I_{d} be the 𝔽_{q}-subspace of f ∈ S_{d} that vanish on Z. Let I_{homog} = ⋃_{d≥0} I_{d}. We want to measure the density of subsets of I_{homog}, but under the definition in [Poo04], the set I_{homog} itself has density 0 whenever dim Z > 0; therefore we use a new definition of density, relative to I_{homog}. Namely, we define the density of a subset P ⊆ I_{homog} by

μ_{Z}(P) := lim_{d→∞} \frac{#(P ∩ I_{d})}{#I_{d}},

if the limit exists. For a scheme X of finite type over 𝔽_{q}, define the zeta function [Wei49]

ζ_{X}(s) = Z_{X}(q^{−s}) := ∏_{closed P ∈X} (1 − q^{−s deg P} )^{−1} = exp (∑_{r=1}^{∞} \frac{#X(𝔽_{q^{r}})}{r}q^{−rs});

the product and sum converge when Re(s) > dim X.

Date: June 29, 2007; typos corrected March 9, 2012.

1991 Mathematics Subject Classification. Primary 14J70; Secondary 11M38, 11M41, 14G40, 14N05.

This research was supported by NSF grant DMS-0301280.

Theorem 1.1. Let X be a smooth quasi-projective subscheme of ℙ^{n} of dimension m ≥ 0 over 𝔽_{q}. Let Z be a closed subscheme of ℙ^{n}. Assume that the scheme-theoretic intersection V := Z ∩ X is smooth of dimension ℓ. (If V is empty, take ℓ = −1.) Define

P := { f ∈ I_{homog} : H_{f} ∩ X is smooth of dimension m − 1 }.

(i) If m > 2ℓ, then

μ_{Z}(P) = \frac{ζ_{V} (m + 1)}{ζ_{V} (m − ℓ) ζ_{X}(m + 1)} = \frac{1}{ζ_{V} (m − ℓ) ζ_{X−V} (m + 1)}.

In this case, in particular, for d ≫ 1, there exists a degree-d hypersurface H containing Z such that H ∩ X is smooth of dimension m − 1.

(ii) If m ≤ 2ℓ, then μ_{Z}(P) = 0.

The proof will use the closed point sieve introduced in [Poo04]. In fact, the proof is parallel to the one in that paper, but changes are required in almost every line.

2. Singular points of low degree

Let I_{Z} ⊆ O_{ℙ^{n}} be the ideal sheaf of Z, so I_{d} = H^{0}(ℙ^{n}, I_{Z}(d)). Tensoring the surjection

O^{⊕(n+1)} → O
(f_{0}, …, f_{n}) ↦ x_{0}f_{0} + ⋯ + x_{n}f_{n}

with I_{Z}, twisting by O(d), and taking global sections shows that S_{1}I_{d} = I_{d+1} for d ≫ 1. Fix c such that S_{1}I_{d} = I_{d+1} for all d ≥ c.

Before proving the main result of this section (Lemma 2.3), we need two lemmas.

Lemma 2.1. Let Y be a finite subscheme of ℙ^{n}. Let

ϕ_{d}: I_{d} = H^{0}(ℙ^{n}, I_{Z}(d)) → H^{0}(Y, I_{Z} · O_{Y} (d))

be the map induced by the map of sheaves I_{Z} → I_{Z} · O_{Y} on ℙ^{n}. Then ϕ_{d} is surjective for d ≥ c + dim H^{0}(Y, O_{Y}),

Proof. The map of sheaves O_{ℙ^{n}} → O_{Y} on ℙ^{n} is surjective so I_{Z} → I_{Z} · O_{Y} is surjective too. Thus ϕ_{d} is surjective for d ≫ 1.

Enlarging 𝔽_{q} if necessary, we can perform a linear change of variable to assume Y ⊆ 𝔸^{n} := {x_{0} ≠ 0}. Dehomogenization (setting x_{0} = 1) identifies S_{d} with the space S′_{d} of polynomials in 𝔽_{q}[x_{1}, …, x_{n}] of total degree ≤ d, and identifies ϕ_{d} with a map

I′_{d} → B := H^{0}(ℙ^{n}, I_{Z} · O_{Y}).

By definition of c, we have S′_{1}I′_{d} = I′_{d+1} for d ≥ c. For d ≥ c, let B_{d} be the image of I′_{d} in B, so S′_{1}B_{d} = B_{d+1} for d ≥ c. Since 1 ∈ S′_{1}, we have I′_{d} ⊆ I′_{d+1}, so

B_{c} ⊆ B_{c+1} ⊆ · · · .

