Proceedings of Symposia in Pure Mathematics

Weak approximation for hypersurfaces of low degree

Brendan Hassett and Yuri Tschinkel

Abstract. We explore the arithmetic of rationally connected varieties over function fields of curves. The main technical issues revolve around the existence of free rational curves with prescribed geometric properties.

Contents

Introduction 1 1. Approximation for hypersurfaces of low degree 2 2. Ordinary singularities reviewed 5 3. Managing free curves 7 4. Weak approximation at places with ordinary double points 9 5. Applications to cubic hypersurfaces 16 Appendix: Recent developments for hypersurfaces of low degree 17 References 18

Introduction

In this paper, we survey arithmetic questions over function fields of curves which have attracted considerable attention recently. The starting point is the theorem of Graber, Harris, and Starr [GHS03]: Every rationally-connected variety over the function field of a curve has a rational point. Once we have one rational point, it is natural to analyze the structure of the set of all rational points, e.g., density in the Zariski topology (or other topologies), weak approximation, etc.

The first result in this direction actually predates the work of Graber, Harris, and Starr. Let π : X → B be a flat proper morphism to a smooth curve, with generic fiber smooth and rationally connected. Kollár, Miyaoka, and Mori proved that the existence of one section guarantees a Zariski-dense collection of sections. More precisely, fix points b_{1}, …, b_{m} ∈ B of good reduction for π, i.e., with X_{b_{i}} = π^{−1}(b_{i}) smooth. Then for any points x_{i} ∈ X_{b_{i}} there exists a section σ : B → X with

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

2000 Mathematics Subject Classification. 14M20,14D06.

The first author is partially supported by the Sloan Foundation and NSF grant 0134259.

σ(b_{i}) = x_{i} for each i. In [HT06], we proved the existence of sections with prescribed jets at these points, i.e., weak approximation at places of good reduction.

Full weak approximation requires the analysis of singular fibers of π. Precisely, weak approximation holds if for any smooth model X → B and points x_{i} in the smooth locus X_{b_{i}}^{sm}, there exists a section σ : B → X with σ(b_{i}) = x_{i} for each i. Reaching smooth points of arbitrary fibers is much harder than reaching arbitrary points of smooth fibers.

Even the case of cubic surfaces is open at present. In [HT08], we analyzed fibrations π : X → B with smooth total space such that the fibers satisfy strong rational-connectedness, i.e., for each b ∈ B, any two points in X_{b}^{sm} can be joined by a proper rational curve contained in X_{b}^{sm}. This suffices to prove weak approximation for generic cubic surfaces over function fields of curves, e.g., those with square-free discriminant.

We now summarize the principal new results of this paper. Theorem 1 establishes full weak approximation for smooth hypersurfaces of very low degree, e.g., for cubic fivefolds. Theorem 18 addresses fibrations π : X → B with ordinary singularities and singular total space. Under suitable technical assumptions (cf. Hypothesis 14), we establish weak approximation. We give applications to cubic hypersurfaces in Theorem 25.

Our goal here is to highlight new techniques for ‘managing’ rational curves: Deformation theory, constructions of combs, specialization arguments, and producing curves with prescribed intersection properties. One overarching question is how to produce ‘good curves’ (e.g., free rational curves or sections of fibrations) in a given homology class. For instance, given an irreducible component E ⊂ X_{b} with multiplicity one, is there a section σ : B → X with σ(b) ∈ E? The section given by the theorem of Graber-Harris-Starr might pass through a different component of X_{b}. How can we induce sections to ‘jump’ from component to component?

Throughout, we work over an algebraically closed ground field K.

Acknowledgments: Jason Starr suggested analyzing weak approximation for hypersurfaces of small degree over function fields. We benefitted from discussions with János Kollár about degenerate fibers of rationally-connected fibrations. We are grateful to Jean-Louis Colliot-Thélène and Trevor Wooley for pointing out references to relevant number-theoretic results.

1. Approximation for hypersurfaces of low degree

Assume K is of characteristic zero.

Theorem 1. There exists a function ϕ : ℕ → ℕ with the following property: Consider a smooth curve B with function field F and a smooth hypersurface X ⊂ ℙ^{n} of degree d over F. If n ≥ ϕ(d) then X satisfies weak approximation over F.

We give a recursive definition for ϕ(d). Let ϕ(1) = 1 and define ϕ(d), d > 1 by the formula

ϕ(d) = \binom{ϕ(d − 1) + d − 1}{ϕ(d − 1)}.

In particular, we have

\begin{matrix}d & 1 & 2 & 3 & 4 & 5 \\ ϕ(d) & 1 & 2 & 6 & 84 & 2331890 \end{matrix} .

Remark 2. The theorem is weaker than one might expect, in light of the strong results obtained over number fields using the circle method [Ski97], [Bir62]: Let F be a number field and X ⊂ ℙ^{n} a smooth hypersurface over F of degree d. Assume n ≥ 2^{d}(d − 1) and for each place ν of F we have X(F_{ν}) ≠ ∅. Then X satisfies weak approximation over F.

By way of comparison

\begin{matrix}d & 1 & 2 & 3 & 4 & 5 \\ 2^{d}(d − 1) & 0 & 4 & 16 & 48 & 128 \end{matrix} ,

which is much smaller than ϕ(d) for d ≫ 0.

Proof. We shall require some basic results on varieties of linear subspaces:

Lemma 3 (cf. 23.3 of [PS92]). Let X ⊂ ℙ^{n} be a hypersurface of degree d over F. Consider the ℓ-dimensional projective subspaces contained in X

Π_{ℓ,X} = {[Λ] ∈ 𝔾(ℓ, n) : Λ ⊂ X},

realized as a subscheme of the Grassmannian. Suppose that its expected dimension

expdim Π_{ℓ,X} := (n − ℓ)(ℓ + 1) − \binom{d + ℓ}{d} > 0.

