The Brauer group of torsors and its arithmetic applications

David Harari and Alexei Skorobogatov

April 29, 2002

Introduction

This paper consists of two parts. In the first part we describe the behaviour of the Brauer group with respect to the pull-back map from the variety to a torsor under a torus over any field of characteristic 0. To a variety X endowed with a surjective morphism π : X → P_{k}^{1} with geometrically integral generic fibre one canonically associates certain X-torsors, called vertical torsors. The second part is devoted to the arithmetic of vertical torsors over a number field k. We prove that the Manin obstruction to the Hasse principle and weak approximation on smooth and proper models of such torsors is the only one provided the same property holds for the k-fibres of π, and the number of ‘bad’ fibres is small. Using the results of the first part we give sufficient conditions for X itself (strictly speaking, for any smooth and proper model of X) to have the afore-mentioned property. To illustrate possible applications we exhibit apparently new classes of conic bundle threefolds and some other varieties with the property that the possible failure of the Hasse principle or weak approximation is accounted for by the Manin obstruction.

Let us now describe the contents of each section in more detail. Let k be a field of characteristic 0, and let X be a variety over k. Denote by k̄ an algebraic closure of k, and set \overline{X} := X ×_{k} k̄, Γ := Gal (k̄/k). Let Br X := H_{ét}^{2}(X, G_{m}) be the cohomological Brauer–Grothendieck group of X. In their fundamental paper [CS87a] Colliot-Thélène and Sansuc defined universal torsors and studied their various properties. For a universal torsor f : Y → X over a variety X such that all invertible functions on \overline{X} are constants, and Pic \overline{X} has no torsion, they prove that the kernel of the map Br Y → Br \overline{Y} is naturally isomorphic to Br k. In this paper we show (Theorem 1.6) that the map Br X → Br Y/Br k defined by f is canonically identified with the map Br X → (Br \overline{X})^{Γ} (provided k is such that H^{3}(Γ, \overline{k}^{∗}) = 0). We also obtain sufficient conditions that ensure that the unramified Brauer group of Y is the image of the unramified Brauer group of X.

The main achievement of the arithmetic part of the paper, seen from the applications point of view, are the following results (Theorems 2.9 and 2.7, proved by a combination of ‘open descent’ and fibration methods). Let X be a smooth and projective variety over a number field k, and let π : X → P_{k}^{1} be a dominant morphism with geometrically integral generic fibre. Assume that the sum of degrees of closed points of P_{k}^{1} corresponding to non-split (bad) fibres is at most 3, and that every non-split fibre contains a multiplicity 1 irreducible component splitting into two over an algebraic closure of the residue field. Then, under some mild additional hypotheses, and provided we are not in an explicitly described exceptional case, the possible failure of the Hasse principle and weak approximation on X is accounted for by the Manin obstruction, provided the same is true for smooth k-fibres of π. We also prove a similar result in the case when there are only two non-split fibres, both over k-points, each containing an irreducible component of multiplicity 1 defined over an extension of k of prime degree. This extends the results of [H94], as well as those of [S90], [S96], [CS00], by allowing a few bad fibres in the theorem on the behaviour of the Manin obstruction regarding the passage from k-fibres to the total space.

For an example of application consider the equation

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

where a_{i}(x, t) ∈ k[x, t] are pairwise coprime, and deg_{x} a_{0} + deg_{x} a_{1} + deg_{x} a_{2} ≤ 4. Assume that there are at most three points t_{0} ∈ k̄ such that a_{l}(x, t_{0}) is identically zero and a_{m}(x, t_{0})a_{n}(x, t) ∈ k̄(x)^{∗2}, for some {l, m, n} = {0, 1, 2}. Provided we are not in an explicitly described exceptional case the Manin obstruction is shown to be the only obstruction to the Hasse principle and weak approximation for any smooth and projective model of the variety given by the displayed equation (see Proposition 2.10 for a precise statement). An explicit example of a different kind can be found in the end of the paper.

