Descent on fibrations over P_{k}^{1} revisited

J.-L. Colliot-Thélène and A. N. Skorobogatov

Résumé. Soient k un corps de nombres et f : X → P_{k}^{1} un k-morphisme surjectif de k-variétés projectives lisses, à fibre générique géométriquement intègre. Supposons que les fibres de f audessus d’un sous-ensemble Hilbertien de P^{1}(k) satisfont le principe de Hasse (resp. le principe de Hasse et l’approximation faible). Supposons que toutes les fibres géométriques de f ont au moins une composante de multiplicité un. Le rang de f est la somme des degrés des points fermés P dont la fibre X_{P} = f^{−1}(P) ne possède pas de composante de multiplicité un géométriquement intègre. Supposons le rang au plus égal à 2. Alors l’obstruction de Brauer-Manin au principe de Hasse (resp. à l’approximation faible) sur X est la seule.

Dans des articles antérieurs nous n’avions obtenu ce résultat que sous des hypothèses plus fortes sur la nature des fibres. Le présent énoncé, obtenu grâce à une descente sur des variétés ouvertes, permet d’étudier des variétés données par des équations affines simples, sans calcul explicite d’un modèle projectif et lisse.

Notation and definitions

Let k be a field, char(k)=0, let \overline{k} be an algebraic closure, G = Gal(\overline{k}/k). A k-variety is a separated k-scheme of finite type. For an integral k-variety X we denote by k(X) the function field of X. For a k-variety X we let \overline{X} = X ×_{k} \overline{k}. In what follows Br(X) = H_{ét}^{2}(X, G_{m}) is the cohomological Brauer-Grothendieck group of X. Let Br_{0}(X) ⊂ Br(X) (resp. Br_{1}(X) ⊂ Br(X)) be the image (resp. the kernel) of the natural map Br(k) → Br(X) (resp. of Br(X) → Br(\overline{X})). We shall constantly use the theorem of Grothendieck that if Y is a dense open subset of a regular scheme X, then the natural restriction Br(X) → Br(Y) is injective.

Let f : X → Z be a dominant k-morphism of regular, geometrically integral k-varieties. Then the vertical Brauer group with respect to f is defined as follows :

Br_{vert}(X) = Br(X) ∩ f^{∗}(Br(k(Z))) ⊂ Br(k(X)).

A k-variety is called split if it contains a non-empty geometrically integral open smooth subscheme. A k-variety X is geometrically split if \overline{X} is split, or, equivalently, if X contains a non-empty open smooth subscheme. Note that a geometrically integral variety is obviously split. Let X and Z be geometrically integral, regular k-varieties and let f : X → Z be a flat surjective morphism with geometrically integral generic fibre. Then one can show that for f proper and P a point of codimension one on Z, the condition that f has a split (resp. geometrically split) fibre at P depends only on the generic fibre of f and not on a particular model X ([12], Cor. 1.2). (The notion of split or geometrically split fibre is thus a more natural candidate for what should be considered to be a “good” fibre, rather than geometrical integrality. The other point is that the split varieties over finite fields form a natural class of varieties to which the Lang–Weil estimate applies.) If P is a point of codimension one on Z the fibre X_{P} = f^{−1}(P) is geometrically split if and only if X_{P} contains an irreducible component of multiplicity one. (The multiplicity is defined with respect to the valuations in the local ring of P in Z and in the local ring of the generic point of the given component in X.) If dim(Z) = 1 we define the rank of f to be the sum of the degrees (over k) of the residue fields of the closed points P ∈ Z such that X_{P} is not split.

When k is a number field, we denote by Ω the set of places of k, by 𝔸_{k} the adèle ring of k, and by O_{k} the ring of integers of k. As usual, k_{v} is the completion of k at a place v, and O_{v} ⊂ k_{v} is the ring of integers of k_{v}. Let X be a k-variety. There is a natural pairing between X(𝔸_{k}) and Br(X) sending ({P_{v}}_{v∈Ω}, A) (with {P_{v}}_{v∈Ω} ∈ X(𝔸_{k}) and A ∈ Br(X)) to ∑_{v∈Ω} inv_{v}(A(P_{v})) ∈ ℚ/ℤ. For B ⊂ Br(X) we write X(𝔸_{k})^{B} for the set of adelic points on X orthogonal to B with respect to this pairing. If X(𝔸_{k})^{B} ≠ ∅, one says that there is no Brauer-Manin obstruction to the Hasse principle associated to B. If X(𝔸_{k})^{B} ≠ ∅ implies X(k) ≠ ∅, one says that the Brauer-Manin obstruction to the Hasse principle associated to B is the only one. If X/k is proper and X(k) is dense in X(𝔸_{k})^{B} ⊂ X(𝔸_{k}) = ∏_{v∈Ω} X(k_{v}), one says that the Brauer-Manin obstruction to weak approximation associated to B is the only one.