Then Π_{ℓ,X} ≠ ∅ and thus

dim Π_{ℓ,X} ≥ expdim Π_{ℓ,X}.

Consider the incidence correspondence

I = {(Λ, X) : Λ ⊂ X} ⊂ 𝔾(ℓ, n) × Hilb

where Hilb ≃ ℙ^{\binom{n+d}{d}−1} is the Hilbert scheme of hypersurfaces of degree d in ℙ^{n}. Since I is a projective bundle over the first factor, it is smooth and irreducible. By Bertini’s Theorem, the generic fiber of the second projection is also smooth, i.e., Π_{ℓ,X} is smooth of the expected dimension for generic X. Moreover, the fibers of the projection onto Hilb have the expected dimension over a subset with complement of codimension ≥ 2.

Lemma 4. Let X ⊂ ℙ^{n} be a hypersurface of degree d over F, with n ≥ ϕ(d). Then

Π_{ℓ,X}(F) ≠ ∅

if ℓ < ϕ(d − 1).

We first consider the special case where Π = Π_{ℓ,X} is smooth of the expected dimension. If S → 𝔾(ℓ, n) is the universal subbundle of rank ℓ + 1 then Π_{ℓ,X} is given by the vanishing of section of Sym^{d}S^{∗}, which has rank \binom{d+ℓ}{d}. The adjunction formula gives

K_{Π} = (K_{𝔾(ℓ,n)} + c_{1}(Sym^{d}S^{∗}))|_{Π} = O_{Π}(−n − 1 + \binom{d + ℓ}{d − 1}),

which is negative provided ℓ < ϕ(d − 1) and n ≥ ϕ(d). Thus Π is Fano, in the sense that its anticanonical divisor is ample. It is therefore rationally connected, and Π(F) ≠ ∅ by [GHS03].

To complete the lemma, we require a well-known specialization result (see §2 of [GHMS05]):

Sublemma 5. Let T = Spec K[[t]] with closed point 0 and geometric generic point τ. Let C → T be a flat proper morphism with geometric generic fiber C_{τ} a smooth connected curve. Let X → C be a flat proper morphism so that X ×_{C} C_{τ} → C_{τ} admits a section. Then for each normal irreducible component C′_{0} ⊂ C_{0}, the induced X ×_{C} C′_{0} → C′_{0} admits a section.

Here is the idea: The section degenerates to a stable map over C_{0} of degree one, which yields a section over each normal component.

Let X be an arbitrary hypersurface over F = K(B) with classifying morphism μ : B → Hilb. (For simplicity, assume now that B is projective.) Let (ρ_{s})_{s∈𝔾_{m}} ⊂ GL\binom{n+d}{d} be a one-parameter subgroup acting without fixed points on B, so that for generic s:

• ρ_{s}(B) is contained in the locus in Hilb parametrizing hypersurfaces Y with Π_{ℓ,Y} of the expected dimension;

• ρ_{s}(B) meets the locus over which Π_{ℓ,Y} is smooth.

Let T be the completion of 𝔾_{m} at the identity and consider the restriction

W := I ×_{Hilb} (B × T) → B × T.

This might not be flat, but there is a flattening blow-up β : C → B × T so that the dominating component of Z ⊂ W ×_{B×T} C is flat over C. We may take C to be nonsingular and β to be an isomorphism over the generic point of T; in particular, C_{0} has a component C′_{0} ≃ B. Combining Sublemma 5 with our result in the case of smooth Fano varieties, we obtain a section of Z ×_{C} B → B. However, the natural morphism over B

Z ×_{C} B → I ×_{Hilb} B

induces a section of I ×_{Hilb} B → B. (Recall we are constructing the fiber product using the projection to Hilb and the classifying morphism μ.) Restricting to the generic point of B, we obtain a rational point in X(F).

The remainder of the proof is an induction on d. We now use the fact that X is smooth. Let Λ ⊂ X be the linear subspace produced in the last lemma. Projecting from Λ gives a morphism

π_{Λ} : Bl_{Λ}X → ℙ^{n−ℓ−1},

with fibers hypersurfaces of degree d − 1 in ℙ^{ℓ+1}; the generic such fiber is smooth. The inductive hypothesis and our assumption that ℓ < ϕ(d − 1) guarantee these fibers satisfy weak approximation. Proposition 1.2 of [CTG04] implies weak approximation holds for X. Its hypothesis demands smoothness of the fibers, but weak approximation is a birational property, so we might as well restrict to the dense open set where π_{Λ} is smooth. □

Remark 6. It should be possible to weaken the smoothness hypothesis: Varieties having log terminal singularities and ample anticanonical class are rationally connected [Zha06].

Remark 7. There are similar inductive proofs of the unirationality of small degree hypersurfaces over algebraically-closed fields [PS92] [HMP98]. The growth of n with respect to d is also an iterated exponential function.

2. Ordinary singularities reviewed

We assume the ground field has characteristic different from two.

Definition 8. A variety V has ordinary singularities if it has a finite number of isolated singularities, each étale-locally isomorphic to a cone over a smooth quadric hypersurface.

Proposition 9. Let T be the spectrum of a complete DVR over the ground field with closed point 0 and geometric generic point τ. Consider V → T with V_{τ} smooth and V_{0} having ordinary singularities. In a formal (or étale) neighborhood of each singular point v ∈ V_{0}, V → T takes the form

x_{1}^{2} + … + x_{n}^{2} = t^{r},

where dim V_{0} = n − 1 and r is a positive integer.

Proof. Over a field of characteristic ≠ 2, each ordinary singularity v ∈ V_{0} is formally-locally isomorphic to

0 ∈ {(x_{1}, …, x_{n}) : x_{1}^{2} + … + x_{n}^{2} = 0} ⊂ 𝔸^{n}.

We use standard properties of the versal deformation space Ver(V_{0}, v): It is smooth of dimension one and the corresponding deformation may be written explicitly

s = y_{1}^{2} + … + y_{n}^{2}.