Notation and conventions

All cohomology groups in this paper are Galois or étale cohomology groups. Let G be a finite group, and M be an F -module. Then �_{ω}^{i}(G, M) denotes the subgroup of H^{i}(G, M) consisting of the classes whose image under the restriction map to every cyclic subgroup of G is zero.

For a Γ-module M we write H^{i}(k, M) for H^{i}(Γ, M). When M, as an abelian group, is of finite type and torsion-free we define �_{ω}^{i}(k, M) := �_{ω}^{i}(G, M), for i = 1, 2, where G is the (finite) image of the action of Γ on M. Note that these groups are finite.

Let k′ the smallest extension of k such that the action of Gal (k̄/k′) on M is trivial. Write F = Gal (k′/k). Then H^{1}(k′, M) = 0 because H^{1}(k′, Z) = 0. The restriction-inflation sequence for M gives H^{1}(k, M) = H^{1}(G, M). There is also an exact sequence

0 → H^{2}(G, M) → H^{2}(k, M) → H^{2}(k′, M),

so that �_{ω}^{i}(k, M) is naturally a subgroup of H^{i}(k, M).

When k is a number field we denote by Ω_{k} the set of places of k, and by k_{v} the completion of k at the place v. Let A_{k} be the ring of adèles of k. It is an easy corollary of Tchebotarev’s density theorem that for a number field k the group �_{ω}^{i}(k, M) consists of α ∈ H^{i}(k, M) such that the localisation α_{v} ∈ H^{i}(k_{v}, M) is trivial for almost all places v ∈ Ω_{k}.

By a variety X over a field k we understand in this paper a separated scheme of finite type over Spec k. The Brauer group Br X = H_{ét}^{2}(X, G_{m}) is equipped with a natural filtration Br_{0} X ⊂ Br_{1} X ⊂ Br X, where Br_{0} X = Im [Br k → Br X] and Br_{1} X = Ker [Br X → Br \overline{X}]. Set k̄[X]^{∗} = H^{0}(\overline{X}, G_{m}). There is a Hochschild–Serre spectral sequence

H^{p}(k, H^{q}(\overline{X}, G_{m})) ⇒ H^{p+q}(X, G_{m}).

It yields a canonical map (Br \overline{X})^{Γ} → H^{2}(k, Pic \overline{X}). When k̄[X]^{∗} = k̄^{∗} we obtain the exact sequence of low degree terms

0 → (Pic \overline{X})^{Γ}/Pic X → Br k → Br_{1} X → H^{1}(k, Pic \overline{X}) → H^{3}(k, k̄^{∗}) → H^{3}(X, G_{m}). (1)

Recall that H^{3}(k, k̄^{∗}) = 0 if k is a number field ([CF], 7.11.4), or if k = k_{0}(t) where k_{0} is a number field (see [H94], proof of Thm. 3.5.1). When k is arbitrary, a k-point on X defines a section of the maps Br k → Br X and H^{3}(k, k̄^{∗}) → H^{3}(X, G_{m}). If this is the case, (1) shows that Br_{0} X = Br k and that the map Br_{1} X → H^{1}(k, Pic \overline{X}) is surjective.

The unramified Brauer group of the field k(X) with respect to k is a subgroup Br_{nr} (k(X)/k) ⊂ Br (k(X)); it is isomorphic to the Brauer group of any smooth and proper model of X ([G], 6.2).

Recall that for any number field k local class field theory gives an injective map j_{v} : Br k_{v} → Q/Z, which is an isomorphism for v finite. Let X be a proper, smooth and geometrically integral k-variety. Then the set of adelic points X(A_{k}) is just ∏_{v∈Ω_{k}} X(k_{v}). Assume that X(A_{k}) ≠ ∅, and let \overline{X(k)} be the closure of the diagonal image of X(k) in X(A_{k}). Set

X(A_{k})^{Br} = {(P_{v}) ∈ X(A_{k})| ∑_{v∈Ω_{k}} j_{v}(A(P_{v})) = 0 for any A ∈ Br X}.