But b := dim B < ∞, so B_{j} = B_{j+1} for some j ∈ [c, c + b]. Then

B_{j+2} = S′_{1}B_{j+1} = S′_{1}B_{j} = B_{j+1}.

Similarly B_{j} = B_{j+1} = B_{j+2} = …, and these eventually equal B by the previous paragraph. Hence ϕ_{d} is surjective for d ≥ j, and in particular for d ≥ c + b. □

Lemma 2.2. Suppose 𝔪 ⊆ O_{X} is the ideal sheaf of a closed point P ∈ X. Let Y ⊆ X be the closed subscheme whose ideal sheaf is 𝔪^{2} ⊆ O_{X}. Then for any d ∈ ℤ_{≥0}.

#H^{0}(Y, I_{Z} · O_{Y} (d)) = \begin{cases}q^{(m−ℓ) deg P} , & if P ∈ V, \\ q^{(m+1) deg P} , & if P ∉ V. \end{cases}

Proof. Since Y is finite, we may now ignore the twisting by O(d). The space H^{0}(Y, O_{Y}) has a two-step filtration whose quotients have dimensions 1 and m over the residue field κ of P. Thus #H^{0}(Y, O_{Y}) = (#κ)^{m+1} = q^{(m+1) deg P}. If P ∈ V (or equivalently P ∈ Z), then H^{0}(Y, O_{Z∩Y}) has a filtration whose quotients have dimensions 1 and ℓ over κ; if P ∉ V, then H^{0}(Y, O_{Z∩Y}) = 0. Taking cohomology of

0 → I_{Z} · O_{Y} → O_{Y} → O_{Z∩Y} → 0

on the 0-dimensional scheme Y yields

#H^{0}(Y, I_{Z} · O_{Y}) = \frac{#H^{0}(Y, O_{Y})}{#H^{0}(Y, O_{Z∩Y})}

= \begin{cases}q^{(m+1) deg P} /q^{(ℓ+1) deg P} , & if P ∈ V, \\ q^{(m+1) deg P} , & if P ∉ V. \end{cases}

□

If U is a scheme of finite type over 𝔽_{q}, let U_{<r} be the set of closed points of U of degree < r. Similarly define U_{>r}.

Lemma 2.3 (Singularities of low degree). Let notation and hypotheses be as in Theorem 1.1, and define

P_{r} := { f ∈ I_{homog} : H_{f} ∩ X is smooth of dimension m − 1 at all P ∈ X_{<r} }.

Then

μ_{Z}(P_{r}) = ∏_{P ∈V_{<r}} (1 − q^{−(m−ℓ) deg P} ) · ∏_{P ∈(X−V)_{<r}} (1 − q^{−(m+1) deg P} ) .

Proof. Let X_{<r} = {P_{1}, …, P_{s}}. Let 𝔪_{i} be the ideal sheaf of P_{i} on X. let Y_{i} be the closed subscheme of X with ideal sheaf 𝔪_{i}^{2} ⊆ O_{X}, and let Y = ⋃ Y_{i}. Then H_{f} ∩ X is singular at P_{i} (more precisely, not smooth of dimension m − 1 at P_{i}) if and only if the restriction of f to a section of O_{Y_{i}}(d) is zero.

By Lemma 2.1, μ_{Z}(P) equals the fraction of elements in H^{0}(I_{Z} · O_{Y} (d)) whose restriction to a section of O_{Y_{i}}(d) is nonzero for every i. Thus

μ_{Z}(P_{r}) = ∏_{i=1}^{s} \frac{#H^{0}(Y_{i}, I_{Z} · O_{Y_{i}}) − 1}{#H^{0}(Y_{i}, I_{Z} · O_{Y_{i}})}

= ∏_{P ∈V_{<r}} (1 − q^{−(m−ℓ) deg P} ) · ∏_{P ∈(X−V)_{<r}} (1 − q^{−(m+1) deg P} ) ,

by Lemma 2.2. □

Corollary 2.4. If m > 2ℓ, then

lim_{r→∞} μ_{Z}(P_{r}) = \frac{ζ_{V} (m + 1)}{ζ_{X}(m + 1) ζ_{V} (m − ℓ)}.

Proof. The products in Lemma 2.3 are the partial products in the definition of the zeta functions. For convergence, we need m − ℓ > dim V = ℓ, which is equivalent to m > 2ℓ. □

Proof of Theorem 1.1(ii). We have P ⊆ P_{r}. By Lemma 2.3,

μ_{Z}(P_{r}) ≤ ∏_{P ∈V_{<r}} (1 − q^{−(m−ℓ) deg P} ) ,

which tends to 0 as r → ∞ if m ≤ 2ℓ. Thus μ_{Z}(P) = 0 in this case. □

From now on, we assume m > 2ℓ.