The local structure of the deformation V → T near v is determined by the classifying morphism

T → Ver(V_{0}, v),

or the induced homomorphism of K-algebras

K[[s]] → K[[t]]
s ↦ c_{r}t^{r} + c_{r+1}t^{r+1} + … c_{r} ≠ 0.

The characteristic is different from two, so we can solve

f(t)^{2} = c_{r} + c_{r+1}t + …

for f(t) ∈ K[[t]]. Substituting y_{i} = f(t)x_{i} for i = 1, …, n gives the desired normal form. □

Remark 10. For later use, we describe what happens when the classifying morphism

T → Ver(V_{0}, v)

is constant. Here, the generic fiber V_{τ} remains singular, which is incompatible with the assumptions of Proposition 9. In this situation, V → T admits an equisingular section σ : T → V with σ(0) = v, i.e., a section along which the fibers all have ordinary double points. In local coordinates, V can be written

x_{1}^{2} + … + x_{n}^{2} = 0

with the section σ equal to the origin. Blowing up along σ gives a simultaneous resolution of the fibers.

Figure 1. Degenerate fiber of the resolution when r = 7

Retain the notation of Proposition 9 and let (V, v) denote the localization of V at v. We describe a resolution of its singularities. If r = 1 then no resolution is necessary. When r > 1, consider the sequence of blow-ups

β : W(⌊r/2⌋) → W(⌊(r − 2)/2⌋) → … → W(1) → W(0) := (V, v),

where each W(j) has one singularity w(j) locally isomorphic to

x_{1}^{2} + … + x_{n}^{2} = t^{r−2j}

and

W(j + 1) = Bl_{w(j)}W(j).

The exceptional locus

β^{−1}(v) = E_{1} ∪_{D_{12}} E_{2} ⋯ E_{j} ∪_{D_{j,j+1}} E_{j+1} ⋯ E_{⌊r/2⌋},

where each E_{j} is the proper transform of the exceptional divisor of W(j) → W(j−1) and

(2.1)  E_{j} ≃ \begin{cases}ℙ(O_{Q_{n−1}}(−1) ⊕ O_{Q_{n−1}}) & if j < ⌊r/2⌋, \\ Cone Q_{n−1} & if j = (r − 1)/2, \\ Q_{n} & if j = r/2, \end{cases}

where Q_{m} ⊂ ℙ^{m} is a smooth quadric hypersurface of dimension m − 1. The components E_{j} and E_{j+1} meet along D_{j,j+1} ≃ Q_{n−1}, embedded in E_{j} as the negative section and in E_{j+1} as a positive section (for j +1 < ⌊r/2⌋), or a generic hyperplane section (for j + 1 = ⌊r/2⌋.) The proper transform of V_{0} meets E_{1} along a variety D_{01} ≃ Q_{n−1}, where D_{01} ⊂ Bl_{v}V_{0} is the exceptional divisor and D_{01} ⊂ E_{1} is a generic hyperplane section.

3. Managing free curves

Definition 11. A smooth separably rationally connected variety Y is strongly rationally connected if any of the following conditions hold:

(1) for each point y ∈ Y, there exists a rational curve f : ℙ^{1} → Y joining y and a generic point in Y;

(2) for each point y ∈ Y, there exists a very free rational curve containing y;

(3) for any finite collection of points y_{1}, …, y_{m} ∈ Y, there exists a very free rational curve containing the y_{j} as smooth points;

(4) for any finite collection of jets

Spec K[ϵ]/ ⟨ϵ^{N+1}⟩ ⊂ Y, i = 1, …, m

supported at distinct points y_{1}, …, y_{m}, there exists a very free rational curve smooth at y_{1}, …, y_{m} and containing the prescribed jets.

For the equivalence of these formulations, see [HT08] (cf. [Kol96] 3.9.4). (In [HT08], the word ‘very’ was left out in the second condition; this omission does not affect the other results stated there.)

Definition 12. A comb with m reducible teeth is a projective nodal curve C which is the union of m + 1 subcurves R, T_{1}, …, T_{m} such that

• R is smooth and irreducible;

• T_{ℓ} ∩ T_{ℓ′} = ∅, for all ℓ ≠ ℓ′;

• each T_{ℓ} meets R transversally in a single point; and

• each T_{ℓ} is a chain of ℙ^{1}’s. Here R is called the handle and the T_{ℓ} the teeth.

We shall use the following technical result repeatedly:

Proposition 13 (cf. Proposition 24 of [HT06]). Let C be a nodal curve with dual graph a tree, Y a smooth algebraic space, ι : C → Y an immersion with nodal image, and 𝔭 = {p_{1}, …, p_{w}} ⊂ C a collection of smooth points. Suppose that for each component C_{ℓ} the normal bundle N_{ι}(−𝔭)|_{C_{ℓ}} is globally generated and has no higher cohomology. Then ι deforms to an immersion of a smooth curve into Y containing ι(𝔭).

We will also use the following standard fact: If C is a treelike nodal curve and V is locally free over C, then V is globally generated (resp. has vanishing higher cohomology) on C iff V |_{C_{ℓ}} is globally generated (resp. has vanishing higher cohomology) over each irreducible component C_{ℓ} ⊂ C. (See Proposition 21 of [HT06] for a proof.)

Hypothesis 14 (Key Hypothesis). Let X be a projective variety with ordinary singularities, and ρ : X̃ → X the minimal divisorial resolution, obtained by blowing up each singularity. Assume that:

• X^{sm} is strongly rationally connected;

• for each exceptional divisor D ⊂ X̃, there exists a rational curve f : ℙ^{1} → X̃ meeting D transversely in one point and avoiding the other exceptional divisors of ρ.

Remark 15. There are Fano hypersurfaces with ordinary singularities that do not satisfy Hypothesis 14. We give an example at the end of Section 5.

