BERTINI THEOREMS OVER FINITE FIELDS

BJORN POONEN

Abstract. Let X be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 0 over F_{q}. Then there exist homogeneous polynomials f over F_{q} for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζ_{X}(m + 1)^{−1}, where ζ_{X}(s) = Z_{X}(q^{−s}) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.

1. Introduction

The classical Bertini theorems say that if a subscheme X ⊆ P^{n} has a certain property, then for a sufficiently general hyperplane H ⊂ P^{n}, H ∩X has the property too. For instance, if X is a quasiprojective subscheme of P^{n} that is smooth of dimension m ≥ 0 over a field k, and U is the set of points u in the dual projective space P̌^{n} corresponding to hyperplanes H ⊂ P_{κ(u)}^{n} such that H ∩ X is smooth of dimension m − 1 over the residue field κ(u) of u, then U contains a dense open subset of P̌^{n}. If k is infinite, then U ∩ P̌^{n}(k) is nonempty, and hence one can find H over k. But if k is finite, then it can happen that the finitely many hyperplanes H over k all fail to give a smooth intersection H ∩ X. See Theorem 3.1.

N. M. Katz [Kat99] asked whether the Bertini theorem over finite fields can be salvaged by allowing hypersurfaces of unbounded degree in place of hyperplanes. (In fact he asked for a little more; see Section 3 for details.) We answer the question affirmatively below. O. Gabber [Gab01, Corollary 1.6] has independently proved the existence of good hypersurfaces of any sufficiently large degree divisible by the characteristic of k.

Let F_{q} be a finite field of q = p^{a} elements. 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, and let S_{homog} = ⋃_{d=0}^{∞} S_{d}. For each f ∈ S_{d}, let H_{f} be the subscheme Proj(S/(f)) ⊆ P^{n}. Typically (but not always), H_{f} is a hypersurface of dimension n − 1 defined by the equation f = 0. Define the density of a subset P ⊆ S_{homog} by

μ(P) := lim_{d→∞} \frac{#(P ∩ S_{d})}{#S_{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}) .

Date: August 31, 2004.

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

This research was supported by NSF grant DMS-9801104 and a Packard Fellowship. Part of the research was done while the author was enjoying the hospitality of the Université de Paris-Sud. This article will appear in Annals of Math.

Theorem 1.1 (Bertini over finite fields). Let X be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 0 over F_{q}. Define

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

Then μ(P) = ζ_{X}(m + 1)^{−1}.

Remarks.

(1) The empty scheme is smooth of any dimension, including −1. Later (for instance, in Theorem 1.3), we will similarly use the convention that if P is a point not on a scheme X, then for any r, the scheme X is automatically smooth of dimension r at P.

(2) In this paper, ∩ denotes scheme-theoretic intersection (when applied to schemes).

(3) If n ≥ 2, the density is unchanged if we insist also that H_{f} be a geometrically integral hypersurface of dimension n − 1. This follows from the easy Proposition 2.7.

(4) The case n = 1, X = A^{1}, is a well known polynomial analogue of the fact that the set of squarefree integers has density ζ(2)^{−1} = 6/π^{2}. See Section 5 for a conjectural common generalization.

(5) The density is independent of the choice of embedding X ↪ P^{n}!

(6) By [Dwo60], ζ_{X} is a rational function of q^{−s}, so ζ_{X}(m + 1)^{−1} ∈ Q.

The overall plan of the proof is to start with all homogeneous polynomials of degree d, and then for each closed point P ∈ X to sieve out the polynomials f for which H_{f} ∩X is singular at P. The condition that P be singular on H_{f} ∩X amounts to m+1 linear conditions on the Taylor coefficients of a dehomogenization of f at P, and these linear conditions are over the residue field of P. Therefore one expects that the probability that H_{f} ∩ X is nonsingular at P will be 1 − q^{−(m+1) deg P}. Assuming that these conditions at different P are independent, the probability that H_{f} ∩ X is nonsingular everywhere should be

∏_{closed P ∈X} (1 − q^{−(m+1) deg P} ) = ζ_{X}(m + 1)^{−1}.

Unfortunately, the independence assumption and the individual singularity probability estimates break down once deg P becomes large relative to d. Therefore we must approximate our answer by truncating the product after finitely many terms, say those corresponding to P of degree < r. The main difficulty of the proof, as with many sieve proofs, is in bounding the error of the approximation, i.e., in showing that when d ≫ r ≫ 1, the number of polynomials of degree d sieved out by conditions at the infinitely many P of degree ≥ r is negligible.

In fact we will prove Theorem 1.1 as a special case of the following, which is more versatile in applications. The effect of T below is to prescribe the first few terms of the Taylor expansions of the dehomogenizations of f at finitely many closed points.

Theorem 1.2 (Bertini with Taylor conditions). Let X be a quasiprojective subscheme of P^{n} over F_{q}. Let Z be a finite subscheme of P^{n}, and assume that U := X − (Z ∩ X) is smooth of dimension m ≥ 0. Fix a subset T ⊆ H^{0}(Z, O_{Z}). Given f ∈ S_{d}, let f|_{Z} be the element of H^{0}(Z, O_{Z}) that on each connected component Z_{i} equals the restriction of x_{j}^{−d}f to Z_{i}, where j = j(i) is the smallest j ∈ {0, 1 …, n} such that the coordinate x_{j} is invertible on Z_{i}. Define

P := { f ∈ S_{homog} : H_{f} ∩ U is smooth of dimension m − 1, and f|_{Z} ∈ T }.

Then

μ(P) = \frac{#T}{#H^{0}(Z, O_{Z})} ζ_{U}(m + 1)^{−1}.

Using a formalism analogous to that of Lemma 20 of [PS99], we can deduce the following even stronger version, which allows us to impose infinitely many local conditions, provided that the conditions at most points are no more stringent than the condition that the hypersurface intersect a given finite set of varieties smoothly.

Theorem 1.3 (Infinitely many local conditions). For each closed point P of P^{n} over F_{q}, let μ_{P} be normalized Haar measure on the completed local ring Ô_{P} as an additive compact group, and let U_{P} be a subset of Ô_{P} whose boundary ∂U_{P} has measure zero. Also for each P, fix a nonvanishing coordinate x_{j}, and for f ∈ S_{d} let f|_{P} be the image of x_{j}^{−d}f in Ô_{P}. Assume that there exist smooth quasiprojective subschemes X_{1}, …, X_{u} of P^{n} of dimensions m_{i} = dim X_{i} over F_{q} such that for all but finitely many P, U_{P} contains f|_{P} whenever f ∈ S_{homog} is such that H_{f} ∩ X_{i} is smooth of dimension m_{i} − 1 at P for all i. Define

P := { f ∈ S_{homog} : f|_{P} ∈ U_{P} for all closed points P ∈ P^{n} }.

Then μ(P) = ∏_{closed P ∈P^{n}} μ_{P} (U_{P}).

Remark. Implicit in Theorem 1.3 is the claim that the product ∏_{P} μ_{P} (U_{P}) always converges, and in particular that its value is zero if and only if μ_{P} (U_{P}) = 0 for some closed point P.

The proofs of Theorems 1.1, 1.2, and 1.3 are contained in Section 2. But the reader at this point is encouraged to jump to Section 3 for applications, and to glance at Section 5, which shows that the abc conjecture and another conjecture imply analogues of our main theorems for regular quasiprojective schemes over Spec Z. The abc conjecture is needed to apply a multivariable generalization [Poo03] of A. Granville’s result [Gra98] about squarefree values of polynomials. For some open questions, see Sections 4 and 5.7, and also Conjecture 5.2.

The author hopes that the technique of Section 2 will prove useful in removing the condition “assume that the ground field k is infinite” from other theorems in the literature.

2. Bertini over finite fields: the closed point sieve

Sections 2.1, 2.2, and 2.3 are devoted to the proofs of Lemmas 2.2, 2.4, and 2.6, which are the main results needed in Section 2.4 to prove Theorems 1.1, 1.2, and 1.3.

2.1. Singular points of low degree. Let A = F_{q}[x_{1}, …, x_{n}] be the ring of regular functions on the subset A^{n} := {x_{0} ≠ 0} ⊆ P^{n}, and identify S_{d} with the set of dehomogenizations A_{≤d} = { f ∈ A : deg f ≤ d }, where deg f denotes total degree.

Lemma 2.1. If Y is a finite subscheme of P^{n} over a field k, then the map

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

is surjective for d ≥ dim H^{0}(Y, O_{Y}) − 1.

Proof. Let I_{Y} be the ideal sheaf of Y ⊆ P^{n}. Then coker(ϕ_{d}) is contained in H^{1}(P^{n}, I_{Y} (d)), which vanishes for d ≫ 1 by Theorem III.5.2b of [Har77]. 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}. Dehomogenize by setting x_{0} = 1, so that ϕ_{d} is identified with a map from A_{≤d} to B := H^{0}(Y, O_{Y}). Let b = dim B. For i ≥ −1, let B_{i} be the image of A_{≤i} in B. Then 0 = B_{−1} ⊆ B_{0} ⊆ B_{1} ⊆ …, so B_{j} = B_{j+1} for some j ∈ [−1, b − 1]. Then

B_{j+2} = B_{j+1} + ∑_{i=1}^{n} x_{i}B_{j+1} = B_{j} + ∑_{i=1}^{n} x_{i}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 ≥ b − 1. □

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 2.2 (Singularities of low degree). Let notation and hypotheses be as in Theorem 1.2, and define

P_{r} := { f ∈ S_{homog} : H_{f} ∩ U is smooth of dimension m − 1 at all P ∈ U_{<r}, and f|_{Z} ∈ T }.