(The sum is well-defined, [CS87a] III, [S01], 5.2). The reciprocity law of global class field theory implies that

\overline{X(k)} ⊂ X(A_{k})^{Br} .

In particular, the condition X(A_{k})^{Br} = ∅ is an obstruction to the existence of a k-rational point on X; this is the Manin (or Brauer–Manin) obstruction to the Hasse principle. The condition X(A_{k})^{Br} ≠ X(A_{k}) is the Brauer–Manin obstruction to weak approximation. One says that the Brauer–Manin obstruction to weak approximation is the only one if \overline{X(k)} = X(A_{k})^{Br} .

1. Torsors and the Brauer group

In this section k is any field of characteristic zero.

1.1. Torsors under tori and their relative compactifications

By T we shall denote an algebraic k-torus with module of characters T̂. Let p : X → Spec k be a smooth and geometrically integral k-variety, and let f : Y → X be a torsor under T. Equivalently, Y is a smooth and geometrically integral k-variety equipped with a (scheme-theoretically) free action of T such that Y/T = X.

We shall frequently use the Leray spectral sequence

H^{p}(X, R^{q}f_{∗}G_{m}) ⇒ H^{p+q}(Y, G_{m}).  (2)

By the relative Rosenlicht lemma ([CS87a], Prop. 1.4.2) we have an exact sequence of sheaves on X in the étale topology:

1 → G_{m} → f_{∗}G_{m} → p^{∗}T̂ → 0.  (3)

Here p^{∗}T̂ is a sheaf on X obtained from the Γ-module T̂. Note that H^{1}(S, Z) = 0 for any normal integral scheme S ([SGA 4], IX 3.6 (ii)). This implies in particular that H^{1}(\overline{X}, p^{∗}T̂) = 0.

Lemma 1.1 Let f : Y → X be a torsor under a torus T. Then (i) R^{1}f_{∗}G_{m} = 0. (ii) If T = G_{m}, then R^{2}f_{∗}G_{m} = 0.

Proof. (i) is established in the proof of ([CS87a], Prop. 2.1.1). For the convenience of the reader we reproduce this argument here in the case T = G_{m} alongside with the proof of (ii).

Let x be a geometric point of X, R be the strictly henselian local ring of X at x, and K be the field of fractions of R. The stalk of R^{1}f_{∗}G_{m} at x is Pic (Y ×_{X} R), and the stalk of R^{2}f_{∗}G_{m} at x is Br (Y ×_{X} R). Note that H^{1}(R, G_{m}) = 0 for any local ring, hence the scheme Y ×_{X} R is isomorphic to G_{m,R}. We have Pic G_{m,S} = Pic S for any normal base scheme S. Now H^{1}(R, G_{m}) = 0 which implies Pic (Y ×_{X} R) = 0. Therefore, R^{1}f_{∗}G_{m} = 0.

Recall that a ring of finite type over an excellent ring (resp. the localization of an excellent ring) is excellent ([EGA4], 7.8.3), hence the local ring O_{X,x} is excellent. Since k is of characteristic zero, Hironaka’s resolution of singularities holds for O_{X,x}. By [EGA4], 7.9.5 and 18.8.17, the ring R, and hence also the scheme G_{m,R}, is excellent. Therefore the Brauer group Br G_{m,R} fits into the following exact sequence of abelian groups ([G], Cor. 6.2):

0 → Br A_{R}^{1} → Br G_{m,R} → H^{1}(R, Q/Z) → 0.