Proposition 16. Suppose X satisfies Hypothesis 14 with minimal resolution ρ : X̃ → X having exceptional divisors {D_{i}}. Then for each D_{i} and p ∈ D_{i}, there exists a very free curve in X̃ meeting D_{i} transversely at p but disjoint from the D_{j}, j ≠ i. Given nonnegative integers {m_{i}}, there exists a very free curve in X̃ meeting D_{i} transversely at m_{i} points.

Proof. We claim there exists a very free curve meeting D_{i} transversely at one point. Let R = f(ℙ^{1}) be the rational curve provided by Hypothesis 14. Construct a comb with handle R and teeth T_{ℓ} consisting of very free rational curves in X^{sm}, which exist since X^{sm} is strongly rationally connected. For a suitable choice of teeth, we can deform our comb to a very free rational curve g : ℙ^{1} → X̃ [Kol96] II.7.9-10. This still meets D_{i} transversally at one point.

For each i, choose m_{i} very free curves meeting D_{i} in distinct points, as described above. We use these as teeth for our comb. As for the handle, since X^{sm} is strongly rationally connected, there exists a very free curve R ⊂ X^{sm} intersecting these curves transversely in distinct points. Adding additional teeth contained in X^{sm}, if necessary, we obtain a comb that deforms to a very free curve, meeting D_{i} transversely in m_{i} points. This proves the last assertion of the proposition.

It remains to show that there are very free curves meeting D_{i} at a prescribed point p. Let g : ℙ^{1} → X̃ be very free with image meeting D_{i} transversely at q ∈ D_{i}. Of course, we may deform g to a very free curve meeting D_{i} at a point nearby q in D_{i}. Recall D_{i} ≃ Q_{n−1}, a smooth quadric hypersurface of dimension n − 2, where n = dim X + 1. Any two points p, q ∈ Q_{n−1} can be joined by a smooth rational curve R ⊂ Q_{n−1} with deg R = 1, 2. Indeed, if the line ℓ joining p and q lies in Q_{n−1}, we take R = ℓ; otherwise, we take R to be a smooth conic. For simplicity, we choose g so that q is not contained in any line ℓ ⊂ Q_{n−1} through p, thus we are in the second case. In particular, N_{R/D_{i}} ≃ O_{ℙ^{1}}(2)^{n−3}.

We have an exact sequence

\begin{aligned}0 & → & N_{R/D_{i}} & → & N_{R/X̃} & → & N_{D_{i}/X̃}|_{R} & → & 0 \\ 0 & → & O_{ℙ^{1}}(2)^{n−3} & → & O_{ℙ^{1}}(2)^{n−3} ⊕ O_{ℙ^{1}}(−2) & → & O_{ℙ^{1}}(−2) & → & 0 \end{aligned} .

Construct a comb C with handle R and teeth very free curves T_{1}, T_{2}, T_{3} meeting D_{i} transversally at distinct points q_{1}, q_{2}, q_{3} ∈ R, different from p. The normal bundle analysis from Proposition 23 of [HT06] yields

N_{C/X̃}|_{R} ≃ O_{ℙ^{1}}(2)^{n−3} ⊕ O_{ℙ^{1}}(+1);

and Proposition 13 guarantees that C smooths to a rational curve through p. Since C · D_{i} = 1, the smoothing meets D_{i} transversally at p. □

Corollary 17. Let X → B be a flat proper morphism. The locus

{b ∈ B : X_{b} = π^{−1}(b) satisfies Hypothesis 14 }

is open.

Proof. Our analysis of versal deformations in Section 2 implies having ordinary singularities is an open condition. Having a very free curve in the smooth locus is also an open condition, as very free curves deform to very free curves (cf. Theorem 3.11 of [Kol96].) It remains to consider the condition that there exists a rational curve meeting each exceptional divisor once transversally. Let x ∈ X_{b} be the corresponding ordinary singularity. If x is smoothed in X then there is nothing to check. Otherwise, X → B admits a simultaneous resolution locally over suitable étale neighborhoods of b and x (see Remark 10). By Proposition 16, the curve in the resolution X̃_{b} meeting the exceptional divisor over x can be taken to be very free, and thus deforms to nearby fibers. □

4. Weak approximation at places with ordinary double points

Let B be a smooth curve with function field F. If X is a proper variety over F, a model of X over B is a flat proper morphism π : X → B with generic fiber X. The existence of such models is a general result of Nagata [Nag63]. For each b ∈ B, let X_{b} = π^{−1}(b) and X_{b}^{sm} ⊂ X_{b} the locus of smooth points.

Theorem 18. Let X be a smooth proper variety over F. Assume that X admits a model π : X → B such that, for each b ∈ B, X_{b} satisfies Hypothesis 14. Then X satisfies weak approximation.

In Section 2, we described the local structure of natural resolutions of oneparameter deformations of ordinary singularities. Let β : Y → X denote the corresponding global resolution and φ : Y → B the induced fibration. Note that each section B → Y factors through the smooth locus of φ, denoted Y^{sm}. Each irreducible component of Y_{b} has multiplicity one, so Y_{b}^{sm} is dense in Y_{b}.

What does proving weak approximation entail? Fix N ∈ ℕ and sections over the completions of B at b_{1}, …, b_{m},

ŝ_{i} : B̂_{b_{i}} → Y ×_{B} B̂_{b_{i}}, B̂_{b_{i}} = Spec \underleftarrow{lim} O_{B,b_{i}}/𝔪_{B,b_{i}}^{N}.

We must produce a section σ : B → Y approximating each ŝ_{i} to order N, i.e.,

σ ≡ ŝ_{i} (mod 𝔪_{B,b_{i}}^{N+1}).

In other words, we need a section with prescribed jet data

J := {ŝ_{i} (mod 𝔪_{B,b_{i}}^{N+1}), i = 1, …, m}.