3. Singular points of medium degree

Lemma 3.1. Let P ∈ X is a closed point of degree e, where e ≤ \frac{d−c}{m+1}. Then the fraction of f ∈ I_{d} such that H_{f} ∩ X is not smooth of dimension m − 1 at P equals

\begin{cases}q^{−(m−ℓ)e}, & if P ∈ V, \\ q^{−(m+1)e}, & if P ∉ V. \end{cases}

Proof. This follows by applying Lemma 2.1 to the Y in Lemma 2.2, and then applying Lemma 2.2. □

Define the upper and lower densities \overline{μ}_{Z}(P), \underline{μ}_{Z}(P) of a subset P ⊆ I_{homog} as μ_{Z}(P) was defined, but using lim sup and lim inf in place of lim.

Lemma 3.2 (Singularities of medium degree). Define

Q_{r}^{medium} := ⋃_{d≥0}\{ f ∈ I_{d} : there exists P ∈ X with r ≤ deg P ≤ \frac{d − c}{m + 1}

such that H_{f} ∩ X is not smooth of dimension m − 1 at P }.

Then lim_{r→∞} \overline{μ}_{Z}(Q_{r}^{medium}) = 0.

Proof. By Lemma 3.1, we have

\frac{#(Q_{r}^{medium} ∩ I_{d})}{#I_{d}} ≤ ∑_{\begin{matrix}P ∈Z \\ r≤deg P ≤ \frac{d−c}{m+1} \end{matrix}} q^{−(m−ℓ) deg P} + ∑_{\begin{matrix}P ∈X−Z \\ r≤deg P ≤ \frac{d−c}{m+1} \end{matrix}} q^{−(m+1) deg P}

≤ ∑_{P ∈Z_{≥r}} q^{−(m−ℓ) deg P} + ∑_{P ∈(X−Z)_{≥r}} q^{−(m+1) deg P} .

Using the trivial bound that an m-dimensional variety has at most O(q^{em}) closed points of degree e, as in the proof of [Poo04, Lemma 2.4], we show that each of the two sums converges to a value that is O(q^{−r}) as r → ∞, under our assumption m > 2ℓ. □

4. Singular points of high degree

Lemma 4.1. Let P be a closed point of degree e in ℙ^{n} − Z. For d ≥ c, the fraction of f ∈ I_{d} that vanish at P is at most q^{− min(d−c,e)}.

Proof. Equivalently, we must show that the image of ϕ_{d} in Lemma 2.1 for Y = P has 𝔽_{q}- dimension at least min(d − c, e). The proof of Lemma 2.1 shows that as d runs through the integers c, c + 1, …, this dimension increases by at least 1 until it reaches its maximum, which is e. □

Lemma 4.2 (Singularities of high degree off V). Define

Q_{X−V}^{high} := ⋃_{d≥0}\{ f ∈ I_{d} : ∃P ∈ (X−V)_{> \frac{d−c}{m+1}} such that H_{f} ∩ X is not smooth of dimension m − 1 at P \}

Then \overline{μ}_{Z}(Q_{X−V}^{high} ) = 0.

Proof. It suffices to prove the lemma with X replaced by each of the sets in an open covering of X − V, so we may assume X is contained in 𝔸^{n} = {x_{0} ≠ 0} ⊆ ℙ^{n}, and that V = ∅. Dehomogenize by setting x_{0} = 1, to identify I_{d} ⊆ S_{d} with subspaces of I′_{d} ⊆ S′_{d} ⊆ A := 𝔽_{q}[x_{1}, …, x_{n}].

Given a closed point x ∈ X, choose a system of local parameters t_{1}, …, t_{n} ∈ A at x on 𝔸^{n} such that t_{m+1} = t_{m+2} = ⋯ = t_{n} = 0 defines X locally at x. Multiplying all the t_{i} by an element of A vanishing on Z but nonvanishing at x, we may assume in addition that all the t_{i} vanish on Z. Now dt_{1}, …, dt_{n} are a O_{𝔸^{n},x}-basis for the stalk Ω_{𝔸^{n}/𝔽_{q},x}^{1}. Let ∂_{1}, …, ∂_{n} be the dual basis of the stalk T_{𝔸^{n}/𝔽_{q},x} of the tangent sheaf. Choose s ∈ A with s(x) ≠ 0 to clear denominators so that D_{i} := s∂_{i} gives a global derivation A → A for i = 1, …, n. Then there is a neighborhood N_{x} of x in 𝔸^{n} such that N_{x} ∩ {t_{m+1} = t_{m+2} = ⋯ = t_{n} = 0} = N_{x} ∩ X, Ω_{N_{x}/𝔽_{q}}^{1} = ⊕_{i=1}^{n}O_{N_{x}}dt_{i}, and s ∈ O(N_{u})^{∗}. We may cover X with finitely many N_{x}, so we may reduce to the case where X ⊆ N_{x} for a single x. For f ∈ I′_{d} ≃ I_{d}, H_{f} ∩ X fails to be smooth of dimension m−1 at a point P ∈ U if and only if f(P) = (D_{1}f)(P) = ⋯ = (D_{m}f)(P) = 0.