Then

μ(P_{r}) = \frac{#T}{#H^{0}(Z, O_{Z})} ∏_{P ∈U_{<r}} (1 − q^{−(m+1) deg P} ) .

Proof. Let U_{<r} = {P_{1}, …, P_{s}}. Let 𝔪_{i} be the ideal sheaf of P_{i} on U, let Y_{i} be the closed subscheme of U corresponding to the ideal sheaf 𝔪_{i}^{2} ⊆ O_{U}, and let Y = ⋃ Y_{i}. Then H_{f} ∩ U 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. Hence P_{r} ∩ S_{d} is the inverse image of

T × ∏_{i=1}^{s} (H^{0}(Y_{i}, O_{Y_{i}}) − {0})

under the F_{q}-linear composition

ϕ_{d} : S_{d} = H^{0}(P^{n}, O_{P^{n}}(d)) → H^{0}(Y ∪ Z, O_{Y ∪Z}(d)) ≃ H^{0}(Z, O_{Z}) × ∏_{i=1}^{s} H^{0}(Y_{i}, O_{Y_{i}}),

where the last isomorphism is the (noncanonical) untwisting, component by component, by division by the d-th powers of various coordinates, as in the definition of f|_{Z}. Applying Lemma 2.1 to Y ∪ Z shows that ϕ_{d} is surjective for d ≫ 1, so

μ(P_{r}) = lim_{d→∞} \frac{# [T × ∏_{i=1}^{s} (H^{0}(Y_{i}, O_{Y_{i}}) − \{0\})]}{# [H^{0}(Z, O_{Z}) × ∏_{i=1}^{s} H^{0}(Y_{i}, O_{Y_{i}})]} = \frac{#T}{#H^{0}(Z, O_{Z})} ∏_{i=1}^{s} (1 − q^{−(m+1) deg P_{i}}) ,

since H^{0}(Y_{i}, O_{Y_{i}}) has a two-step filtration whose quotients O_{U,P_{i}}/𝔪_{U,P_{i}} and 𝔪_{U,P_{i}}/𝔪_{U,P_{i}}^{2} are vector spaces of dimensions 1 and m respectively over the residue field of P_{i}. □

2.2. Singular points of medium degree.

Lemma 2.3. Let U be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 0 over F_{q}. If P ∈ U is a closed point of degree e, where e ≤ d/(m + 1), then the fraction of f ∈ S_{d} such that H_{f} ∩ U is not smooth of dimension m − 1 at P equals q^{−(m+1)e}.

Proof. Let 𝔪 be the ideal sheaf of P on U, and let Y be the closed subscheme of U corresponding to 𝔪^{2}. The f ∈ S_{d} to be counted are those in the kernel of ϕ_{d} : H^{0}(P^{n}, O(d)) → H^{0}(Y, O_{Y} (d)). We have dim H^{0}(Y, O_{Y} (d)) = dim H^{0}(Y, O_{Y}) = (m + 1)e ≤ d, so ϕ_{d} is surjective by Lemma 2.1, and the F_{q}-codimension of ker ϕ_{d} equals (m + 1)e. □

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

Lemma 2.4 (Singularities of medium degree). Let U be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 0 over F_{q}. Define

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

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

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

Proof. Using Lemma 2.3 and the crude bound #U(F_{q^{e}}) ≤ cq^{em} for some c > 0 depending only on U [LW54], we obtain

\frac{#(Q_{r}^{medium} ∩ S_{d})}{#S_{d}} ≤ ∑_{e=r}^{⌊d/(m+1)⌋} (number of points of degree e in U) q^{−(m+1)e}

≤ ∑_{e=r}^{⌊d/(m+1)⌋} #U(F_{q^{e}})q^{−(m+1)e}

≤ ∑_{e=r}^{∞} cq^{em}q^{−(m+1)e},

= \frac{cq^{−r}}{1 − q^{−1}} .

Hence \overline{μ}(Q_{r}^{medium}) ≤ cq^{−r}/(1 − q^{−1}), which tends to zero as r → ∞. □

2.3. Singular points of high degree.

Lemma 2.5. Let P be a closed point of degree e in A^{n} over F_{q}. Then the fraction of f ∈ A_{≤d} that vanish at P is at most q^{− min(d+1,e)}.

Proof. Let ev_{P} : A_{≤d} → F_{q^{e}} be the evaluation-at-P map. The proof of Lemma 2.1 shows that dim_{F_{q}} ev_{P} (A_{≤d}) strictly increases with d until it reaches e, so dim_{F_{q}} ev_{P} (A_{≤d}) ≥ min(d+1, e). Equivalently, the codimension of ker(ev_{P}) in A_{≤d} is at least min(d + 1, e). □

Lemma 2.6 (Singularities of high degree). Let U be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 0 over F_{q}. Define

Q^{high} := ⋃_{d≥0}{ f ∈ S_{d} : ∃P ∈ U_{>d/(m+1)} such that H_{f} ∩ U is not smooth of dimension m − 1 at P }.

Then \overline{μ}(Q^{high}) = 0.

Proof. If the lemma holds for U and for V, it holds for U ∪ V, so we may assume U ⊆ A^{n} is affine.

Given a closed point u ∈ U, choose a system of local parameters t_{1}, …, t_{n} ∈ A at u on A^{n} such that t_{m+1} = t_{m+2} = ⋯ = t_{n} = 0 defines U locally at u. Then dt_{1}, …, dt_{n} are a O_{A^{n},u}-basis for the stalk Ω_{A^{n}/F_{q},u}^{1}. Let ∂_{1}, …, ∂_{n} be the dual basis of the stalk T_{A^{n}/F_{q},u} of the tangent sheaf. Choose s ∈ A with s(u) ≠ 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_{u} of u in A^{n} such that N_{u} ∩ {t_{m+1} = t_{m+2} = ⋯ = t_{n} = 0} = N_{u} ∩ U, Ω_{N_{u}/F_{q}}^{1} = ⊕_{i=1}^{n}O_{N_{u}}dt_{i}, and s ∈ O(N_{u})^{∗}. We may cover U with finitely many N_{u}, so by the first sentence of this proof, we may reduce to the case where U ⊆ N_{u} for a single u. For f ∈ A_{≤d}, H_{f} ∩ U 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.

Now for the trick. Let τ = max_{i}(deg t_{i}), γ = ⌊(d − τ)/p⌋, and η = ⌊d/p⌋. If f_{0} ∈ A_{≤d}, g_{1} ∈ A_{≤γ}, …, g_{m} ∈ A_{≤γ}, and h ∈ A_{≤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 A_{≤d}. We will bound the probability that an f constructed in this way has a point P ∈ U_{>d/(m+1)} where f(P) = (D_{1}f)(P) = ⋯ = (D_{m}f)(P) = 0. By writing f in this way, we partially decouple the D_{i}f from each other: 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} := U ∩ {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 U and the D_{i}.)

Proof of Claim 1: Let V_{1}, …, V_{ℓ} be the (m − i)-dimensional F_{q}-irreducible components of (W_{i})_{red}. By Bézout’s theorem [Ful84, p. 10],

ℓ ≤ (deg \overline{U})(deg D_{1}f) … (deg D_{i}f) = O(d^{i})

as d → ∞, where \overline{U} is the projective closure of U. Since dim V_{k} ≥ 1, there exists a coordinate x_{j} depending on k such that the projection x_{j}(V_{k}) has dimension 1. We need to bound the set

G_{k}^{bad} := { g_{i+1} ∈ A_{≤γ} : D_{i+1}f = (D_{i+1}f_{0}) + g_{i+1}^{p}s vanishes identically on V_{k} }.

If g, g′ ∈ G_{k}^{bad}, then by taking the difference and multiplying by s^{−1}, we see that g − g′ vanishes on V_{k}. Hence if G_{k}^{bad} is nonempty, it is a coset of the subspace of functions in A_{≤γ} vanishing on V_{k}. The codimension of that subspace, or equivalently the dimension of the image of A_{≤γ} in the regular functions on V_{k}, exceeds γ + 1, since a nonzero polynomial in x_{j} alone does not vanish on V_{k}. Thus the probability that D_{i+1}f vanishes on some V_{k} is at most ℓq^{−γ−1} = O(d^{i}q^{−(d−τ)/p}) = o(1) as d → ∞. This proves Claim 1.

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

Proof of Claim 2: The Bézout theorem argument in the proof of Claim 1 shows that #W_{m} = O(d^{m}). For a given point P ∈ W_{m}, the set H^{bad} of h ∈ A_{≤η} for which H_{f} passes through P is either ∅ or a coset of ker(ev_{P} : A_{≤η} → κ(P)), where κ(P) is the residue field of P. If moreover deg P > d/(m + 1), then Lemma 2.5 implies #H^{bad}/#A_{≤η} ≤ q^{−ν} where ν = min (η + 1, d/(m + 1)). Hence

Prob(H_{f} ∩ W_{m} ∩ U_{>d/(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 of Lemma 2.6: Choose f ∈ S_{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} ∩ U_{>d/(m+1)} = ∅. But H_{f} ∩ W_{m} is the subvariety of U cut out by the equations f(P) = (D_{1}f)(P) = ⋯ = (D_{m}f)(P) = 0, so H_{f} ∩ W_{m} ∩ U_{>d/(m+1)} is exactly the set of points of H_{f} ∩ U of degree > d/(m + 1) where H_{f} ∩ U is not smooth of dimension m − 1. □

2.4. Proofs of theorems over finite fields.

Proof of Theorem 1.2. As mentioned in the proof of Lemma 2.4, the number of closed points of degree r in U is O(q^{rm}); this guarantees that the product defining ζ_{U}(s)^{−1} converges at s = m + 1. By Lemma 2.2,

lim_{r→∞} μ(P_{r}) = \frac{#T}{#H^{0}(Z, O_{Z})} ζ_{U}(m + 1)^{−1}.