We use the iterated blow-up construction from Section 2.3 of [HT06]. Let

γ : Y(J) → Y

be obtained by blowing up N times along each ŝ_{i}; more precisely, at each stage we blow up the point of the proper transform of ŝ_{i} lying over b_{i}. Fix y_{i} = ŝ′_{i}(b_{i}) ∈ Y(J)_{b_{i}}, where ŝ′_{i} is the proper transform of ŝ_{i} in Y(J). Any section σ′ : B → Y(J) with σ′(b_{i}) = y_{i}, i = 1, …, m yields a section σ := γ ∘ σ′ : B → Y with the desired jet data J.

To simplify notation, we will argue one place of B at a time: First resolve singularities and make iterated blow-ups in the fiber over b_{1}, then in the fiber over b_{2}, etc. until all m jet conditions are satisfied. Our main theorem then follows from the following local statement:

Theorem 19. Let X be a smooth proper variety of dimension n − 1 over F with model π : V → B. Let b ∈ B be a place over which π satisfies Hypothesis 14 and {b_{i}} ⊂ B a (possibly empty) set of additional places distinct from b. Fix a section σ′ : B → V. Then for each local section ŝ at b and each N there exists a section σ : B → V with σ ≡ ŝ (mod 𝔪_{B,b}^{N+1}) and σ(b_{i}) = σ′(b_{i}) for each i.

Figure 2. Moving sections from E_{3} to E_{2}

Let v ∈ V_{b} be an ordinary singularity. Let W → V be the minimal resolution of V at v with exceptional divisors E_{1}, …, E_{⌊r/2⌋}, as described in Section 2. We retain the notation introduced there and use E_{0} to denote the proper transform of V_{b}.

Suppose that V → B admits a section σ : B → V. Its proper transform σ′ : B → W meets one of the divisors E_{j}.

Case I: Moving sections down There exists a section τ : B → W with τ(b) ∈ E_{j−1} and τ(b_{i}) = σ′(b_{i}) for each i.

Suppose σ′ : B → W is a section with σ′(b) ∈ E_{j}, j > 0. We may assume that N_{σ′(B)/W}(−b_{1}−…−b_{m}) is globally generated and has no higher cohomology: After attaching very free curves in generic fibers, we obtain a comb with these properties, which smooths to a section with the same properties, cf.[GHS03] §2. Claim 1: There exists a smooth rational curve T_{j} ⊂ E_{j} with the following properties:

(1) σ′(b) ∈ T_{j};

(2) T_{j} meets D_{j−1,j} in two points transversely;

(3) N_{T_{j}/E_{j}} ≃ O_{ℙ^{1}}(+2)^{n−2};

(4) T_{j} is disjoint from D_{j,j+1}. By the description of fibers in (2.1) in Section 2, there exists a birational morphism

ψ : E_{j} → Q′ ⊂ ℙ^{n}

onto a quadric hypersurface. This contracts D_{j,j+1} if j < ⌊r/2⌋ and is an isomorphism otherwise; the image of D_{j−1,j} is a hyperplane section of Q′. Since σ′(b) is a smooth point of W_{b}, σ′(b) ∉ D_{j,j+1} and q = ψ(σ′(b)) is a smooth point of Q′. We

Figure 3. Moving sections from E_{2} to E_{3}

take T_{j} to be the proper transform of a generic plane section of Q′ containing q. It is straightforward to check this has the properties listed above.

Let r_{j−1,j} be one of the points of T_{j}∩D_{j−1,j}. The description of the components E_{k} at the end of Section 2 also gives: Claim 2: There exist unique smooth rational curves T_{j−1}, …, T_{1} with the following properties:

(1) each T_{k} is a ruling of E_{k}, with respect to the ℙ^{1}-bundle structure specified in (2.1);

(2) T_{j−1} contains r_{j−1,j} and T_{k} ∩ D_{k,k+1} = T_{k+1} ∩ D_{k,k+1} := r_{k,k+1} for each

k = 1, …, j − 1.

Moreover we have N_{T_{k}/E_{k}} ≃= O_{ℙ^{1}}^{n−2}. Finally, Proposition 16 yields Claim 3: There exists a very free rational curve T_{0} ⊂ E_{0} meeting D_{01} transversely at r_{01} := T_{1} ∩ D_{01}.

Consider the nodal curve

C = σ′(B) ∪_{q} T_{j} ∪_{r_{j−1,j}} T_{j} ∪ … ∪_{r_{01}} T_{0}.

The restriction of N_{C/W}(−b_{1}−…−b_{m}) to each component of C is globally generated and has no higher cohomology. By Proposition 13, C deforms to a section τ : B → W containing {σ′(b_{i})}. Computing intersection numbers, we deduce that τ(B) ∩ E_{j−1} ≠ ∅.

Case II: Moving sections up There exists a section τ : B → W with τ(b) ∈ E_{j+1} and τ(b_{i}) = σ′(b_{i}) for each i.

Suppose σ′ : B → W is a section with σ′(b) ∈ E_{j}, j < ⌊r/2⌋, chosen so that N_{σ′(B)/W}(−b_{1} − … − b_{m}) is globally generated and has no higher cohomology.

Figure 4. Moving sections within E_{3}

As in case I, we construct a nodal curve

C = σ′(B) ∪_{q} T_{j} ∪ … ∪ T_{0}

where T_{k} ⊂ E_{k} is a ruling for k = 1, …, j. This differs from the previous case in that T_{j} is also a ruling of E_{j}, and thus intersects D_{j,j+1} transversely in one point. Repeating the deformation argument as above gives the desired section τ.

Remark 20. Since the sections we construct have globally generated normal bundles, they deform to sections passing through the generic point of each E_{j}. We will tacitly assume this below.

Case III: Moving sections within a component Let W → V denote the minimal resolution of all the singularities of V in the fiber V_{b}. Before giving the details of our argument, we put our present problem in a more general context:

Question 21. For b ∈ B let W → B be a model with nonsingular total space over b. Choose an irreducible component E ⊂ W_{b} with multiplicity 1, and suppose there exists a section τ : B → W with τ(b) ∈ E. Does weak approximation hold for jets of sections supported in E?