For f : X → Z as above, the vertical Brauer-Manin obstructions are the Brauer-Manin obstructions associated to Br_{vert}(X).

1. Brauer-Manin obstruction on open varieties

The following simple proposition will be the key technical tool in the present note.

Proposition 1.1. Let X be a smooth geometrically integral variety over a number field k, Y ⊂ X be a dense open set. Let B be a subgroup of Br(Y) such that B/(B ∩ Br_{0}(Y)) is finite. Then X(𝔸_{k})^{B∩Br(X)} ≠ ∅ if and only if Y (𝔸_{k})^{B} ≠ ∅. If X/k is proper, then Y (𝔸_{k})^{B} is dense in the closed subset X(𝔸_{k})^{B∩Br(X)} of X(𝔸_{k}) = ∏_{v∈Ω} X(k_{v}).

Proof. Let X_{c} be a smooth compactification of X (i.e. X_{c} is geometrically integral, smooth and X is an open set in X_{c}). We have obvious inclusions

Y (𝔸_{k})^{B} ⊂ X(𝔸_{k})^{B∩Br(X)} ⊂ X_{c}(𝔸_{k})^{B∩Br(X_{c})},

so we only have to prove that X_{c}(𝔸_{k})^{B∩Br(X_{c})} ≠ ∅ implies Y (𝔸_{k})^{B} ≠ ∅. Let

A_{1}, …, A_{n} ∈ B ⊂ Br(Y)

be a set of representatives for the finite group B/(B ∩ Br_{0}(Y)). Over a dense open subset U ⊂ Spec(O_{k}) the variety Y/k has a smooth integral model Y/U such that Y(O_{v}) ≠ ∅ for each v ∈ U, and such that each A_{i} is contained in Br(Y) ⊂ Br(Y).

Let {M_{v}}_{v∈Ω} ∈ X_{c}(𝔸_{k})^{B∩Br(X)}. Let S be a finite set of places of k which contains all places not in U (hence contains the archimedean places). For each place v ∈ S, let U_{v} ⊂ X(k_{v}) be an open set. Harari’s “formal lemma” ([7], Cor. 2.6.1) implies that there exist a finite set T of places of k such that S ∩ T = ∅, and points P_{v} ∈ Y (k_{v}) for v ∈ S ∪ T, with P_{v} ∈ U_{v} for v ∈ S, such that

∑_{v∈S∪T} inv_{v}(A_{i}(P_{v})) = 0, i = 1, …, n.

Now pick up any set of integral points P_{v} ∈ Y(O_{v}) for v ∉ S ∪ T. Then A_{i}(P_{v}) ∈ Br(O_{v}) = 0, hence we obtain

∑_{v∈Ω} inv_{v}(A(P_{v})) = 0,

for any A ∈ B. This means that {P_{v}}_{v∈Ω} ∈ Y (𝔸_{k})^{B}. This proves the first part of the proposition. The second part also follows, because if X/k is proper, hence X = X_{c}, then a basis of open sets for the topology of X(𝔸_{k}) is given by sets ∏_{v∈S} U_{v} × ∏_{v∉S} X(k_{v}).

The following corollary records a simple case, not explicitly used in the present paper (but see [9]), where the finiteness assumption on the Brauer group is satisfied.

Corollary 1.2. Let X be a smooth geometrically integral variety over a number field k, Y ⊂ X be a dense open set such that H^{0}(\overline{Y} , G_{m}) = \overline{k}^{∗}, Pic(\overline{Y}) is of finite type and torsion free, and Br(\overline{Y}) is finite. Then X(𝔸_{k})^{Br} ≠ ∅ if and only if Y (𝔸_{k})^{Br} ≠ ∅. If moreover X/k is proper, then Y (𝔸_{k})^{Br} is dense in X(𝔸_{k})^{Br}.

Proof. From the Hochschild–Serre spectral sequence

(1)  H^{p}(G, H^{q}(\overline{Y} , G_{m})) ⇒ H^{p+q}(Y, G_{m}),

we obtain the exact sequence

H^{2}(G, H^{0}(\overline{Y} , G_{m})) → Ker[Br(Y) → Br(\overline{Y})] → H^{1}(G, Pic(\overline{Y})).

Our assumptions imply that B = Br(Y) satisfies the finiteness hypothesis in Proposition 1.1.