On the other hand, the definitions imply P ⊆ P_{r} ⊆ P ∪ Q_{r}^{medium} ∪ Q^{high}, so \overline{μ}(P) and \underline{μ}(P) each differ from μ(P_{r}) by at most \overline{μ}(Q_{r}^{medium}) + \overline{μ}(Q^{high}). Applying Lemmas 2.4 and 2.6 and letting r tend to ∞, we obtain

μ(P) = lim_{r→∞} μ(P_{r}) = \frac{#T}{#H^{0}(Z, O_{Z})} ζ_{U}(m + 1)^{−1}.

□

Proof of Theorem 1.1. Take Z = ∅ and T = {0} in Theorem 1.2. □

Proof of Theorem 1.3. The existence of X_{1}, …, X_{u} and Lemmas 2.4 and 2.6 let us approximate P by the set P_{r} defined only by the conditions at closed points P of degree less than r, for large r. For each P ∈ P_{<r}^{n}, the hypothesis μ_{P} (∂U_{P}) = 0 lets us approximate U_{P} by a union of cosets of an ideal I_{P} of finite index in Ô_{P}. (The details are completely analogous to those in the proof of Lemma 20 of [PS99].) Finally, Lemma 2.1 implies that for d ≫ 1, the images of f ∈ S_{d} in ∏_{P ∈P_{<r}^{n}} Ô_{P} /I_{P} are equidistributed. □

Finally let us show that the densities in our theorems do not change if in the definition of density we consider only f for which H_{f} is geometrically integral, at least for n ≥ 2.

Proposition 2.7. Suppose n ≥ 2. Let R be the set of f ∈ S_{homog} for which H_{f} fails to be a geometrically integral hypersurface of dimension n − 1. Then μ(R) = 0.

Proof. We have R = R_{1} ∪ R_{2} where R_{1} is the set of f ∈ S_{homog} that factor nontrivially over F_{q}, and R_{2} is the set of f ∈ S_{homog} of the form N_{F_{q^{e}}/F_{q}}(g) for some homogeneous polynomial g ∈ F_{q^{e}}[x_{0}, …, x_{n}] and e ≥ 2. (Note: if our base field were an arbitrary perfect field, an irreducible polynomial that is not absolutely irreducible would be a constant times a norm, but the constant is unnecessary here, since N_{F_{q^{e}}/F_{q}} : F_{q^{e}} → F_{q} is surjective.)

We have

\frac{#(R_{1} ∩ S_{d})}{#S_{d}} ≤ \frac{1}{#S_{d}} ∑_{i=1}^{⌊d/2⌋}(#S_{i})(#S_{d−i}) = ∑_{i=1}^{⌊d/2⌋} q^{−N_{i}},

where

N_{i} = \binom{n + d}{n} − \binom{n + i}{n} − \binom{n + d − i}{n}.

For 1 ≤ i ≤ d/2 − 1,

N_{i+1} − N_{i} = [\binom{n + d − i}{n} − \binom{n + d − i − 1}{n}] − [\binom{n + i + 1}{n} − \binom{n + i}{n}]

= \binom{n + d − i − 1}{n − 1} − \binom{n + i}{n − 1}

> 0.

Similarly, for d ≫ n,

N_{1} = \binom{n + d − 1}{n − 1} − \binom{n + 1}{n} ≥ \binom{n + d − 1}{1} − \binom{n + 1}{1} = d − 2.

Thus

\frac{#(R_{1} ∩ S_{d})}{#S_{d}} ≤ ∑_{i=1}^{⌊d/2⌋} q^{−N_{i}} ≤ ∑_{i=1}^{⌊d/2⌋} q^{2−d} ≤ dq^{2−d},

which tends to zero as d → ∞.

The number of f ∈ S_{d} that are norms of homogeneous polynomials of degree d/e over F_{q^{e}} is at most (q^{e})^{\binom{d/e+n}{n}}. Therefore

\frac{#(R_{2} ∩ S_{d})}{#S_{d}} ≤ ∑_{e|d,e>1} q^{−M_{e}}

where M_{e} = \binom{d+n}{n} − e\binom{d/e+n}{n}. For 2 ≤ e ≤ d,

\frac{e\binom{d/e+n}{n}}{\binom{d+n}{n}} = \frac{e (\frac{d}{e} + n) (\frac{d}{e} + n − 1) ⋯ (\frac{d}{e} + 1)}{(d + n)(d + n − 1) ⋯ (d + 1)}

≤ \frac{e (\frac{d}{e} + n) (\frac{d}{e} + n − 1)}{(d + n)(d + n − 1)}

≤ \frac{e (\frac{d}{e} + n)^{2}}{d^{2}}

= \frac{1}{e} + \frac{2n}{d} + \frac{en^{2}}{d^{2}}

≤ \frac{1}{2} + \frac{2n^{2}}{d} + \frac{dn^{2}}{d^{2}}

≤ 2/3,

once d ≥ 18n^{2}. Hence in this case, M_{e} ≥ \frac{1}{3} \binom{d+n}{n} ≥ d^{2}/6 for large d, so

\frac{#(R_{2} ∩ S_{d})}{#S_{d}} ≤ ∑_{e|d,e>1} q^{−M_{e}} ≤ dq^{−d^{2}/6},

which tends to zero as d → ∞. □

Another proof of Proposition 2.7 is given in Section 3.2, but that proof is valid only for n ≥ 3.

3. Applications

3.1. Counterexamples to Bertini. Ironically, we can use our hypersurface Bertini theorem to construct counterexamples to the original hyperplane Bertini theorem! More generally, we can show that hypersurfaces of bounded degree do not suffice to yield a smooth intersection.

Theorem 3.1 (Anti-Bertini theorem). Given a finite field F_{q} and integers n ≥ 2, d ≥ 1, there exists a smooth projective geometrically integral hypersurface X in P^{n} over F_{q} such that for each f ∈ S_{1} ∪ ⋯ ∪ S_{d}, H_{f} ∩ X fails to be smooth of dimension n − 2.

Proof. Let H^{(1)}, …, H^{(ℓ)} be a list of the H_{f} arising from f ∈ S_{1} ∪⋯∪S_{d}. For i = 1, …, ℓ in turn, choose a closed point P_{i} ∈ H^{(i)} distinct from P_{j} for j < i. Using a T as in Theorem 1.2, we can express the condition that a hypersurface in P^{n} be smooth of dimension n − 1 at P_{i} and have tangent space at P_{i} equal to that of H^{(i)} whenever the latter is smooth of dimension n − 1 at P_{i}. Theorem 1.2 (with Proposition 2.7) implies that there exists a smooth projective geometrically integral hypersurface X ⊆ P^{n} satisfying these conditions. Then for each i, X ∩ H^{(i)} fails to be smooth of dimension n − 2 at P_{i}. □

Remark. Katz [Kat99, p. 621] remarks that if X is the hypersurface

∑_{i=1}^{n+1}(X_{i}Y_{i}^{q} − X_{i}^{q}Y_{i}) = 0

in P^{2n+1} over F_{q} with homogeneous coordinates X_{1}, …, X_{n+1}, Y_{1}, …, Y_{n+1}, then H ∩ X is singular for every hyperplane H in P^{2n+1} over F_{q}.

3.2. Singularities of positive dimension. Let X be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 0 over F_{q}. Given f ∈ S_{homog}, let (H_{f} ∩ X)_{sing} be the closed subset of points where H_{f} ∩ X is not smooth of dimension m − 1.

Although Theorem 1.1 shows that for a nonempty smooth quasiprojective subscheme X ⊆ P^{n} of dimension m ≥ 0, there is a positive probability that (H_{f} ∩ X)_{sing} ≠ ∅, we now show that the probability that dim(H_{f} ∩ X)_{sing} ≥ 1 is zero.

Theorem 3.2. Let X be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 0 over F_{q}. Define

S := { f ∈ S_{homog} : dim(H_{f} ∩ X)_{sing} ≥ 1 }.

Then μ(S) = 0.

Proof. This is a corollary of Lemma 2.6 with U = X, since S ⊆ Q^{high}. □

Remark. If f ∈ S_{homog} is such that H_{f} is not geometrically integral of dimension n − 1, then dim(H_{f})_{sing} ≥ n − 2. Hence Theorem 3.2 with X = P^{n} gives a new proof of Proposition 2.7, at least when n ≥ 3.

3.3. Space-filling curves. We next use Theorem 1.2 to answer affirmatively all the open questions in [Kat99]. In their strongest forms, these are

Question 10: Given a smooth projective geometrically connected variety X of dimension m ≥ 2 over F_{q}, and a finite extension E of F_{q}, is there always a closed subscheme Y in X, Y ≠ X, such that Y (E) = X(E) and such that Y is smooth and geometrically connected over F_{q}?

Question 13: Given a closed subscheme X ⊆ P^{n} over F_{q} that is smooth and geometrically connected of dimension m, and a point P ∈ X(F_{q}), is it true for all d ≫ 1 that there exists a hypersurface H ⊆ P^{n} of degree d such that P lies on H and H ∩ X is smooth of dimension m − 1?