It is known that Br A_{R}^{1} = Br R = 0 ([M], IV.2.13). Similarly H^{1}(R, Z/n) = 0 for any n > 0 by [M], III.3.11 (because the residue field is algebraically closed), hence H^{1}(R, Q/Z) = 0. This proves that Br G_{m,R} = 0, hence R^{2}f_{∗}G_{m} = 0. QED

From (2), (3) and statement (i) of this lemma we obtain an exact sequence of Γ-modules ([CS87a], Prop. 2.1.1, see also [S01], 1.3):

1 → k̄[X]^{∗}/k̄^{∗} ⟶^{f^{∗}} k̄[Y]^{∗}/k̄^{∗} → T̂ → Pic \overline{X} ⟶^{f^{∗}} Pic \overline{Y} → 0.  (4)

The map T̂ → Pic \overline{X} in this exact sequence is called the type of the torsor f : Y → X. The torsor f : Y → X is called universal if its type is an isomorphism. In this case we have Pic \overline{Y} = 0. When k̄[X]^{∗} = k̄^{∗} the type of the torsor f : Y → X is injective if and only if k̄[Y]^{∗} = k̄^{∗}. In particular, this property holds for universal torsors, and implies Br_{1} Y = Br_{0} Y = Br k by (1).

In this paper we define a relative smooth compactification of a torsor under a torus as follows.

Definition 1.2 Let f : Y → X be a torsor under a torus. A relative smooth compactification of f : Y → X is a geometrically integral variety Z endowed with a smooth and proper morphism g : Z → X with connected and reduced geometric fibres, together with an embedding i : Y ↪ Z such that i(Y) is open and dense in Z, and f = g ∘ i.

The existence of a relative smooth compactification in characteristic 0 follows from Hironaka’s theorem.

Recall [CS77] that an exact sequence of Γ-modules

0 → M → P → F → 0,

such that M, P, F are free abelian groups of finite rank, is called a flasque resolution of M if the following conditions hold.

(a) P is a permutation Γ-module, that is, P has a Z-basis permuted by Γ.

(b) F is flasque, that is, the Tate cohomology group H^{−1}(G′, F) is trivial for any subgroup G′ ⊂ G, where G is the image of the action of Γ on F.

Lemma 1.3 Let f : Y → X be a torsor under a torus T of type τ : T̂ → Pic \overline{X}. Let g : Z → X be a relative smooth compactification of f : Y → X. Then

(i) there is a natural injection of Γ-modules T̂ → Div_{\overline{Z}\\overline{Y}} (\overline{Z}); it defines a flasque resolution of T̂:

0 → T̂ → Div_{\overline{Z}\\overline{Y}} (\overline{Z}) → Div_{\overline{Z}\\overline{Y}} (\overline{Z})/T̂ → 0;

(ii) we have

Ker[g^{∗} : H^{2}(k, Pic \overline{X}) → H^{2}(k, Pic \overline{Z})] = τ_{∗}(�_{ω}^{2}(k, T̂)).

Proof. We prove (i) and (ii) simultaneously. Since Z is regular we have an exact sequence of sheaves on Z:

0 → G_{m,Z} → i_{∗}G_{m,Y} → Div_{Z\Y} → 0.  (5)

The sheaf Div_{Z\Y} is defined as the direct sum of j_{∗}Z, where j is the embedding into Z of a closed integral subset of codimension 1 contained in Z \ Y, for all such closed subsets of Z. The sequence (5) is exact since Z is regular, and hence the Weil divisors coincide with Cartier divisors (see [M], II, Example 3.9). Let D^{+}(X) (resp. D^{+}(Z)) be the derived category of bounded below complexes of étale scheaves on X (resp. on Z). The derived functor Rg_{∗} : D^{+}(Z) → D^{+}(X) gives an exact triangle in D^{+}(X):

Rg_{∗}G_{m,Z} → Rg_{∗}(i_{∗}G_{m,Y}) → Rg_{∗}Div_{Z\Y} .