We shall need the following result which is a slightly more general version of one of the main theorems of the descent theory developed in [2]. Let X/k be a smooth geometrically integral variety and let λ : M → Pic(\overline{X}) be a homomorphism of G-modules, where M is of finite type (as an abelian group). Let F be the k-group of multiplicative type with module of characters M. There exists a canonical map χ : H^{1}(X, F) → Hom_{G}(M, Pic(\overline{X})) ([2], 1.5.1, 1.5.2). If T → X is a torsor under F, then χ(T ) is called the type of T . Define Br_{λ}(X) ⊂ Br_{1}(X) as the preimage of λ_{∗}(H^{1}(G, M)) ⊂ H^{1}(G, Pic(\overline{X})) under the canonical map Br_{1}(X) → H^{1}(G, Pic(\overline{X})) in the Hochschild–Serre spectral sequence (1) (with X in place of Y).

Proposition 1.3. Let X be a smooth geometrically integral variety over a number field k, such that H^{0}(\overline{X}, G_{m}) = \overline{k}^{∗}. If X(𝔸_{k})^{Br_{λ}(X)} ≠ ∅, then there exists an X-torsor T under F of type λ such that T (𝔸_{k}) ≠ ∅. More precisely, given {M_{v}}_{v∈Ω} ∈ X(𝔸_{k})^{Br_{λ}(X)}, there exists an X-torsor T under F of type λ and a point {P_{v}}_{v∈Ω} ∈ T (𝔸_{k}) sent to {M_{v}}_{v∈Ω} under the map induced by the structural map T → X.

Proof. See [13], Section 3, Theorem 3 (a), where the proof closely follows the pattern of [2]. (The corresponding results in [2] had been proved under the simplifying assumption that X/k is proper and M torsion free.)

2. Main results

The possibility to reduce the study of the Brauer-Manin obstruction to an appropriate open subset has important consequences for the fibration theorems. These are the results which allow one, under certain restrictions, to conclude that the Brauer-Manin obstruction for a variety fibred over P_{k}^{1} is the only obstruction to the Hasse principle if one knows that the Hasse principle holds for sufficiently many k-fibres. The following result improves on the main theorem of [12] by dispensing with some superfluous restrictions (the condition that all the components of the non-split fibres are of multiplicity one, as well as the condition of the existence of a section over \overline{k}). It also improves on [4], Thm. 2.2.1, by dispensing with the abelianity condition. This is quite convenient in practice, as will be demonstrated on examples further below.

Theorem A. Let X be a smooth projective geometrically integral variety over a number field k, and let f : X → P_{k}^{1} be a faithfully flat morphism with geometrically split fibres and geometrically integral generic fibre. Suppose that rank(f) ≤ 2.

(a) Suppose that the k-fibres over a Hilbertian subset of k ∪ ∞ = P^{1}(k) satisfy the Hasse principle. Suppose that there is no vertical Brauer-Manin obstruction to the Hasse principle. Then there are k-rational points on X, and the image of X(k) in P^{1}(k) is Zariski-dense in P_{k}^{1}.

(b) Suppose that the k-fibres over a Hilbertian subset of k ∪ ∞ = P^{1}(k) satisfy the Hasse principle and weak approximation. Then the vertical Brauer-Manin obstruction to weak approximation on X is the only obstruction.

Theorem 4.2 of [10] has the following generalization :

Theorem B. Let X be a smooth projective geometrically integral variety over a number field k, and let f : X → P_{k}^{1} be a faithfully flat morphism with geometrically integral generic fibre. Suppose that rank(f) ≤ 3. Suppose that every non-split fibre X_{P} contains a multiplicity one component V_{P} such that the integral closure of the residue field k(P) in the function field k(V_{P}) is a quadratic extension of k(P). Then the same statements as in Theorem A hold.

Applying the theorem : a recipe

In practice we shall be given a geometrically integral quasi-projective k-variety X_{0} and a dominant (hence flat) morphism g_{0} : X_{0} → P_{k}^{1} with geometrically integral generic fibre. The following property will hold : The smooth locus of the k-fibres of g_{0} over a Hilbertian subset of k ∪ ∞ = P_{k}^{1}(k) satisfies the Hasse principle (resp. the Hasse principle and weak approximation).

At the outset, the variety X_{0} need not be projective, and it need not be smooth. The generic fibre itself need not be smooth.

Patching and taking schematic closures, one easily realizes X_{0} as an open set of an explicit geometrically integral quasi-projective k-variety X_{1} equipped with a faithfully flat morphism g_{1} : X_{1} → P_{k}^{1} extending g_{0} (recall that flatness is automatic for a dominating map from an integral variety to a regular curve, so the only issue is surjectivity of g_{1} as a morphism of schemes). Let X′_{1} ⊂ X_{1} be the open set of smooth points of X_{1}. Let g′_{1} : X′_{1} → P_{k}^{1} be the induced morphism. We shall assume that the morphism g′_{1} is surjective. This is a fairly mild hypothesis. It is fulfilled if X_{1} is normal (in that case, the generic points of components of fibres of g_{1} are in the smooth locus).