Both of these questions are answered by the following:

Theorem 3.3. Let X be a smooth quasiprojective subscheme of P^{n} of dimension m ≥ 1 over F_{q}, and let F ⊂ X be a finite set of closed points. Then there exists a smooth projective geometrically integral hypersurface H ⊂ P^{n} such that H ∩ X is smooth of dimension m − 1 and contains F.

Remarks.

(1) If m ≥ 2 and if X in Theorem 3.3 is geometrically connected and projective in addition to being smooth, then H ∩X will be geometrically connected and projective too. This follows from Corollary III.7.9 in [Har77].

(2) Recall that if a variety is geometrically connected and smooth, then it is geometrically integral.

(3) Question 10 and (partially) Question 13 were independently answered by Gabber [Gab01].

Proof of Theorem 3.3. Let T_{P,X} be the Zariski tangent space of a point P on X. At each P ∈ F choose a codimension 1 subspace V_{P} ⊂ T_{P,P^{n}} not equal to T_{P,X}. We will apply Theorem 1.3 with the following local conditions: for P ∈ F, U_{P} is the condition that the hypersurface H_{f} passes through P and T_{P,H} = V_{P}; for P ∉ F, U_{P} is the condition that H_{f} and H_{f} ∩ X be smooth of dimensions n − 1 and m − 1, respectively, at P. Theorem 1.3 (with Proposition 2.7) implies the existence of a smooth projective geometrically integral hypersurface H ⊂ P^{n} satisfying these conditions. □

Remark. If we did not insist in Theorem 3.3 that H be smooth, then in the proof, Theorem 1.2 would suffice in place of Theorem 1.3. This weakened version of Theorem 3.3 is already enough to imply Corollaries 3.4 and 3.5, and Theorem 3.7. Corollary 3.6 also follows from Theorem 1.2.

Corollary 3.4. Let X be a smooth, projective, geometrically integral variety of dimension m ≥ 1 over F_{q}, let F be a finite set of closed points of X, and let y be an integer with 1 ≤ y ≤ m. Then there exists a smooth, projective, geometrically integral subvariety Y ⊆ X of dimension y such that F ⊂ Y.

Proof. Use Theorem 3.3 with reverse induction on y. □

Corollary 3.5 (Space-filling curves). Let X be a smooth, projective, geometrically integral variety of dimension m ≥ 1 over F_{q}, and let E be a finite extension of F_{q}. Then there exists a smooth, projective, geometrically integral curve Y ⊆ X such that Y (E) = X(E).

Proof. Apply Corollary 3.4 with y = 1 and F the set of closed points corresponding to X(E). □

In a similar way, we prove the following:

Corollary 3.6 (Space-avoiding varieties). Let X be a smooth, projective, geometrically integral variety of dimension m over F_{q}, and let ℓ and y be integers with ℓ ≥ 1 and 1 ≤ y < m. Then there exists a smooth, projective, geometrically integral subvariety Y ⊆ X of dimension y such that Y has no points of degree less than ℓ.

Proof. Repeat the arguments used in the proof of Theorem 3.3 and Corollary 3.4, but in the first application of Theorem 1.3, instead force the hypersurface to avoid the finitely many points of X of degree less than ℓ. □

3.4. Albanese varieties. For a smooth, projective, geometrically integral variety X over a field, let Alb X be its Albanese variety. As pointed out in [Kat99], a positive answer to Question 13 implies that every positive dimensional abelian variety A over F_{q} contains a smooth, projective, geometrically integral curve Y such that the natural map Alb Y → A is surjective. We generalize this slightly in the next result, which strengthens Theorem 11 of [Kat99] in the finite field case.

Theorem 3.7. Let X be a smooth, projective, geometrically integral variety of dimension m ≥ 1 over F_{q}. Then there exists a smooth, projective, geometrically integral curve Y ⊆ X such that the natural map Alb Y → Alb X is surjective.

Proof. Choose a prime ℓ not equal to the characteristic. Represent each ℓ-torsion point in (Alb X)(\overline{F}_{q}) by a zero-cycle of degree zero on X, and let F be the finite set of closed points appearing in these. Use Corollary 3.4 to construct a smooth, projective, geometrically integral curve Y passing through all points of F. The image of Alb Y → Alb X is an abelian subvariety of Alb X containing all the ℓ-torsion points, so the image equals Alb X. (The trick of using the ℓ-torsion points is due to Gabber [Kat99].) □

Remarks.

(1) A slightly more general argument proves Theorem 3.7 over an arbitrary field k [Gab01, Proposition 2.4].

(2) It is also true that any abelian variety over a field k can be embedded as an abelian subvariety of the Jacobian of a smooth, projective, geometrically integral curve over k [Gab01].

3.5. Plane curves. The probability that a projective plane curve over F_{q} is nonsingular equals

ζ_{P^{2}}(3)^{−1} = (1 − q^{−1})(1 − q^{−2})(1 − q^{−3}).

(We interpret this probability as the density given by Theorem 1.1 for X = P^{2} in P^{2}.) Theorem 1.3 with a simple local calculation shows that the probability that a projective plane curve over F_{q} has at worst nodes as singularities equals

ζ_{P^{2}}(4)^{−1} = (1 − q^{−2})(1 − q^{−3})(1 − q^{−4}).

For F_{2}, these probabilities are 21/64 and 315/512.

Remark. Although Theorem 1.1 guarantees the existence of a smooth plane curve of degree d over F_{q} only when d is sufficiently large relative to q, in fact such a curve exists for every d ≥ 1 and every finite field F_{q}. Moreover, the corresponding statement for hypersurfaces of specified dimension and degree is true [KS99, §11.4.6]. In fact, for any field k and integers n ≥ 1, d ≥ 3 with (n, d) not equal to (1, 3) or (2, 4), there exists a smooth hypersurface X over k of degree d in P^{n+1} such that X has no nontrivial automorphisms over \overline{k} [Poo00]. This last statement is false for (1, 3); whether or not it holds for (2, 4) is an open question.

4. An open question

In response to Theorem 1.1, Matt Baker has asked the following:

Question 4.1. Fix a smooth quasiprojective subscheme X of dimension m over F_{q}. Does there exist an integer n_{0} > 0 such that for n ≥ n_{0}, if ι : X → P^{n} is an embedding such that no connected component of X is mapped by ι into a hyperplane in P^{n}, then there exists a hyperplane H ⊆ P^{n} over F_{q} such that H ∩ ι(X) is smooth of dimension m − 1?

Theorem 1.1 proves that the answer is yes, if one allows only the embeddings ι obtained by composing a fixed initial embedding X → P^{n} with d-uple embeddings P^{n} → P^{N}. Nevertheless, we conjecture that for each X of positive dimension, the answer to Question 4.1 is no.

5. An arithmetic analogue

We formulate an analogue of Theorem 1.1 in which the smooth quasiprojective scheme X over F_{q} is replaced by a regular quasiprojective scheme X over Spec Z, and we seek hyperplane sections that are regular. The reason for using regularity instead of the stronger condition of being smooth over Z is discussed in Section 5.7.

Fix n ∈ N = Z_{≥0}. Redefine S as the homogeneous coordinate ring Z[x_{0}, …, x_{n}] of P_{Z}^{n}, let S_{d} ⊂ S be the Z-submodule of homogeneous polynomials of degree d, and let S_{homog} = ⋃_{d=0}^{∞} S_{d}. If p is prime, let S_{d,p} be the set of homogeneous polynomials in F_{p}[x_{0}, …, x_{n}] of degree d. For each f ∈ S_{d}, let H_{f} be the subscheme Proj(S/(f)) ⊆ P_{Z}^{n}. Similarly, for f ∈ S_{d,p}, let H_{f} be Proj(F_{p}[x_{0}, …, x_{n}]/(f)) ⊆ P_{F_{p}}^{n}.

If P is a subset of Z^{N} for some N ≥ 1, define the upper density

\overline{μ}(P) := max_{σ} lim sup_{B_{σ(1)}→∞} ⋯ lim sup_{B_{σ(N)}→∞} \frac{#(P ∩ Box)}{#Box},

where σ ranges over permutations of {1, 2, …, N} and

Box = {(x_{1}, …, x_{N}) ∈ Z^{N} : |x_{i}| ≤ B_{i} for all i}.

(In other words, we take the lim sup only over growing boxes whose dimensions can be ordered so that each is very large relative to the previous dimensions.) Define lower density \underline{μ}(P) similarly using min and lim inf. Define upper and lower densities \overline{μ}_{d} and \underline{μ}_{d} of subsets of a fixed S_{d} by identifying S_{d} with Z^{N} using a Z-basis of monomials. If P ⊆ S_{homog}, define \overline{μ}(P) = lim sup_{d→∞} \overline{μ}_{d}(P ∩ S_{d}) and \underline{μ}(P) = lim inf_{d→∞} \underline{μ}_{d}(P ∩ S_{d}). Finally, if P is a subset of S_{homog}, define μ(P) as the common value of \overline{μ}(P) and \underline{μ}(P) if \overline{μ}(P) = \underline{μ}(P). The reason for choosing this definition is that it makes our proof work; aesthetically, we would have preferred to prove a stronger statement by defining density as the limit over arbitrary boxes in S_{d} with min{d, B_{1}, …, B_{N}} → ∞; probably such a statement is also true but extremely difficult to prove.