We solve this in our specific situation; however, our analysis applies more generally.

Proposition 22. Fix b ∈ B and let W → B be flat proper morphism with rationally connected generic fiber so that W is nonsingular over b. Suppose W_{b} satisfies the following: For each p ∈ W_{b}^{sm} and q a generic point of the irreducible component E ⊂ W_{b} containing p, there exists a connected nodal curve C of genus zero, distinguished smooth points 0, ∞ ∈ C in the same irreducible component, and a differential-geometric immersion ι : C → W_{b} with the following properties

(i) ι(0) = p and ι(∞) = q; (ii) each irreducible component of W_{b} intersects C with degree zero; (iii) ι takes C to the open subset of W_{b} with normal crossings singularities of multiplicity at most two; ι^{−1}(W_{b}^{sing}) ⊂ C^{sing} and at points of W_{b}^{sing}∩ι(C) there is one branch of C through each component of W_{b};

(iv) N_{ι}(−p) is globally generated. Then weak approximation holds for jets of sections supported in E.

As usual, if we start with a section σ′ : B → W with prescribed values {x_{i}} at {b_{i}}, the sections produced by Proposition 22 can be chosen with the same values.

Remark 23. Assumption (iii) implies that ι is a local complete intersection morphism over points of W_{b}^{sing} ∩ ι(C). In particular, the normal sheaf N_{ι} is locally free on C.

In suitable local coordinates centered at such a point

\begin{aligned}C & → & W_{b} \\ \{x_{3} = x_{4} = … = x_{n} = 0\} & ⊂ & \{x_{1}x_{2} = 0\} \end{aligned} .

If C_{1} and C_{2} are branches of C and W_{1} and W_{2} are the corresponding branches of W_{b} then N_{ι}|_{C_{k}} = N_{C_{k}/W_{k}}.

The assumptions of Proposition 22 guarantee the analogous conditions when the two points are ‘infinitely near’:

Lemma 24. For each p ∈ W_{b}^{sm} and tangent direction v ∈ ℙ(T_{p}W_{b}), there exists a connected nodal curve C of genus zero with one distinguished smooth point 0 ∈ C lying on C_{1} and an immersion ι with the properties enumerated above, except that condition (i) is replaced by:

i’. dι(T_{0}C_{1}) = v, i.e., C_{1} has the prescribed tangency at p.

We sketch the argument, following the analysis of strongly rationally connected varieties in [HT08]. The hypotheses of the proposition guarantee the existence of two curves C and C′ smooth at p with distinct tangent directions w and w′, so that {w, w′, v} lie on a line ℓ ⊂ ℙ(T_{p}W_{b}). Let W̃_{b} = Bl_{p}W_{b} and consider the proper transforms C, C′ → W̃_{b}. Concatenate to get the genus-zero curve

C′′ := C ∪_{w} ℓ ∪_{w′} C′

with an immersion ι′′ : C′′ → W̃_{b}. By Proposition 23 of [HT06] we have

N_{ι′′}(−v)|_{ℓ} ≃ O_{ℙ^{1}}^{n−2};

the assumptions of Proposition 22 guarantee N_{ι′′}(−v)|_{C} and N_{ι′′}(−v)|_{C′} are also globally generated.

We see that N_{ι′′}(−v) is globally generated–in particular, there are sections which smooth w and w′ to first order. Since C′′ is a nodal connected curve of genus zero, it follows that N_{ι′′}(−v) has vanishing higher cohomology. By a variant of Proposition 13 applicable to normal crossings varieties, we can deform ι′′ to an immersion ῑ : C̄ → W̃_{b} which is smooth near the exceptional locus of W̃_{b} → W_{b} and has v ∈ ῑ(C̄). An intersection computation shows this curve meets the exceptional locus of W̃_{b} → W_{b} with multiplicity one. The resulting curve C̄ → W_{b} has the desired tangency; since properties (ii)-(iv) are open conditions (in the Hilbert scheme parametrizing immersed genus-zero curves in W_{b}), they hold as well.

We now prove Proposition 22 by induction on N, the order of the jet we seek to approximate. The N = 0 case entails producing a section through an arbitrary point p ∈ E ∩ W_{b}. Let σ′ : B → W_{b} be a section through q, the generic point of E and C the genus-zero curve guaranteed by the hypothesis of Proposition 22. We claim the nodal curve

C′ := σ′(B) ∪_{q} C

has the following properties:

(1) the restriction of N_{C′/W}(−p) to each irreducible component of C′ is globally generated and has no higher cohomology;

(2) C′ intersects fibers of W → B with multiplicity one.

Proposition 13 then implies C′ deforms to a smooth curve containing p, which is a section of W → B.

The second property is an immediate consequence of assumption (ii); C meets each component of W_{b} trivially, so C′ only meets the component E. For the first property, we have the exact sequence

0 → N_{C/W_{b}} → N_{C/W} → N_{W_{b}/W}|_{C} → 0

which yields

0 → N_{C/W_{b}}(−p) → N_{C/W}(−p) → O_{C}(−p) → 0.

(For consistency, we write N_{C/W_{b}} for N_{ι}.) We also have the extension

0 → N_{C/W} → N_{C′/W}|_{C} → Q → 0

where Q is a length-one torsion sheaf supported at q. By Proposition 23 of [HT06], N_{C/W_{b}} is saturated in this extension and thus

N_{C′/W}(−p)|_{C}/N_{C/W_{b}}(−p) ≃ O_{C}(−p + q) ≃ O_{C}.

The last isomorphism uses the fact that p and q lie on the same irreducible component of a nodal curve of genus zero. We deduce N_{C′/W}(−p)|_{C} is an extension of locally free sheaves that are globally generated with no higher cohomology; hence it also is globally generated with no higher cohomology.