One may further easily realize X_{1} as an open set of an explicit geometrically integral projective k-variety X_{2} equipped with a faithfully flat projective morphism g_{2} : X_{2} → P_{k}^{1}.

Hironaka’s theorem ensures that there exists a projective birational k-morphism φ : X → X_{2} such that the projective k-variety X is smooth and such that moreover φ induces an isomorphism between φ^{−1}(X′_{1}) and X′_{1}. Indeed X is obtained from X_{2} by successive blow-ups whose support is in the singular locus. Let f : X → P_{k}^{1} be the composite map X → X_{2} → P_{k}^{1}.

We would like to apply Theorem A to f : X → P_{k}^{1} . Making X and f explicit may be a very hard labour. Let us explain why there is no need to do so.

If all the fibres of g′_{1} are geometrically split, then such are also the fibres of f. One immediately sees that the rank of f is less than or equal to the rank of g′_{1}. Hence if the rank of g′_{1}, which is easy to compute, is at most 2, then the rank of f is at most 2.

For many fibrations X/P_{k}^{1} whose fibres satisfy the Hasse principle, the following hypothesis holds. The generic fibre X_{η} of f over k(P^{1}) acquires a rational point over the C_{1}-field \overline{k}(P^{1}). Then f has a section over \overline{k}, hence all fibres of f are geometrically split.

The hypothesis that the k-fibres of g_{0} over a Hilbertian subset of k ∪ ∞ = P_{k}^{1}(k) satisfy the Hasse principle (resp. the Hasse principle and weak approximation) implies the same statement for the fibres of f.

Applying Theorem A to the (not explicit) fibration f : X → P_{k}^{1} also involves the vertical Brauer group

Br_{vert}(X) = Br(X) ∩ f^{∗}(Br(k(P^{1}))) ⊂ Br(k(X)),

which is a finite group (Lemma 3.1). This finite group is a birational invariant, it does not depend on the model X → P_{k}^{1} with X/k smooth and projective. In a number of concrete examples, one may manage to compute that group without explicitly producing a model X (cf. [3], §2.2 and §2.3).

Finally, given an element A ∈ Br(X), the image of X(𝔸_{k}) → ℚ/ℤ under the map which sends an adèle {P_{v}}_{v∈Ω} to the sum of all local invariants inv_{v}(A(P_{v})) can be computed by restricting attention to adèles all components of which belong to a given open set (say X′_{1}). It is now clear how one may “apply” Theorem A to g_{0} : X_{0} → P_{k}^{1}.

There is an analogous discussion for Theorem B.

Examples

1) Let A be a central simple algebra over the field k(t) = k(P^{1}). Let {M_{i}}_{i∈I} be the set of closed points in P_{k}^{1} where A has a non-trivial residue. Let Z/k(t) be the Severi-Brauer variety associated to A, and let f : X → P_{k}^{1} be a projective map from a smooth integral k-variety X such that the generic fibre is isomorphic to Z/k(t). Assume that the sum of the degrees [k(M_{i}) : k] is equal to two (for example assume that there exists a single closed point with non-trivial residue, and that this closed point is quadratic). Then the Brauer-Manin obstruction to the Hasse principle for X is the only obstruction. Indeed, over the complement of the points M_{i}, the associated relative Severi-Brauer variety is a model with smooth geometrically integral fibres. Resolution of singularities provides the existence of a model f : X → P_{k}^{1} of rank at most 2 (an explicit model is available, see [5]). Over \overline{k}, the map f acquires a section (Tsen’s theorem). Hence f has geometrically split fibres. Moreover the smooth fibres are Severi-Brauer varieties, hence satisfy the Hasse principle, by Châtelet’s theorem. All the hypotheses of Theorem A are fulfilled.

2) Theorem 4.5 of [1], which handles certain cubic threefolds with three conjugate singular points, is now recovered as a particular case of Theorem A. This was in principle covered by the main theorem of [12] but to apply that theorem one needed to construct an explicit smooth proper model and check that all components of non-split fibres had multiplicity one. This is no longer necessary – checking this for one component in each fibre is now enough. The original proof ([1], §4) involved a rather ad hoc descent, and ended with a small miracle.

One starts with the affine cubic threefold X_{0} defined in four-dimensional affine space by the equation

N_{K/k}(x_{1} + ωx_{2} + ω^{2}x_{3}) + Q(t)(L(x_{1}, x_{2}, x_{3}) + t) = 0,

where K = k(ω) is a cubic extension of k, N_{K/k} is the norm map, L is a non-zero linear form with coefficients in k, and Q(t) is a separable polynomial in k[t], of degree two, which does not vanish for t = 0. There is an obvious nonsingular k-point x_{1} = x_{2} = x_{3} = t = 0 and the cubic threefold is k-unirational, hence k-rational points are Zariski dense ([1], Lemma 4.2). The issue here is weak approximation.