For a scheme X of finite type over Z, define the zeta function [Ser65, §1.3]

ζ_{X}(s) := ∏_{closed P ∈X} (1 − #κ(P)^{−s})^{−1} ,

where κ(P) is the (finite) residue field of P. This generalizes the definition of Section 1, since a scheme of finite type over F_{q} can be viewed as a scheme of finite type over Z. On the other hand, ζ_{Spec Z}(s) is the Riemann zeta function.

The abc conjecture, formulated by D. Masser and J. Oesterlé in response to insights of R. C. Mason, L. Szpiro, and G. Frey, is the statement that for any ϵ > 0, there exists a constant C = C(ϵ) > 0 such that if a, b, c are coprime positive integers satisfying a + b = c,

then c < C( ∏_{p|abc} p)^{1+ϵ}.

For convenience, we say that a scheme X of finite type over Z is regular of dimension m if for every closed point P of X, the local ring O_{X,P} is regular of dimension m. For a scheme X of finite type over Z, this is equivalent to the condition that O_{X,P} be regular for all P ∈ X and all irreducible components of X have Krull dimension m. If X is smooth of relative dimension m − 1 over Spec Z, then X is regular of dimension m, but the converse need not hold. The empty scheme is regular of every dimension.

Theorem 5.1 (Bertini for arithmetic schemes). Assume the abc conjecture and Conjecture 5.2 below. Let X be a quasiprojective subscheme of P_{Z}^{n} that is regular of dimension m ≥ 0. Define

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

Then μ(P) = ζ_{X}(m + 1)^{−1}.

Remark . The case X = P_{Z}^{0} = Spec Z in P_{Z}^{0} of Theorem 5.1 is the statement that the density of squarefree integers is ζ(2)^{−1}, where ζ is the Riemann zeta function. The proof of Theorem 5.1 in general will involve questions about squarefree values of multivariable polynomials.

Given a scheme X, let X_{Q} = X × Q, and let X_{p} = X × F_{p} for each prime p.

Conjecture 5.2. Let X be an integral quasiprojective subscheme of P_{Z}^{n} that is smooth over Z of relative dimension r. There exists c > 0 such that if d and p are sufficiently large, then

\frac{#\{ f ∈ S_{d,p} : dim(H_{f} ∩ X_{p})_{sing} ≥ 1 \}}{#S_{d,p}} < \frac{c}{p^{2}} .

Heuristically one expects that Conjecture 5.2 is true even if c/p^{2} is replaced by c/p^{k} for any fixed k ≥ 2. On the other hand, for the application to Theorem 5.1, it would suffice to prove a weak form of Conjecture 5.2 with the upper bound c/p^{2} replaced by any ϵ_{p} > 0 such that ∑_{p} ϵ_{p} < ∞. We used c/p^{2} only to simplify the statement.

If d is sufficiently large relative to p, then Theorem 3.2 provides a suitable upper bound on the ratio in Conjecture 5.2. If p is sufficiently large relative to d, then one can derive a suitable upper bound from the Weil Conjectures. (In particular, the truth of Conjecture 5.2 is unchanged if we drop the assumption that d and p are sufficiently large.) The difficulty lies in the case where d is comparable to p.

See Section 5.4, for a proof of Conjecture 5.2 in the case where the closure of X_{Q} in P_{Q}^{n} has at most isolated singularities.

5.1. Singular points with small residue field. We begin the proof of Theorem 5.1 with analogues of results in Section 2.1. If M is a finite abelian group, let length_{Z}M be its length as a Z-module.

Lemma 5.3. If Y is a zero-dimensional closed subscheme of P_{Z}^{n}, then the map ϕ_{d} : S_{d} = H^{0}(P_{Z}^{n}, O(d)) → H^{0}(Y, O_{Y} (d)) is surjective for d ≥ length_{Z}H^{0}(Y, O_{Y}) − 1.

Proof. Assume d ≥ length_{Z}H^{0}(Y, O_{Y}) − 1. The cokernel C of ϕ_{d} is finite, since it is a quotient of the finite group H^{0}(Y, O_{Y} (d)). Moreover, C has trivial p-torsion for each prime p, by Lemma 2.1 applied to Y_{F_{p}} in P_{F_{p}}^{n}. Thus C = 0. Hence ϕ_{d} is surjective. □

Lemma 5.4. If P ⊆ Z^{N} is a union of c distinct cosets of a subgroup G ⊆ Z^{N} of index m, then μ(P) = c/m.

Proof. Without loss of generality, we may replace G with its subgroup (mZ)^{N} of finite index. The result follows, since any of the boxes in the definition of μ can be approximated by a box of dimensions that are multiples of m, with an error that becomes negligible compared with the number of lattice points in the box as the box dimensions tend to infinity. □

If X is a scheme of finite type over Z, define X_{<r} as the set of closed points P with #κ(P) < r. (This conflicts with the corresponding definition before Lemma 2.2; forget that one.) Define X_{≥r} similarly. We say that X is regular of dimension m at a closed point P of P_{Z}^{n} if either P ∉ X or O_{X,P} is a regular local ring of dimension m.

Lemma 5.5 (Small singularities). Let X be a quasiprojective subscheme of P_{Z}^{n} that is regular of dimension m ≥ 0. Define

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

Then

μ(P_{r}) = ∏_{P ∈X_{<r}} (1 − #κ(P)^{−(m+1)}) .

Proof. Given Lemmas 5.3 and 5.4, the proof is the same as that of Lemma 2.2 with Z = ∅. □

5.2. Reductions. Theorem 1 of [Ser65] shows that ∏_{P ∈X_{<r}} (1 − #κ(P)^{−(m+1)}) converges to

ζ_{X}(m+1)^{−1} as r → ∞. Thus Theorem 5.1 follows from Lemma 5.5 and the following, whose proof will occupy the rest of Section 5.

Lemma 5.6 (Large singularities). Assume the abc conjecture and Conjecture 5.2. Let X be a quasiprojective subscheme of P_{Z}^{n} that is regular of dimension m ≥ 0. Define

Q_{r}^{large} := { f ∈ S_{homog} : there exists P ∈ X_{≥r} such that

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

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

Lemma 5.6 holds for X if it holds for each subscheme in an open cover of X, since by quasicompactness any such open cover has a finite subcover. In particular, we may assume that X is connected. Since X is also regular, X is integral. If the image of X → Spec Z is a closed point (p), then X is smooth of dimension m over F_{p}, and Lemma 5.6 for X follows from Lemmas 2.4 and 2.6. Thus from now on, we assume that X dominates Spec Z.

Since X is regular, its generic fiber X_{Q} is regular. Since Q is a perfect field, it follows that X_{Q} is smooth over Q, of dimension m − 1. By [EGA IV(4), 17.7.11(iii)], there exists an integer t ≥ 1 such that X × Z[1/t] is smooth of relative dimension m − 1 over Z[1/t].

5.3. Singular points of small residue characteristic.

Lemma 5.7 (Singularities of small characteristic). Fix a nonzero prime p ∈ Z. Let X be an integral quasiprojective subscheme of P_{Z}^{n} that dominates Spec Z and is regular of dimension m ≥ 0. Define

Q_{p,r} := { f ∈ S_{homog} : there exists P ∈ X_{p} with #κ(P) ≥ r

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

Then lim_{r→∞} \overline{μ}(Q_{p,r}) = 0.

Proof. We may assume that X_{p} is nonempty. Then, since X_{p} is cut out in X by a single equation p = 0, and since p is neither a unit nor a zerodivisor in H^{0}(X, O_{X}), dim X_{p} = m−1.

Let

Q_{p,r}^{medium} := ⋃_{d≥0}{ f ∈ S_{d} : there exists P ∈ X_{p} with r ≤ #κ(P) ≤ p^{d/(m+1)}

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

and

Q_{p}^{high} := ⋃_{d≥0}{ f ∈ S_{d} : there exists P ∈ X_{p} with #κ(P) > p^{d/(m+1)}

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

Since Q_{p,r} = Q_{p,r}^{medium} ∪ Q_{p}^{high}, it suffices to prove lim_{r→∞} \overline{μ}(Q_{p,r}^{medium}) = 0 and \overline{μ}(Q_{p}^{high}) = 0. We will adapt the proofs of Lemmas 2.4 and 2.6.

If P is a closed point of X, let 𝔪_{X,P} ⊆ O_{X} be the ideal sheaf corresponding to P, and let Y_{P} be the closed subscheme of X corresponding to the ideal sheaf 𝔪_{X,P}^{2} . For fixed d, the set Q_{p,r}^{medium} ∩ S_{d} is contained in the union over P with r ≤ #κ(P) ≤ p^{d/(m+1)} of the kernel of the restriction ϕ_{P} : S_{d} → H^{0}(Y_{P} , O(d)). Since H^{0}(Y_{P} , O(d)) ≃ H^{0}(Y_{P} , O_{Y_{P}}) has length (m+1)[κ(P) : F_{p}] ≤ d as a Z-module, ϕ_{P} is surjective by Lemma 5.3, and Lemma 5.4 implies \overline{μ}(ker ϕ_{P}) = #κ(P)^{−(m+1)}. Thus

\overline{μ}(Q_{p,r}^{medium} ∩ S_{d}) ≤ ∑_{P} \overline{μ}(ker ϕ_{P}) = ∑_{P} #κ(P)^{−(m+1)}.

where the sum is over P ∈ X_{p} with r ≤ #κ(P) ≤ p^{d/(m+1)}. The crude form #X_{p}(F_{p^{e}}) = O(p^{e(m−1)}) of the bound in [LW54] implies that

lim_{r→∞} \overline{μ}(Q_{p,r}^{medium}) = lim_{r→∞} lim_{d→∞} \overline{μ}(Q_{p,r}^{medium} ∩ S_{d}) = 0.