In D^{+}(X) there is also an exact triangle defined by (3). We have a natural isomorphism f_{∗}G_{m,Y} = g_{∗}i_{∗}G_{m,Y}, and a canonical morphism G_{m,X} → g_{∗}G_{m,Z} (which is also an isomorphism since g is proper). These morphisms define a morphism of exact triangles in D^{+}(X):

\begin{aligned}G_{m,X} & → & f_{∗}G_{m,Y} & → & p^{∗}T̂ \\ ↓ & & ↓ & & ↓ \\ Rg_{∗}G_{m,Z} & → & Rg_{∗}(i_{∗}G_{m,Y}) & → & Rg_{∗}Div_{Z\Y} \end{aligned}  (6)

Let K be the function field \overline{k}(X), K_{0} = k(X). We have the following identifications: H^{1}(\overline{X}, f_{∗}G_{m,Y}) = Pic \overline{Y} in view of (2) and Lemma 1.1 (i); H^{1}(\overline{Z}, i_{∗}G_{m,Y}) = Pic \overline{Y} since R^{1}i_{∗}G_{m,Y} = 0 ([CS87a], p. 393); H^{0}(\overline{Z}, Div_{Z\Y}) = Div_{\overline{Z}\\overline{Y}} (\overline{Z}) ([CS87a], (i) on p. 393), likewise H^{0}(Spec k̄(Z), Div_{Z\Y}) = Div_{Z_{K}\Y_{K}}(Z_{K}); H^{1}(\overline{Z}, Div_{Z\Y}) = 0 ([CS87a], (ii) on p. 393).

From (6) we obtain the following commutative diagram of Γ-modules with exact rows and columns, which is the definition of F:

0  0
↓  ↓
k̄[Y]^{∗}/k̄^{∗} →  T̂  → Pic \overline{X} → Pic \overline{Y} → 0
||  ↓  ↓  ||
k̄[Y]^{∗}/k̄^{∗} → Div_{\overline{Z}\\overline{Y}} (\overline{Z}) → Pic \overline{Z} → Pic \overline{Y} → 0
↓  ↓
F  = F
↓  ↓
0  0

(7)

The arrows in the right hand square are the natural maps. The top row comes from the exact sequence (4); the map T̂ → Pic \overline{X} is the type τ of the torsor f : Y → X. The injective map T̂ → Div_{\overline{Z}\\overline{Y}} (\overline{Z}) is the one whose existance was claimed in (i).

Let us now show that F is a flasque Γ-module. Let k′ ⊂ \overline{k} be the smallest extension of k such that Gal (k̄/k′) acts trivially on the modules in diagram (7). Call G the finite group Gal (k′/k).

Note that the generic fibre Z_{K_{0}} = Z ×_{X} K_{0} of g : Z → X is a smooth compactification of the principal homogeneous space Y_{K_{0}} = Y ×_{X} K_{0} of the torus T_{K_{0}} = T ×_{X} K_{0}. We identify Γ = Gal (\overline{k}/k) with Gal (K/K_{0}). Then we have a canonical isomorphism of Γ-modules K[Y_{K_{0}}]^{∗}/K^{∗} = T̂ (see, e.g. [S01], Lemma 2.4.3). This leads to an exact sequence of Γ-modules

0 → T̂ → Div_{Z_{K}\Y_{K}}(Z_{K}) → Pic Z_{K} → 0.  (8)

The exact sequence (8) will not change if we replace the extension K = \overline{k}(X) of K_{0} = k(X) by the extension L = \overline{k}(Y) of L_{0} = k(Y). Indeed, since K_{0} is integrally closed in L_{0} the Galois group Gal (L/L_{0}) is canonically isomorphic to Γ = Gal (K/K_{0}). On the other hand, we are dealing with Γ-modules which are free abelian groups of finite type that do not change after the passage from K to L. The advantage now is that Y has a canonical L_{0}-point given by the embedding of the generic point into Y. Hence the Galois module Pic Z_{L} = Pic Z_{K} is the Picard module of a smooth compactification of the torus T_{L}. The Galois group Γ acts on Pic Z_{L} via its quotient G. By a theorem of Voskresenskiǐ ([V], Thm. 4.6) the G-module Pic Z_{L} is flasque.