Projection to the t-coordinate defines a map g_{0} : X_{0} → A_{k}^{1} ⊂ P_{k}^{1}. Writing s = 1/t, making obvious changes of coordinates and patching, one embeds X_{0} into a variety X_{1} equipped with a faithfully flat map g_{1} : X_{1} → P_{k}^{1} extending g_{0}. One easily checks that all fibres of g_{1} except those over Q(t) = 0 are geometrically integral ([1], Lemma 2.2), and the fibres over Q(t) = 0 are geometrically split. The fibres of g_{0} at k-points of A_{k}^{1} away from Q(t) = 0 are cubic surfaces of a special type, over a number field their open set of smooth points have long been known to satisfy the Hasse principle and weak approximation (Skolem, Segre, Coray-Tsfasman, see [1], Thm. 1.7).

The general recipe described above applies without further ado : on any smooth projective model X of X_{0}, the Brauer-Manin obstruction to weak approximation is the only one.

In this example, for a given model X/P_{k}^{1} of X_{0}/P_{k}^{1} with X/k smooth and projective, the whole Brauer group of X is vertical (with respect to the map f : X → P_{k}^{1}). It is trivial (reduced to the Brauer group of k) unless Q(t) is reducible over the discriminant extension of K/k ([1], Thm. 9.6 (a)). Thus except in this special case, X satisfies weak approximation.

3) Theorem B naturally applies to fibrations the generic fibre of which is a (positive-dimen-sional) quadric. In this case also the whole Brauer group is vertical. Let Y ⊂ A_{k}^{4} be given by the equation

a_{1}(t)x_{1}^{2} + a_{2}(t)x_{2}^{2} + a_{3}(t)x_{3}^{2} = a_{0}(t),

where a_{i}(t) are non-zero square-free polynomials. If all of them have even degree, and their product has at most 3 multiple roots in \overline{k}, then the Brauer-Manin obstruction to (the Hasse principle and) weak approximation is the only one on any smooth, proper model of Y. Without restrictions on the degrees the same conclusion holds if the product of the a_{i}(t) has at most 2 multiple roots in \overline{k}. The hypotheses on the polynomials ensure that the rank of f for a model f : X → P_{k}^{1} of Y/A_{k}^{1} (where A_{k}^{1} = Spec(k[t])), with X/k proper and smooth, is at most three. The “worst case” occurs when the rank of f is three ; this case could be handled in [11] only under an additional restriction (Theorem 4.2 and Corollary 4.2 ; condition (*) of Theorem 4.2 appears on p.63).

Remark. Let K/k be a cyclic extension of number fields, N_{K/k} be the norm map, and let {ω_{i}}_{i=1,...,n} be a k-base of K over k. Let P (t) be a polynomial of degree 2. Consider the equation

N_{K/k}(ω_{1}x_{1} + … + ω_{n}x_{n}) + P (t) = 0.

Let k(t) = k(P^{1}). There is a model f : X → P_{k}^{1} with X/k smooth and f of rank at most 3. The geometric fibres are split (there is a section over \overline{k}), and the smooth fibres satisfy the Hasse principle. However, since the rank may be three, we do not know whether the Brauer-Manin obstruction to the Hasse principle for X is the only obstruction. The first open case occurs for n = 4. (Under the Schinzel hypothesis, the result would hold for K/k cyclic of arbitrary degree and P (t) of arbitrary degree, see [4].)

3. Proofs

Lemma 3.1. Let k be a field of characteristic zero. Let X and Z be regular and geometrically integral k-varieties, and f : X → Z be a flat surjective morphism with geometrically split fibres at points of Z of codimension one, and with geometrically integral generic fibre. Then Br_{vert}(X)/f^{∗}(Br(Z)) is finite.

Proof (cf. [12], pp. 920-922). Since f is flat and surjective, all the components of the fibre of f at a point of Z of codimension one also have codimension one. By the classical results of Grothendieck [6] (see the summary in [3], §1) we have the following commutative diagram :

0 ⟶ Br(Z) ⟶ Br(k(Z)) ⟶^{res} ⊕_{P} H^{1}(k(P), ℚ/ℤ)
f^{∗} ↓  f^{∗} ↓  e_{P ′/P} .f^{∗} ↓
0 ⟶ Br(X) ⟶ Br(k(X)) ⟶^{res} ⊕_{P′}H^{1}(k(P′), ℚ/ℤ)

where P (resp. P′) ranges over all the points of codimension one of Z (resp. of X). In this diagram the maps f^{∗} are the obvious ones, and for P,P′ points of codimension one in Z, resp. X, such that f(P′) = P, we let e_{P′/P} be the multiplicity in the fibre X_{P} = f^{−1}(P) of the irreducible component whose generic point is P′.