Next we turn to Q_{p}^{high}. Since we are free to pass to an open cover of X, we may assume that X is contained in the subset A_{Z}^{n} := {x_{0} ≠ 0} of P_{Z}^{n}. Let A = Z[x_{1}, …, x_{n}] be the ring of regular functions on A_{Z}^{n}. Identify S_{d} with the set of dehomogenizations A_{≤d} = { f ∈ A : deg f ≤ d }, where deg f denotes total degree.

Let Ω be the sheaf of differentials Ω_{X_{p}/F_{p}}. For P ∈ X_{p}, define the dimension of the fiber

ϕ(P) = dim_{κ(P)} Ω ⊗_{O_{Xp}} κ(P).

Let 𝔪_{X_{p},P} be the maximal ideal of the local ring O_{X_{p},P}. If P is a closed point of X_{p}, the isomorphism

Ω ⊗_{O_{Xp}} κ(P) ≃ \frac{𝔪_{X_{p},P}}{𝔪_{X_{p},P}^{2}}

of Proposition II.8.7 of [Har77] shows that ϕ(P) = dim_{κ(P)} 𝔪_{X_{p},P} /𝔪_{X_{p},P}^{2} ; moreover

pO_{X,P} → \frac{𝔪_{X,P}}{𝔪_{X,P}^{2}} → \frac{𝔪_{X_{p},P}}{𝔪_{X_{p},P}^{2}} → 0

is exact. Since X is regular of dimension m, the middle term is a κ(P)-vector space of dimension m. But the module on the left is generated by one element. Hence ϕ(P) equals m − 1 or m at each closed point P.

Let Y = { P ∈ X_{p} : ϕ(P) ≥ m }. By Exercise II.5.8(a) of [Har77], Y is a closed subset, and we give Y the structure of a reduced subscheme of X_{p}. Let U = X_{p} −Y. Thus for closed points P ∈ X_{p},

ϕ(P) = \begin{cases}m − 1, & if P ∈ U \\ m, & if P ∈ Y. \end{cases}

If U is nonempty, then dim U = dim X_{p} = m − 1, so U is smooth of dimension m − 1 over F_{p}, and Ω|_{U} is locally free. At a closed point P ∈ U, we can find t_{1}, …, t_{n} ∈ A such that dt_{1}, …, dt_{m−1} represent an O_{X_{p},P} -basis for the stalk Ω_{P}, and dt_{m}, …, dt_{n} represent a basis for the kernel of Ω_{A^{n}/F_{p}} ⊗ O_{X_{p},P} → Ω_{P}. Let ∂_{1}, …, ∂_{n} ∈ T_{A^{n}/F_{p},P} be the basis of derivations dual to dt_{1}, …, dt_{n}. Choose s ∈ A nonvanishing at P such that s∂_{i} extends to a global derivation D_{i} : A → A for i = 1, 2, …, m − 1. In some neighborhood V of P in A_{F_{p}}^{n}, dt_{1}, …, dt_{n} form a basis of Ω_{V/F_{p}}, and dt_{1}, …, dt_{m−1} form a basis of Ω_{U∩V/F_{p}}, and s ∈ O(V)^{∗}. By compactness, we may pass to an open cover of X to assume U ⊆ V. If H_{f} ∩ X is not regular at a closed point Q ∈ U, then the image of f in 𝔪_{U,Q}/𝔪_{U,Q}^{2} must be zero, and it follows that D_{1}f, …, D_{m−1}f, f all vanish at Q. The set of f ∈ S_{d} such that there exists such a point in U can be bounded using the induction argument in the proof of Lemma 2.6.

It remains to bound the f ∈ S_{d} such that H_{f} ∩ X is not regular at some closed point P ∈ Y. Since Y is reduced, and since the fibers of the coherent sheaf Ω ⊗ O_{Y} on Y all have dimension m, the sheaf is locally free by Exercise II.5.8(c) in [Har77]. By the same argument as in the preceding paragraph, we can pass to an open cover of X, and find t_{1}, …, t_{n}, s ∈ A such that dt_{1}, …, dt_{n} are a basis of the restriction of Ω_{A^{n}/F_{p}} to a neighborhood of Y in A_{F_{p}}^{n}, and dt_{1}, …, dt_{m} are an O_{Y} -basis of Ω ⊗ O_{Y}, and s ∈ O(Y)^{∗} is such that if ∂_{1}, …, ∂_{n} is the dual basis to dt_{1}, …, dt_{n}, then s∂_{i} extends to a derivation D_{i} : A → A for i = 1, …, m − 1. (We could also define D_{i} for i = m, but we already have enough.) We finish again by using the induction argument in the proof of Lemma 2.6. □

5.4. Singular points of midsized residue characteristic. While examining points of larger residue characteristic, we may delete the fibers above small primes of Z. Hence in this section and the next, our lemmas will suppose that X is smooth over Z.

Lemma 5.8 (Singularities of midsized characteristic). Assume Conjecture 5.2. Let X be an integral quasiprojective subscheme of A_{Z}^{n} that dominates Spec Z and is smooth over Z of relative dimension m − 1. For d, L, M ≥ 1, define

Q_{d,L<·<M} := { f ∈ S_{d} : there exist p satisfying L < p < M and P ∈ X_{p} such that

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

Given ϵ > 0, if d and L are sufficiently large, then \overline{μ}(Q_{d,L<·<M}) < ϵ.

Proof. If P is a closed point of degree at most d/(m + 1) over F_{p} where L < p < M, then the set of f ∈ S_{d} such that H_{f} ∩ X is not regular of dimension m − 1 at P has upper density #κ(P)^{−(m+1)}, as in the argument for Q_{p,r}^{medium} in Lemma 5.7. The sum over #κ(P)^{−(m+1)} over all such P is small if L is sufficiently large: this follows from [LW54], as usual. By Conjecture 5.2, the upper density of the set of f ∈ S_{d} such that there exists p with L < p < M such that dim(H_{f} ∩ X_{p})_{sing} ≥ 1 is bounded by ∑_{L<p<M} c/p^{2}, which again is small if L is sufficiently large.

Let E_{d,p} be the set of f ∈ S_{d} for which (H_{f} ∩ X_{p})_{sing} is finite and H_{f} ∩ X fails to be regular of dimension m − 1 at some closed point P ∈ X_{p} of degree greater than d/(m + 1) over F_{p}. It remains to show that if d and L are sufficiently large, ∑_{L<p<M} \overline{μ}(E_{d,p}) is small. Write f = f_{0} + pf_{1} where f_{0} has coefficients in {0, 1, …, p − 1}. Once f_{0} is fixed, (H_{f} ∩ X_{p})_{sing} is determined, and in the case where it is finite, we let P_{1}, …, P_{ℓ} be its closed points of degree greater than d/(m + 1) over F_{p}. Now H_{f} ∩ X is not regular of dimension m − 1 at P_{i} if and only if the image of f in O_{X,P_{i}}/𝔪_{X,P_{i}}^{2} is zero; for fixed f_{0}, this is a condition only on the image of f_{1} in O_{X_{p},P_{i}}/𝔪_{X_{p},P_{i}}. It follows from Lemma 2.5 that the fraction of f_{1} for this holds is at most p^{−ν} where ν = d/(m+1). Thus \overline{μ}(E_{d,p}) ≤ ℓp^{−ν}. As usual, we may assume we have reduced to the case where (H_{f} ∩X_{p})_{sing} is cut out by D_{1}f, …, D_{m−1}f, f for some derivations D_{i}, and hence by Bézout’s Theorem, ℓ = O(d^{m}) = O(p^{ν−2}) as d → ∞, so \overline{μ}(E_{d,p}) = O(p^{−2}). Hence ∑_{L<p<M} \overline{μ}(E_{d,p}) is small whenever d and L are large. □

The following lemma and its proof were suggested by the referee.

Lemma 5.9. Conjecture 5.2 holds when the closure \overline{X}_{Q} of X_{Q} in P_{Q}^{n} has at most isolated singularities.

Proof. We use induction on n. Let \overline{X} be the closure of X in P_{Z}^{n}. Since \overline{X}_{Q} has at most isolated singularities, a linear change of coordinates over Q makes \overline{X}_{Q} ∩ \{x_{0} = 0\} is smooth of dimension r − 1. Since the statement of Conjecture 5.2 is unchanged by deleting fibers of X → Spec Z above small primes, we may assume that \overline{X} ∩ \{x_{0} = 0\} is smooth over Z of relative dimension r − 1. Next, we may enlarge X to assume that X is the smooth locus of \overline{X} → Spec Z, since this only makes the desired conclusion harder to prove. The smooth Z-scheme \overline{X} ∩ \{x_{0} = 0\} is contained in the smooth locus X of the Z-scheme \overline{X}, so X ∩ \{x_{0} = 0\} = \overline{X} ∩ \{x_{0} = 0\}.