For the inductive step, let j_{b} := ŝ (mod 𝔪_{B,b}^{N+1}) denote the N-jet we seek to approximate, and j′_{b} := ŝ (mod 𝔪_{B,b}^{N}) its truncation to order N −1. Let σ : B → W be a section with

σ ≡ j′_{b} (mod 𝔪_{B,b}^{N})

and let γ : W′ → W be the iterated blow-up: We blow-up successively N times along the proper transform of σ(B) over the point b. Let σ′ : B → W′ denote the proper transform of σ and G_{1}, …, G_{N} the exceptional components in W′, in the order they appear. We have the following:

(1) G_{N} ≃ ℙ^{n−1} and G_{k} ≃ Bl_{u_{k}}ℙ^{n−1} for 1 ≤ k < N; here u_{k} is the evaluation of the k-th proper transform of σ′ at b;

(2) σ′(b) ∈ G_{N};

(3) there exists a point x ∈ G_{N} so that if τ : B → W′ is a section with τ(b) = x then γ ∘ τ : B → W has the desired jet j_{b}.

We construct rational curves T_{k} ⊂ G_{k}, k = 1, …, N recursively as follows. Let T_{N} ⊂ G_{N} ≃ ℙ^{n−1} denote the line joining x and σ′(b) (if these happen to coincide we have nothing to prove). Let T_{N−1} ⊂ G_{N−1} ≃ Bl_{u_{N−1}}ℙ^{n−1} denote the proper transform of the line in ℙ^{n−1} joining u_{N−1} and T_{N} ∩ G_{N−1}. For each k > 1, let T_{k−1} ⊂ G_{k−1} denote the proper transform of the line in ℙ^{n−1} joining u_{k−1} and T_{k} ∩ G_{k−1}.

Let E′ denote the proper transform of E in W′; we have E′ = Bl_{p}E. Let v denote the intersection of T_{1} with E′, which corresponds to an element of ℙ(T_{p}E). Let C be the curve guaranteed by Lemma 24 and T_{0} its proper transform in W′. The nodal chain of curves

C′ = σ′(B) ∪ T_{N} ∪ T_{N−1} … ∪ T_{0}

satisfies

(1) the restriction of N_{C′/W′}(−x) to each irreducible component of C′ is globally generated and has no higher cohomology;

(2) C′ intersects fibers of W′ → B with multiplicity one.

These properties are verified by computations analogous to those in the base case N = 0. By Proposition 13, C′ deforms to a smooth curve containing x, which is a section of W′ → B. The image γ(C′) has the specified N-jet at b.

We complete the proof of Theorem 19. We just have to verify that there exist genus-zero curves satisfying the assumptions of Proposition 22. Again E_{j} ⊂ W_{b}, j = 0, …, ⌊r/2⌋ denote the components of our degenerate fiber. As in Case I, we have a morphism ψ : E_{j} → Q′ to a quadric surface. We also make reference to the description of components in Equation (2.1).

When j = 0 we take C to be a very free curve in E_{0} ∩ W_{b}^{sm} = V_{b}^{sm} joining p and q, which exists since V_{b}^{sm} is strongly rationally connected. For j > 0, C takes the form

T′_{0} ∪ T′_{1} ∪ … ∪ T′_{j−1} ∪ T_{j} ∪ T′′_{j−1} ∪ … T′′_{0},

where the T′_{k} and T′′_{k} are recursively defined as follows:

(1) T_{j} is a plane section of E_{j} joining p and q. Since q ∈ E_{j} is generic, we may assume T_{j} ∩ D_{j−1,j} = {r′_{j−1,j}, r′′_{j−1,j}}. We have N_{T_{j}/E_{j}} ≃ O_{ℙ^{1}}(+2)^{n−2}.

(2) For each k ≥ 1, T′_{k} (resp. T′′_{k}) is the ruling of E_{k} passing through r′_{k,k+1} := T′_{k+1} ∩ D_{k,k+1} (resp. r′′_{k,k+1} := T′′_{k+1} ∩ D_{k,k+1}). Its normal bundle in E_{k} is isomorphic to O_{ℙ^{1}}^{n−2}.

(3) T′_{0} ≃ ℙ^{1} (resp. T′′_{0}) is a very free rational curve in E_{0} meeting D_{01} at r′_{01} := T′_{1} ∩ D_{01} (resp. r′′_{01} := T′′_{1} ∩ D_{01}). This exists by Proposition 16; its normal bundle is positive.

The T′_{k} and T′′_{k} have normal bundles which are globally generated and have vanishing higher cohomology. Furthermore, N_{T_{j}/E_{j}}(−p) has the same properties. An intersection computation shows that each component of W_{b} meets C with degree zero. Each component of C is immersed in the corresponding component of W_{b}. Finally, the nodes of C are embedded in W_{b} at points where two irreducible components meets. Proposition 22 applies and weak approximation for jets supported along E_{j} follows. This completes the proof of Theorem 19.

5. Applications to cubic hypersurfaces

Weak approximation is known for cubic surfaces over the function field of a curve, provided there exists a nonsingular model X → B such that the singular fibers are cubic surfaces with rational double points [HT08]. Of course, not all rational double points can arise–see [Cor96] for a comprehensive analysis of possible models of cubic surfaces. Here we focus on the case where the fibers have only ordinary singularities:

Theorem 25. Let X be a smooth cubic hypersurface of dimension at least two over the function field of a curve B. Suppose that X admits a (possibly singular) model X → B with fibers cubic hypersurfaces with only ordinary singularities. When the dimension is two we also assume that no fiber is isomorphic to the Cayley cubic surface

wxy + xyz + yzw + zwx = 0.

Then X satisfies weak approximation over B.

We emphasize here that B need not be compact; we can omit places where the singularities are worse than ordinary double points.