Since the fibres f^{−1}(P) are geometrically split, for each P of codimension one on Z, we can choose a point P̃ of codimension 1 on X such that f(P̃) = P and e_{P̃/P} = 1. Let kP̃ be the integral closure of k(P) in k(P̃) ; this is a finite, separable extension of k(P). Then the map e_{P̃/P} .f^{∗} on H^{1}(k(P), ℚ/ℤ) decomposes as the restriction map r_{P,P̃} : H^{1}(k(P), ℚ/ℤ) → H^{1}(kP̃, ℚ/ℤ) followed by the inclusion H^{1}(kP̃, ℚ/ℤ) ↪ H^{1}(k(P̃), ℚ/ℤ) (this map is injective since kP̃ is integrally closed in k(P̃)). By the commutativity of the diagram we get the inclusion Br_{vert}(X)/f^{∗}(Br(Z)) ⊂ ⊕_{P} Ker(r_{P,P̃}). Note that if X_{P} is geometrically integral, then Ker(r_{P,P̃}) = 0. Since the generic fibre of f is geometrically integral, this is the case for all but a finite number of codimension one points P ∈ Z. In general Ker(r_{P,P̃}) is a finite group. This proves the lemma.

Corollary 3.2. With assumptions as in Lemma 3.1, let Y ⊂ X be an open set such that the composite (flat) map Y → X → Z is surjective and has geometrically split fibres at points of codimension one of Z. Then Br_{vert}(X) is a subgroup of Br_{vert}(Y) of finite index. If moreover Z is proper and k-birational to a projective space, then the group Br_{vert}(Y)/Br_{0}(Y) is finite.

Proof. This follows from Lemma 3.1. For the last statement note that if Z is proper, smooth and k-rational then Br(Z) = Br(k).

Proposition 3.3. Let X be a smooth, proper, geometrically integral variety over a number field k, let f : X → P_{k}^{1} be a dominant morphism with geometrically split fibres and geometrically integral generic fibre. Let Y ⊂ X be a dense open set as in Corollary 3.2. Then Y (𝔸_{k})^{Brvert(Y)} is dense in X(𝔸_{k})^{Brvert(X)} ⊂ X(𝔸_{k}).

Proof. We have Br_{vert}(X) = Br_{vert}(Y) ∩ Br(X). Using the last assertion of Corollary 3.2 we can apply Proposition 1.1.

Proof of Theorem A. The first part of the proof is purely algebraico-geometric. We may assume that the fibre of f over ∞ is smooth. For each non-smooth or non-irreducible fibre X_{P}, choose a component E_{P} of multiplicity one of X_{P}, let F_{P} be the closed set which is the union of all the other components, let G_{P} be the singular locus of E_{P}. Let Y ⊂ X be the complement in X of the union of the F_{P} ’s and G_{P} ’s over all closed points P ∈ P_{k}^{1}. Thus Y → P_{k}^{1} is smooth and surjective, its generic fibre is proper and geometrically integral, and all its fibres are smooth and connected, hence integral. A fibre Y_{P} is split if and only if it is geometrically integral. Let T be the set of closed points P ∈ P_{k}^{1} such that Y_{P} is not split. For such a P, let us denote by k_{P} the integral closure of k(P) in the function field k(Y_{P}). Since the map Y → P_{k}^{1} is flat and surjective and has a proper and geometrically integral generic fibre, we have H^{0}(\overline{Y} , G_{m}) = \overline{k}^{∗}. Let M = Ker[Pic(\overline{Y}) → Pic(\overline{X}_{η})], where \overline{X}_{η} is the generic fibre of f : \overline{X} → P1 ; \overline{k}. Since f has geometrically split fibres, the generic fibre is proper and geometrically integral and the basis of the fibration f is P1 ; \overline{k}, we conclude that M is torsion free (see [12], 3.2.3). Let F be the algebraic k-torus dual to M. A Y -torsor under F is called vertical if its type is the natural injection λ_{M} : M ↪ Pic(\overline{Y}). As just noticed, we have H^{0}(\overline{Y} , G_{m}) = \overline{k}^{∗}. This is enough to ensure the existence of a vertical torsor ([2], Thm. 2.3.1 and Prop. 2.2.8 (v) ; see also [12], §3, where the initial assumption X/k proper may be replaced throughout the paragraph by the assumption H^{0}(\overline{X}, G_{m}) = \overline{k}^{∗}, which is the standing assumption (2.0.1) in [2]). By [12], Cor. 4.5 (which also only uses H^{0}(\overline{Y} , G_{m}) = \overline{k}^{∗}), the group Br_{vert}(Y) coincides with Br_{λ_{M}} (Y) as defined before Proposition 1.3.