If f ∈ S_{d,p} is such that dim(H_{f} ∩ X_{p})_{sing} ≥ 1, then the closure of (H_{f} ∩ X_{p})_{sing} intersects \{x_{0} = 0\}. But X ∩ \{x_{0} = 0\} = \overline{X} ∩ \{x_{0} = 0\}, so (H_{f} ∩ X_{p})_{sing} itself intersects \{x_{0} = 0\}. Thus it suffices to prove

\frac{#\{f ∈ S_{d,p} : (H_{f} ∩ X_{p})_{sing} ∩ \{x_{0} = 0\} ≠ ∅\}}{#S_{d,p}} < \frac{c}{p^{2}} .

For a closed point y of degree ≤ d/(r + 1) of X_{p} ∩ {x_{0} = 0}, the probability that y ∈ (H_{f} ∩ X_{p})_{sing} is #κ(y)^{−r−1} and the sum over such points is treated as in the proof of Lemma 5.8.

It remains to count f ∈ S_{d,p} such that H_{f} ∩ X_{p} is singular at a closed point y of degree > d/(r+1) of X_{p}∩{x_{0} = 0}. Note that (H_{f} ∩X_{p})_{sing}∩{x_{0} = 0} is contained in the subscheme Σ_{f} := (H_{f} ∩ X_{p} ∩ {x_{0} = 0})_{sing}. By the inductive hypothesis applied to X ∩ {x_{0} = 0}, we may restrict the count to the f for which H_{f} ∩ X_{p} ∩ {x_{0} = 0} is of pure dimension r − 2 and Σ_{f} is finite. Then by Bézout, #Σ_{f} = O(d^{r}), where the implied constant depends only on X. If we write f = f_{0} + f_{1}x_{0} + f_{2}x_{0}^{2} + … with f_{i} ∈ F_{p}[x_{1}, …, x_{n}], then Σ_{f} depends only on f_{0}. For fixed f_{0} and y ∈ Σ_{f} = Σ_{f_{0}}, whether or not y ∈ (H_{f} ∩X_{p})_{sing} depends only on the “value” of f_{1} at y (which is in Γ(y, O(d − 1)|_{y})), and at most one value corresponds to a singularity. The F_{p}-vector space of possible values of f_{1} at y has dimension ≥ min(deg(y), d), so if we restrict to y of degree > d/(r + 1), the probability that y ∈ (H_{f} ∩ X_{p})_{sing} is at most p^{−d/(r+1)}. Thus, for fixed f_{0}, the probability that H_{f} ∩ X_{p} is singular at some such y is O(d^{r}p^{−d/(r+1)}), which is O(p^{−2}) for d large enough. Finally, the implied constant is independent of f_{0}, so the overall probability is again O(p^{−2}). □

5.5. Singular points of large residue characteristic. We continue to identify homogeneous polynomials in x_{0}, …, x_{n} with their dehomogenizations obtained by setting x_{0}, when needed to consider them as functions on A_{Z}^{n} ⊆ P_{Z}^{n}.

Lemma 5.10. Let X be an integral quasiprojective subscheme of A_{Z}^{n} that dominates Spec Z and is smooth over Z of relative dimension m−1. Fix d ≥ 1. Let f ∈ Z[c_{0}, …, c_{N}][x_{0}, …, x_{n}] be the generic homogeneous polynomial in x_{0}, …, x_{n} of total degree d, having the indeterminates c_{0}, …, c_{N} as coefficients (so N + 1 is the number of homogeneous monomials in x_{0}, …, x_{n} of total degree d). Then there exists an integer M > 0 and a squarefree polynomial R(c_{0}, …, c_{N}) ∈ Z[c_{0}, …, c_{N}] such that if f̄ is obtained from f by specializing the coefficients c_{i} to integers γ_{i}, and if Hf̄ ∩ X fails to be regular at a closed point in the fiber X_{p} for some prime p ≥ M, then p^{2} divides the value R(γ_{0}, …, γ_{N}).

Proof. By using a “d-uple embedding” of X (i.e., mapping A^{n} to A^{N} using all homogeneous monomials in x_{0}, …, x_{n} of total degree d), we reduce to the case of intersecting X instead with an affine hyperplane H_{f} ⊂ A_{Z}^{n} defined by (the dehomogenization of) f = c_{0}x_{0} + ⋯ + c_{n}x_{n}. Let A^{n+1} = A_{Z}^{n+1} be the affine space whose points correspond to such homogeneous linear forms. Thus c_{0}, …, c_{n} are the coordinates on A^{n+1}.

If X has relative dimension n over Spec Z (so X is a nonempty open subset of A^{n}), we may trivially take R = c_{0} if n = 0 and R = c_{0}c_{1} if n > 0. Therefore we assume that the relative dimension is strictly less than n in what follows.

Let Σ ⊆ X × A^{n+1} be the reduced closed subscheme of points (x, f) such that the variety H_{f} ∩ X over the residue field of (x, f) is not smooth of dimension m − 2 at x. Then, because we have excluded the degenerate case of the previous paragraph, Σ_{Q} is the closure in X_{Q} × A_{Q}^{n+1} of the inverse image under X_{Q} × A_{Q}^{n+1} −−→ X_{Q} × P_{Q}^{n} of the conormal variety CX ⊆ X_{Q} × P_{Q}^{n} as defined in [Kle86, I-2] (under slightly different hypotheses). Concretely, Σ is the subscheme of X × A^{n+1} locally cut out by the equations D_{1}f = ⋯ D_{m−1}f = f = 0 where the D_{i} are defined locally on X as in the penultimate paragraph of the proof of Lemma 5.7.

Let I be the scheme-theoretic image of Σ under the projection π : Σ → A^{n+1}. Thus I_{Q} ⊆ A_{Q}^{n+1} is the cone over the dual variety X̌, defined as the scheme-theoretic image of the corresponding projection CX → P_{Q}^{n}. By [Kle86, p. 168], we have dim CX = n − 1, so dim X̌ ≤ n − 1.

Case 1. dim X̌ = n − 1.

Then I_{Q} is an integral hypersurface in A_{Q}^{n}, say given by the equation R_{0}(c_{0}, …, c_{n}) = 0, where R_{0} is an irreducible polynomial with content 1. After inverting a finite number of nonzero primes of Z, we may assume that R_{0} = 0 is also the equation defining I in A_{Z}^{n}. Choose M greater than all the inverted primes.

Since dim X̌ = n − 1, the projection CX̌ → X̌ is a birational morphism. By duality (see the Monge-Segre-Wallace criterion on p. 169 of [Kle86]), CX = CX̌, so CX → X̌ is a birational map. It follows that π : Σ → I is a birational morphism. Thus we may choose an open dense subset I′ of I such that the birational morphism π : Σ → I induces an isomorphism Σ′ → I′, where Σ′ = π^{−1}(I′). By Hilbert’s Nullstellensatz, there exists R_{1} ∈ Z[c_{0}, …, c_{n}] such that R_{1} vanishes on the closed subset I − I′ but not on I. We may assume that R_{1} is squarefree. Define R = R_{0}R_{1}. Then R is squarefree.

Suppose that Hf̄ ∩ X fails to be regular at a point P ∈ X_{p} with p ≥ M. Let γ be the closed point of A^{n+1} defined by c_{0} − γ_{0} = ⋯ = c_{n} − γ_{n} = p = 0. Then the point (P, γ) of X × A^{n+1} is in Σ. Hence γ ∈ I, so R_{0}(γ_{0}, …, γ_{n}) is divisible by p. If γ ∈ I − I′, then R_{1}(γ_{0}, …, γ_{n}) is divisible by p as well, so R(γ_{0}, …, γ_{n}) is divisible by p^{2}, as desired.

Therefore we assume from now on that γ ∈ I′, so (P, γ) ∈ Σ′. Let W be the inverse image of I′ under the closed immersion Spec Z → A^{n+1} defined by the ideal (c_{0} − γ_{0}, …, c_{n} − γ_{n}). Let V be the inverse image of Σ′ under the morphism X ↪ X × A^{n+1} induced by the previous closed immersion. Thus we have a cube in which the top, bottom, front, and back faces are cartesian:

\xymatrix{V \ar[rr] \ar[dd] \ar[dr] &  & Σ′ \ar[dd] \ar[dr] \\  & X \ar[rr] \ar[dd] &  & X × A^{n+1} \ar[dd] \\ W \ar[rr] \ar[dr] &  & I′ \ar[dr] \\  & Spec Z \ar[rr] &  & A^{n+1}}

Near (P, γ) ∈ X × A^{n+1} the functions D_{1}f, ⋯, D_{m−1}f, f cut out Σ (and hence also its open subset Σ′) locally in X × A^{n+1}. Then O_{V,P} = O_{X,P} /(D_{1}f̄, D_{2}f̄, …, D_{m−1}f̄, f̄). By assumption, Hf̄ ∩ X is not regular at P, so f̄ maps to zero in O_{X,P} /𝔪_{X,P}^{2} . Now O_{X,P} is a regular local ring of dimension m, D_{i}f̄ ∈ 𝔪_{X,P}, and f̄ ∈ 𝔪_{X,P}^{2} , so the quotient O_{V,P} has length at least 2. Since Σ′ → I′ is an isomorphism, the cube shows that V → W is an isomorphism too. Hence the localization of W at p has length at least 2. On the other hand I′ is an open subscheme of I, whose ideal is generated by R_{0}(c_{0}, …, c_{n}) (after some primes were inverted), so W is an open subscheme of Z/(R_{0}(γ_{1}, …, γ_{N})). Thus R_{0}(γ_{0}, …, γ_{n}) is divisible by p^{2} at least. Thus R(γ_{0}, …, γ_{n}) is divisible by p^{2}.

Case 2. dim X̌ < n − 1.