Let τ = max_{i}(deg t_{i}), γ = ⌊(d − τ)/p⌋, and η = ⌊d/p⌋. If f_{0} ∈ I′_{d}, g_{1} ∈ S′_{γ}, …, g_{m} ∈ S′_{γ}, and h ∈ I′_{η} are selected uniformly and independently at random, then the distribution of

f := f_{0} + g_{1}^{p}t_{1} + ⋯ + g_{m}^{p}t_{m} + h^{p}

is uniform over I′_{d}, because of f_{0}. We will bound the probability that an f constructed in this way has a point P ∈ X_{> \frac{d−c}{m+1}} where f(P) = (D_{1}f)(P) = ⋯ = (D_{m}f)(P) = 0. We have D_{i}f = (D_{i}f_{0}) + g_{i}^{p}s for i = 1, …, m. We will select f_{0}, g_{1}, …, g_{m}, h one at a time. For 0 ≤ i ≤ m, define

W_{i} := X ∩ {D_{1}f = ⋯ = D_{i}f = 0}.

Claim 1: For 0 ≤ i ≤ m − 1, conditioned on a choice of f_{0}, g_{1}, …, g_{i} for which dim(W_{i}) ≤ m − i, the probability that dim(W_{i+1}) ≤ m − i − 1 is 1 − o(1) as d → ∞. (The function of d represented by the o(1) depends on X and the D_{i}.)

Proof of Claim 1: This is completely analogous to the corresponding proof in [Poo04].

Claim 2: Conditioned on a choice of f_{0}, g_{1}, …, g_{m} for which W_{m} is finite, Prob(H_{f} ∩ W_{m} ∩ X_{> \frac{d−c}{m+1}} = ∅) = 1 − o(1) as d → ∞.

Proof of Claim 2: By Bézout’s theorem as in [Ful84, p. 10], we have #W_{m} = O(d^{m}). For a given point P ∈ W_{m}, the set H^{bad} of h ∈ I′_{η} for which H_{f} passes through P is either ∅ or a coset of ker(ev_{P} : I′_{η} → κ(P)), where κ(P) is the residue field of P, and ev_{P} is the evaluation-at-P map. If moreover deg P > \frac{d−c}{m+1}, then Lemma 4.1 implies #H^{bad}/#I′_{η} ≤ q^{−ν} where ν = min (η, \frac{d−c}{m+1}). Hence

Prob(H_{f} ∩ W_{m} ∩ X_{> \frac{d−c}{m+1}} ≠ ∅) ≤ #W_{m}q^{−ν} = O(d^{m}q^{−ν}) = o(1)

as d → ∞, since ν eventually grows linearly in d. This proves Claim 2.

End of proof: Choose f ∈ I_{d} uniformly at random. Claims 1 and 2 show that with probability ∏_{i=0}^{m−1}(1 − o(1)) · (1 − o(1)) = 1 − o(1) as d → ∞, dim W_{i} = m − i for i = 0, 1, …, m and H_{f} ∩ W_{m} ∩ X_{> \frac{d−c}{m+1}} = ∅. But H_{f} ∩ W_{m} is the subvariety of X cut out by the equations f(P) = (D_{1}f)(P) = ⋯ = (D_{m}f)(P) = 0, so H_{f} ∩ W_{m} ∩ X_{> \frac{d−c}{m+1}} is exactly the set of points of H_{f} ∩ X of degree > \frac{d−c}{m+1} where H_{f} ∩ X is not smooth of dimension m − 1. Thus \overline{μ}_{Z}(Q_{X−V}^{high} ) = 0. □

Lemma 4.3 (Singularities of high degree on V). Define

Q_{V}^{high} := ⋃_{d≥0}\{ f ∈ I_{d} : ∃P ∈ V_{> \frac{d−c}{m+1}} such that H_{f} ∩ X is not smooth of dimension m − 1 at P \}.