The group H^{1}(k, F) is the quotient of ∏_{P ∈T} k(P)^{∗}/N_{k_{P} /k(P)}(k_{P}^{∗}) by the diagonal image of k^{∗}. Let {α_{P} }_{P ∈T} ∈ ∏_{t∈T} k(P)^{∗} be a representative of α ∈ H^{1}(k, F).

Let us recall the explicit description ([12], 3.3) of the (smooth) auxiliary k-variety W_{α}. Its equations here simply read :

α_{P} (u − a_{P} v) = N_{k_{P} /k(P)}(z_{P}), (u, v) ≠ (0, 0), P ∈ T.

Here we have fixed an affine coordinate on A_{k}^{1} ⊂ P_{k}^{1}, a_{P} is the coordinate of P over the field k(P), z_{P} is a parameter in k_{P} (corresponding to [k_{P} : k(P)] variables over the field k(P)), u and v are parameters in k, and the whole system of equations is to be viewed as defining a k-variety.

When the rank of f is 2, which we henceforth assume, this W_{α} is a dense open subset of affine space A_{k}^{n} (for some integer n) whose complement is of codimension 2, equipped with a flat surjective morphism ξ : W_{α} → P_{k}^{1}. For any vertical torsor T over Y there exists α ∈ H^{1}(k, F) such that T is k-birationally equivalent to the fibre product Z_{α} = Y ×_{P_{k}^{1}} W_{α} taken with respect to the morphisms f : Y → P_{k}^{1} and ξ : W_{α} → P_{k}^{1} (the last one induced by projection to (u, v) ; same references as above, which are all right since H^{0}(\overline{Y} , G_{m}) = \overline{k}^{∗}). The second projection p : Z_{α} = Y ×_{P_{k}^{1}} W_{α} → W_{α} is smooth since Y → P_{k}^{1} is smooth. The map p is also clearly surjective, hence faithfully flat. Its generic fibre is projective and geometrically integral. The explicit description of W_{α} and a comparison of the maps f and ξ allows us, just as in in [12], Thm. 3.3.3, to apply [12], Cor. 1.4 : we conclude that all fibres of p at points of codimension one of W_{α} are split. As a consequence, away from a closed set of codimension at least 2 in W_{α}, all the fibres of p at points of W_{α} are split.

Now begins the arithmetic part of the proof. We shall prove statements (a) and (b) simultaneously. Let {M_{v}}_{v∈Ω} ∈ X(𝔸_{k})^{Brvert(X)} ⊂ X(𝔸_{k}), let S be a finite set of places of k, and for each v ∈ S let U_{v} ⊂ X(k_{v}) be an open set with M_{v} ∈ U_{v}. By Proposition 3.3 there exists {N_{v}}_{v∈Ω} ∈ Y (𝔸_{k})^{Brvert(Y)} = Y (𝔸_{k})^{Br_{λ_{M}} (Y)} with N_{v} ∈ U_{v} for v ∈ S. By Proposition 1.3, there exists a vertical torsor T → Y and an adèle {P_{v}}_{v∈Ω} ∈ T (𝔸_{k}) whose image under T → Y is {N_{v}}_{v∈Ω}. Let α ∈ H^{1}(k, F) be such that T is birationally equivalent to the smooth k-variety Z_{α} = Y ×_{P_{k}^{1}} W_{α}.

After possibly moving each N_{v} slightly (in particular keeping N_{v} ∈ U_{v}), one finds a family {Q_{v}}_{v∈Ω} ∈ ∏_{v∈Ω} Z_{α}(k_{v}) which maps to {N_{v}}_{v∈Ω} ∈ ∏_{v∈Ω} Y (k_{v}) under the map induced by Z_{α} → Y.

The explicit description of the map ξ : W_{α} → P_{k}^{1} also shows that the generic fibre of this dominant map is geometrically integral (see [12], Lemma 3.3.1). If R ⊂ P^{1}(k) is a Hilbert set (the complement of a thin set [8]), this implies that ξ^{−1}(R) ⊂ W_{α}(k) is also the complement of a thin set, hence a Hilbert set in W_{α}(k) ⊂ A^{n}(k).

From hypothesis (a) of Theorem A we conclude that there exists a Hilbert set of points M ∈ W_{α}(k) whose fibre p^{−1}(M) satisfies the Hasse principle. From hypothesis (b) of Theorem A we conclude that there exists a Hilbert set of points M ∈ W_{α}(k) whose fibre p^{−1}(M) satisfies the Hasse principle and weak approximation.