Then I_{Q} is of codimension ≥ 2 in A_{Q}^{n+1}. Inverting finitely many primes if necessary, we can find a pair of distinct irreducible polynomials R_{1}, R_{2} ∈ Z[c_{0}, …, c_{n}] vanishing on I. Let R = R_{1}R_{2}. As in Case 1, if Hf̄ ∩ X fails to be regular at P ∈ X_{p} with p ≥ M, then the values of R_{1} and R_{2} both vanish modulo p, so the value of R is divisible by p^{2}. □

Because of Lemma 5.10, we would like to know that most values of a multivariable polynomial over Z are almost squarefree (that is, squarefree except for prime factors less than M). It is here that we need to assume the abc conjecture.

Theorem 5.11 (Almost squarefree values of polynomials). Assume the abc conjecture. Let F ∈ Z[x_{1}, …, x_{n}] be squarefree. For M > 0, define

S_{M} := { (a_{1}, …, a_{n}) ∈ Z^{n} | F (a_{1}, …, a_{n}) is divisible by p^{2} for some prime p ≥ M }.

Then \overline{μ}(S_{M}) → 0 as M → ∞.

Proof. The n = 1 case is in [Gra98]. The general case follows from Lemma 6.2 of [Poo03], in the same way that Corollary 3.3 there follows from Theorem 3.2 there. Lemma 6.2 there is proved there by reduction to the n = 1 case. □

Remarks.

(1) These results assume the abc conjecture, but the special case where F factors into one-variable polynomials of degree ≤ 3 is known unconditionally [Hoo67]. Other unconditional results are contained in [GM91].

(2) Theorem 5.11 together with a simple sieve lets one show that the naive heuristic (multiplying probabilities for each prime p) correctly predicts the density of (a_{1}, …, a_{n}) ∈ Z^{n} for which F (a_{1}, …, a_{n}) is squarefree, assuming the abc conjecture.

Lemma 5.12 (Singularities of large characteristic). Assume the abc conjecture. Let X be an integral quasiprojective subscheme of A_{Z}^{n} that dominates Spec Z and is smooth over Z of relative dimension m − 1. Define

Q_{d,≥M} := { f ∈ S_{d} : there exists p ≥ M and P ∈ X_{p} such that

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

If d is sufficiently large, then lim_{M→∞} \overline{μ}(Q_{d,≥M}) = 0.

Proof. We may assume that d is large enough for Lemma 5.10. Apply Theorem 5.11 to the squarefree polynomial R provided by Lemma 5.10 for X. □

5.6. End of proof. We are now ready to prove Theorem 5.1. Recall that in Section 5.2 we reduced to the problem of proving Lemma 5.6 in the case where X is an integral quasiprojective subscheme of A_{Z}^{n} such that X dominates Spec Z and is regular of dimension m ≥ 0. In Lemma 5.6, d tends to infinity for each fixed r, and then r tends to infinity. We choose L depending on r, and M depending on r and d, such that 1 ≪ L ≪ r ≪ d ≪ M. (The precise requirement implied by each ≪ is whatever is needed below for the applications of the lemmas below.) Then

(1)  Q_{r}^{large} ∩ S_{d} ⊆ ( ⋃_{p≤L}(Q_{p,r} ∩ S_{d})) ∪ Q_{d,L<·<M} ∪ Q_{d,≥M},

and we will bound the upper density of each term on the right. Recall from the end of Section 5.2 that X has a subscheme of the form X′ = X ×Spec Z[1/t] that is smooth over Z. We may assume L > t. By Lemma 5.7, lim_{r→∞} \overline{μ}(Q_{p,r}) = 0 for each p, so \overline{μ} (⋃_{p≤L} Q_{p,r}) is small (by which we mean tending to zero) if r sufficiently large relative to L. By Lemma 5.8 applied to X′, if L and d are sufficiently large, then \overline{μ}(Q_{d,L<·<M}) is small. By Lemma 5.12 applied to X′, if d is sufficiently large, and M is sufficiently large relative to d, then \overline{μ}(Q_{d,≥M}) is small. Thus by (1), \overline{μ}(Q_{r}^{large}) is small whenever r is large and d is sufficiently large relative to r. This completes the proof of Lemma 5.6 and hence of Theorem 5.1. □

Remark. Arithmetic analogues of Theorems 1.2 and 1.3, and of many of the applications in Section 3 can be proved as well.

5.7. Regular versus smooth. One might ask what happens in Theorem 5.1 if we ask for H_{f} ∩ X to be not only regular, but also smooth over Z. We now show unconditionally that this requirement is so strict, that at most a density zero subset of polynomials f satisfies it, even if the original scheme X is smooth over Z.

Theorem 5.13. Let X be a nonempty quasiprojective subscheme of P_{Z}^{n} that is smooth of relative dimension m ≥ 0 over Z. Define

P^{smooth} := { f ∈ S_{homog} : H_{f} ∩ X is smooth of relative dimension m − 1 over Z }.

Then μ(P^{smooth}) = 0.

Proof. Let

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

Suppose P ∈ X_{<r} lies above the prime (p) ∈ Spec Z. Let Y be the closed subscheme of X_{p} corresponding to the ideal sheaf 𝔪^{2} where 𝔪 is the ideal sheaf of functions on X_{p} vanishing at P. Then for f ∈ S_{d}, H_{f} ∩ X is smooth of relative dimension m − 1 over Z at P if and only if the image of f in H^{0}(Y, O(d)) is nonzero. Applying Lemma 5.3 to the union of such Y over all P ∈ X_{<r}, and using #H^{0}(Y, O(d)) = #κ(P)^{m+1}, we find

μ(P_{r}^{smooth}) = ∏_{P ∈X_{<r}} (1 − #κ(P)^{−(m+1)}) .

Since dim X = m+1, ζ_{X}(s) has a pole at s = m+1 and our product diverges to 0 as r → ∞. (See Theorems 1 and 3(a) in [Ser65].) But P^{smooth} ⊆ P_{r}^{smooth} for all r, so μ(P^{smooth}) = 0. □

A density zero subset of S_{homog} can still be nonempty or even infinite. For example, if X = Spec Z[1/2, x] ↪ P_{Z}^{1}, then P^{smooth} ∩ S_{d} is infinite for infinitely many d: H_{f} ∩ X is smooth over Z whenever f is the homogenization of (x − a)^{2^{b}} − 2 for some a, b ∈ Z with b ≥ 0.

On the other hand, N. Fakhruddin has given the following two examples in which P^{smooth}∩ S_{d} is empty for all d > 0.

Example 5.14. Let X be the image of the 4-uple embedding P_{Z}^{1} → P_{Z}^{4}. Then X is smooth over Z. If f ∈ P^{smooth} ∩ S_{d} for some d > 0, then H_{f} ∩ X ≃ ∐ Spec A_{i} where each A_{i} is the ring of integers of a number field K_{i} unramified above all finite primes of Z, such that ∑[K_{i} : Q] = 4d. The only absolutely unramified number field is Q, so each A_{i} is Z, and H_{f} ∩ X ≃ ∐_{i=1}^{4d} Spec Z. Then 4d = #(H_{f} ∩ X)(F_{2}) ≤ #X(F_{2}) = #P^{1}(F_{2}) = 3, a contradiction.

Example 5.15. Let X be the image of the 3-uple embedding P_{Z}^{2} → P_{Z}^{9}. Then X is smooth over Z. If f ∈ P^{smooth} ∩ S_{d} for some d > 0, then H_{f} ∩ X is isomorphic to a smooth proper geometrically connected curve in P_{Z}^{2} of degree 3d, hence of genus at least 1, so its Jacobian contradicts the main theorem of [Fon85].

Despite these counterexamples, P. Autissier has proved a positive result for a slightly different problem. An arithmetic variety of dimension m is an integral scheme X of dimension m that is projective and flat over Z, such that X_{Q} is regular (of dimension m − 1). If O_{K} is the ring of integers of a finite extension K of Q, then an arithmetic variety over O_{K} is an O_{K}-scheme X such that X is an arithmetic variety and whose generic fiber X_{K} is geometrically irreducible over K. The following is a part of Théorème 3.2.3 of [Aut01]:

Let X be an arithmetic variety over O_{K} of dimension m ≥ 3. Then there exists a finite extension L of K and a closed subscheme X′ of X_{O_{L}} such that (1) The subscheme X′ is an arithmetic variety over O_{L} of dimension m − 1.

(2) Whenever the fiber X_{𝔭} of X above 𝔭 ∈ Spec O_{K} is smooth, the fiber X′_{𝔭′} of X′ above 𝔭′ is smooth for all 𝔭′ ∈ Spec O_{L} lying above 𝔭.

Actually Autissier proves more, that one can also control the height of X′. (He uses the theory of heights developed by Bost, Gillet, and Soulé, generalizing Arakelov’s theory.)

The most significant difference between Autissier’s result and the phenomenon exhibited by Fakhruddin’s examples is the finite extension of the base allowed in the former.

Acknowledgements

I thank Ernie Croot for a conversation at an early stage of this research (in 2000), David Eisenbud for conversations about Section 5.5, and Matt Baker, Brian Conrad, and Najmuddin Fakhruddin for some other comments. I thank Pascal Autissier and Ofer Gabber for sharing their (now published) preprints with me, and Jean-Pierre Jouanolou for sharing some unpublished notes about resultants and discriminants. Finally, I thank the referee for an extremely helpful report, which included among other things the statement and proof of Lemma 5.9.

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

  E-mail address: poonen@math.berkeley.edu