Then \overline{μ}_{Z}(Q_{V}^{high}) = 0.

Proof. As before, we may assume X ⊆ 𝔸^{n} and we may dehomogenize. Given a closed point x ∈ X, choose a system of local parameters t_{1}, …, t_{n} ∈ A at x on 𝔸^{n} such that t_{m+1} = t_{m+2} = ⋯ = t_{n} = 0 defines X locally at x, and t_{1} = t_{2} = ⋯ = t_{m−ℓ} = t_{m+1} = t_{m+2} = ⋯ = t_{n} = 0 defines V locally at x. If 𝔪_{w} is the ideal sheaf of w on ℙ^{n}, then I_{Z} → \frac{𝔪_{w}}{𝔪_{w}^{2}} is surjective, so we may adjust t_{1}, …, t_{m−ℓ} to assume that they vanish not only on V but also on Z.

Define ∂_{i} and D_{i} as in the proof of Lemma 4.2. Then there is a neighborhood N_{x} of x in 𝔸^{n} such that N_{x} ∩ {t_{m+1} = t_{m+2} = ⋯ = t_{n} = 0} = N_{x} ∩ X, Ω_{N_{x}/𝔽_{q}}^{1} = ⊕_{i=1}^{n}O_{N_{x}}dt_{i}, and s ∈ O(N_{u})^{∗}. Again we may assume X ⊆ N_{x} for a single x. For f ∈ I′_{d} ≃ I_{d}, H_{f} ∩ X fails to be smooth of dimension m − 1 at a point P ∈ V if and only if f(P) = (D_{1}f)(P) = ⋯ = (D_{m}f)(P) = 0.

Again let τ = max_{i}(deg t_{i}), γ = ⌊(d − τ)/p⌋, and η = ⌊d/p⌋. If f_{0} ∈ I′_{d}, g_{1} ∈ S′_{γ}, …, g_{ℓ+1} ∈ S′_{γ}, are chosen uniformly at random, then

f := f_{0} + g_{1}^{p}t_{1} + ⋯ + g_{ℓ+1}^{p}t_{ℓ+1}

is a random element of I′_{d}, since ℓ + 1 ≤ m − ℓ.

For i = 0, …, ℓ + 1, the subscheme

W_{i} := V ∩ {D_{1}f = ⋯ = D_{i}f = 0}

depends only on the choices of f_{0}, g_{1}, …, g_{i}. The same argument as in the previous proof shows that for i = 0, …, ℓ, we have

Prob(dim W_{i} ≤ ℓ − i) = 1 − o(1)

as d → ∞. In particular, W_{ℓ} is finite with probability 1 − o(1).

To prove that \overline{μ}_{Z}(Q_{V}^{high}) = 0, it remains to prove that conditioned on choices of f_{0}, g_{1}, …, g_{ℓ} making dim W_{ℓ} finite,

Prob(W_{ℓ+1} ∩ V_{> \frac{d−c}{m+1}} = ∅) = 1 − o(1).

By Bézout’s theorem, #W_{ℓ} = O(d^{ℓ}). The set H^{bad} of choices of g_{ℓ+1} making D_{ℓ+1}f vanish at a given point P ∈ W_{ℓ} is either empty or a coset of ker(ev_{P} : S′_{γ} → κ(P)). Lemma 2.5 of [Poo04] implies that the size of this kernel (or its coset) as a fraction of #S′_{γ} is at most q^{−ν} where ν := min (γ, \frac{d−c}{m+1}). Since #W_{ℓ}q^{ν} = o(1) as d → ∞, we are done. □

5. Conclusion

Proof of Theorem 1.1(i). We have

P ⊆ P_{r} ⊆ P ∪ Q_{r}^{medium} ∪ Q_{X−V}^{high} ∪ Q_{V}^{high},

so \overline{μ}_{Z}(P) and \underline{μ}_{Z}(P) each differ from μ_{Z}(P_{r}) by at most \overline{μ}_{Z}(Q_{r}^{medium})+\overline{μ}_{Z}(Q_{X−V}^{high} )+\overline{μ}_{Z}(Q_{V}^{high}). Applying Corollary 2.4 and Lemmas 3.2, 4.2, and 4.3, we obtain

μ_{Z}(P) = lim_{r→∞} μ_{Z}(P_{r}) = \frac{ζ_{V} (m + 1)}{ζ_{V} (m − ℓ) ζ_{X}(m + 1)}.

□

Acknowledgements

I thank Shuji Saito for asking the question answered by this paper, and for pointing out [KA79].

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Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

E-mail address: poonen@math.berkeley.edu
URL: http://math.berkeley.edu/~poonen