In case (a) we shall prove that the image of Z_{α}(k) → P^{1}(k) is Zariski dense in P^{1}. This will imply the same statement for the image of Y (k) → P^{1}(k). In case (b), we shall show that one may find Q ∈ Z_{α}(k) arbitrarily close simultaneously to each Q_{v} for v ∈ S.

We would like to apply the basic fibration theorem to the morphism p : Z_{α} → W_{α} ⊂ A_{k}^{n}, namely Thm. 2.1 of [12], which is itself a slight generalization of the main result of [11], except that we run into two obstacles : the map p is not projective, more precisely it is projective only over an open set of A_{k}^{n}, and the restriction of p to a general line in An ; \overline{k} need not have a section. Therefore we must go through the proof of Theorem 1 in [11] and make the necessary modifications alongside with those already made in §2 of [12].

The first step of the quoted theorem is the reduction to the case n = 1. As in [11], pp. 206-207, using a standard argument (cf. [1], p. 526) and a suitable Bertini theorem (see also [8], §3), one shows that there exists an affine line L ⊂ W_{α}, L ≃ A_{k}^{1} satisfying the following properties. We have ∏_{v∈Ω} p^{−1}(L)(k_{v}) ≠ ∅. For each v ∈ S, there exists a point in L(k_{v}) which is in the image of p^{−1}(L)(k_{v}) under the map p and which is very close to p(Q_{v}). The line L contains a Hilbert set H of k-points whose fibre under p satisfies the Hasse principle (resp. the Hasse principle and weak approximation).

Moreover, because the line L may be chosen in a Zariski open set of the space of lines in A_{k}^{n}, we may assume that ξ(L) is a dense open subset of P_{k}^{1}. Let L′ be the projective line containing L as the complement to a point ∞ ∈ L′, then the restriction of ξ to L extends to a dominant morphism ξ′ : L′ → P_{k}^{1}. Define V := Y ×_{P_{k}^{1}} L′, and let g be the second projection V → L′. Since f is a smooth map, so is g, and V is a smooth k-variety. The open set p^{−1}(L) = Y ×_{P_{k}^{1}} L ⊂ Y ×_{P_{k}^{1}} L′ = V is the complement to the fibre g^{−1}(∞). This last fibre is isomorphic to the fibre of Y → P_{k}^{1} at ξ′(∞), and hence is geometrically split as is any fibre of Y → P_{k}^{1}. Let V_{c} be a smooth compactification of V equipped with a map g_{c} : V_{c} → L′ ≃ P_{k}^{1} extending g. Clearly all fibres of g_{c} are geometrically split and are actually split except possibly one at a k-point. This is essentially the situation discussed in the second paragraph of subsection 2.1 of [4].

The proof given there shows that the set of points of L′(k) whose fibre has points everywhere locally is dense in the product ∏_{v∈Ω} g_{c}(V_{c}(k_{v})) ⊂ P^{1}(𝔸_{k}) (hence in particular is Zariski dense in P_{k}^{1}). As in [10], the same proof shows that already the intersection of this set with H has the same property (this uses Ekedahl’s version of Hilbert’s irreducibility theorem). This is enough to complete the proof of Theorem A.

Proof of Theorem B. The proof follows that of Thm. 4.2 of [10] with modifications made in the proof of Theorem A, so we only concentrate on the points in which it differs from the last proof. We choose Y ⊂ X as in Prop. 3.3 such that every non-split fibre of Y → P_{k}^{1} is integral and is split by a quadratic extension of the residue field. The variety W_{α} is now the punctured affine cone over the complement to a closed subset of codimension 2 in a smooth projective quadric (of dimension 4). A vertical torsor T such that T (𝔸_{k}) ≠ ∅ is birationally equivalent to Y ×_{P_{k}^{1}} W_{α} for some α (the second projection having split fibres at points of W_{α} of codimension one). Hence ∏_{v∈Ω} W (k_{v}) ≠ ∅, which implies W_{α}(k) ≠ ∅ by the Minkowski–Hasse theorem. Then W_{α} is k-rational and contains smooth k-rational curves L in generic position. The rest of the proof is identical.

Acknowledgements. We thank Nick Shepherd-Barron for a question which led to the results in §1. Part of the work for this paper was done while the authors were guests of the Isaac Newton Institute, Cambridge.

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J.-L. Colliot-Thélène C.N.R.S. URA D0752, Mathématiques Bâtiment 425 Université de Paris-Sud F-91405 Orsay France e-mail : colliot@math.u-psud.fr

A. N. Skorobogatov Imperial College Department of Mathematics Huxley Building 180 Queen’s Gate London SW7 2BZ United Kingdom e-mail : a.skorobogatov@ic.ac.uk