Proof. The proof in [HT08] shows that each fiber X_{b}^{sm} is strongly rationally connected in relative dimension two. In dimensions > 2, X_{b}^{sm} is strongly rationally connected if, for each x ∈ X_{b}^{sm}, there exists a hyperplane section of X_{b}^{sm} through x which is strongly rationally connected. However, the generic such hyperplane section is nonsingular and thus (strongly) rationally connected.

It remains to verify the second part of Hypothesis 14. Cutting by generic hyperplane sections as above, we are reduced to the case of cubic surfaces with a single double point. (A generic hyperplane section of an ordinary singularity has an ordinary singularity.) It is a classical fact that through each ordinary singularity of a cubic surface, there passes a line. We are done if at least one of these lines does not pass through a second singularity of the surface.

Unfortunately this is not always true, but we can classify the problematic cases:

Lemma 26. Let S be a cubic surface with ordinary singularities. Assume that each line containing one singularity of S also contains a second singularity. Then S is isomorphic to the Cayley cubic surface.

We prove the lemma. The multiplicities of the lines on a singular cubic surface are well known (e.g., [GH78], pp. 640). Let s_{1} ∈ S be an ordinary double point. There are six lines ℓ ⊂ S containing s_{1}, counted with multiplicity. The lines containing a second double point have multiplicity two, and any line contains at most two ordinary double points. If every line through s_{1} also passes through a second ordinary double point, then there are three such lines ℓ_{2}, ℓ_{3}, ℓ_{4} and three double points s_{2}, s_{3}, s_{4} ∈ S besides s_{1}. In particular, S contains four ordinary double points and the six lines joining pairs of these. But the Cayley cubic is the unique cubic surface with four ordinary singularities [BW79]. This completes the proof of the lemma and the theorem. □

Hypothesis 14 fails for the Cayley surface. Let S̃ → S denote the minimal resolution of the Cayley cubic, with exceptional curves D_{1}, D_{2}, D_{3}, D_{4}. Let R be a class in the Néron-Severi group of S̃ with intersection numbers m_{i} = (R · D_{i}). The divisor class D_{1} + D_{2} + D_{3} + D_{4} is two-divisible in the Néron-Severi group, so we have

m_{1} + m_{2} + m_{3} + m_{4} ≡ 0 (mod 2).

In particular, R cannot meet D_{1} once without meeting one of the other D_{i}.

Suppose X → B is a cubic surface fibration with X_{b} a Cayley cubic and generic fiber a smooth cubic. Suppose that for each of the four singularities of X_{b}, the local equation of X is of the form

t^{3} = x_{0}^{2} + x_{1}^{2} + x_{2}^{2}.

The resolution W → X of the singularities over b has four exceptional divisors E^{1}, E^{2}, E^{3}, E^{4}, each of which is the cone over a smooth plane conic. Let E_{0} denote the proper transform of X_{b}.

Case III of the proof of Theorem 19 shows that, given a section through E^{i} or E_{0}, weak approximation holds for jets supported in that component. However, the methods of Cases I and II do not apply in this example. In particular, we cannot deduce the existence of a section through each component of W_{b}.

Appendix: Recent developments for hypersurfaces of low degree

We summarize progress since this paper was first written in March 2006. Work of de Jong and Starr yields weak approximation results for hypersurfaces substantially stronger than those presented in Section 1.

The key ingredient is the notion of rational simple connectedness. Rational connectedness in algebraic geometry plays a rôle analogous to path connectedness in topology. This suggests formulating algebro-geometric conditions analogous to simple connectedness. While there are a number of competing definitions, several have useful applications to weak approximation.

Let X be a smooth projective complex variety with H_{2}(X, ℤ) ≃ ℤ. For each e > 0, let \overline{M}_{0,n}(X, e) denote the Kontsevich space of stable maps (C, p_{1}, …, p_{n}, h), where C is a connected nodal proper curve of arithmetic genus zero, p_{1}, …, p_{n} ∈ C are smooth points, and h : C → X is a morphism of degree e with finitely-many automorphisms. There is an evaluation morphism

\begin{matrix}ev : \overline{M}_{0,n}(X, e) & → & \underbrace{X × … × X}_{n times} \\ (C, p_{1}, …, p_{n}, h) & ↦ & (h(p_{1}), …, h(p_{n})). \end{matrix}

Definition 27. X is strongly rationally simply connected if, given n ≥ 2, for each e sufficiently large there is a canonically-defined irreducible component M_{e,n} ⊂ \overline{M}_{0,n}(X, e) such that

ev : M_{e,n} → \underbrace{X × … × X}_{n times}

is dominant with rationally-connected generic fiber. X is rationally simply connected if this holds for n = 2.

Here ‘canonically defined’ means defined over any field of definition of X. We explain the relevance of these notions to weak approximation: Let X be

a smooth projective variety defined over F = ℂ(B), where B is a complex curve. Assume that H_{2}(X_{ℂ}, ℤ) ≃ ℤ. The first author has shown that if X is strongly rationally simply connected then X satisfies weak approximation over F; Jason Starr has shown that it suffices to assume X is rationally simply connected. (These theorems will be proven elsewhere.) De Jong and Starr have established the following general criteria for rational simple connectedness:

Theorem 28. [dJS06, Theorems 1.1 and 1.2] Let X be a smooth complete intersection of hypersurfaces of degrees (d_{1}, …, d_{c}) in ℙ^{n}, with each d_{j} ≥ 2. Then X is rationally simply connected if

n + 1 ≥ ∑_{j=1}^{c} d_{j}^{2},

with the exception of a quadric surface in ℙ^{3}. It is strongly rationally simply connected if

n + 1 ≥ ∑_{j=1}^{c}(2d_{j}^{2} − d_{j}).

As a corollary, we deduce weak approximation for complete intersections in these degrees.

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Department of Mathematics, Rice University, MS 136, Houston, TX 77251-1892, USA

E-mail address: hassett@rice.edu

Courant Institute, New York University, New York, NY 10012, USA

E-mail address: tschinkel@cims.nyu.edu