The G-module Div_{\overline{Z}\\overline{Y}} (\overline{Z}) is the direct sum of the module of ‘vertical’ divisors Div_{v}, that is, the divisors on \overline{Z} that do not intersect the generic fibre Z_{K}, and the module of ‘horizontal’ divisors, that is, the divisors on \overline{Z} that are Zariski closures of the divisors on Z_{K} with support in Z_{K} \ Y_{K}. The G-module of horizontal divisors is thus identified with Div_{Z_{K}\Y_{K}}(Z_{K}). Restriction to the generic fibre defines a map of G-modules Pic \overline{Z} → Pic Z_{K}. This map is surjective because Z is regular. Its kernel obviously contains Pic \overline{X}, hence it factors through a surjective map F → Pic Z_{K}.

The composition of the map T̂ → Div_{\overline{Z}\\overline{Y}} (\overline{Z}) from diagram (7) with the map Div_{\overline{Z}\\overline{Y}} (\overline{Z}) → Div_{Z_{K}\Y_{K}}(Z_{K}) given by the restriction of divisors to the generic fibre Z_{K}, is no other than the second map from (8). Indeed, the right hand vertical arrow in (6) gives rise to a commutative diagram

H^{0}(\overline{X}, p^{∗}T̂) → H^{0}(Spec k̄(X), p^{∗}T̂)
↓  ↓
H^{0}(\overline{Z}, Div_{Z\Y}) → H^{0}(Spec k̄(Z), Div_{Z\Y})

which is precisely what we need:

T̂  =  T̂
↓  ↓
Div_{\overline{Z}\\overline{Y}} (\overline{Z}) → Div_{Z_{K}\Y_{K}}(Z_{K})

Putting everything together we obtain a commutative diagram of G-modules with exact rows

0  0
↓  ↓
T̂ = T̂
↓  ↓
0 → Div_{v} → Div_{\overline{Z}\\overline{Y}} (\overline{Z}) → Div_{Z_{K}\Y_{K}}(Z_{K}) → 0
||  ↓  ↓
0 → Div_{v} →  F  →  Pic Z_{K}  → 0
↓  ↓
0  0

(9)

Since Div_{v} is a permutation module, it is flasque by Shapiro’s lemma. Now the bottom line of the diagram shows that F is flasque. This establishes (ii).

In particular, we have H^{−1}(C, F) = 0 for any cyclic subgroup C ⊂ G. By the periodicity of Tate cohomology of cyclic groups we obtain H^{1}(C, F) = 0 for any cyclic subgroup C ⊂ G.

We now go back to diagram (7). The just established property of F, and the fact that Div_{\overline{Z}\\overline{Y}} (\overline{Z}) is a permutation G-module, and hence

H^{1}(k, Div_{\overline{Z}\\overline{Y}} (\overline{Z})) = �_{ω}^{2}(k, Div_{\overline{Z}\\overline{Y}} (\overline{Z})) = 0,

enable us to conclude that �_{ω}^{2}(k, T̂) coincides with the injective image of H^{1}(k, F) in H^{2}(k, T̂) (cf. [CS87b], the proof of Prop. 9.5 (ii)). The statement (ii) now follows from diagram (7). QED Remark 1.4 We have associated a canonical flasque resolution of T̂ to any relative smooth compactification of a torsor under T. This generalizes the construction due to Voskresenskiǐ (see [V]) and Colliot-Thélène–Sansuc [CS77] corresponding to the case when the base scheme X is the spectrum of a field.

One can give a shorter proof under the assumption k̄[Y]^{∗} = k̄^{∗} as follows. Define F = Pic \overline{Z}/Pic \overline{X}, then formally define the left hand column of (7) by taking the kernels of the restriction maps to Pic \overline{Y}. Now observe that the restriction to Z_{K} gives a map F → Pic Z_{K} with kernel Div_{v}. Indeed, this kernel is generated by the classes of divisors from Div_{v}. If (ϕ) ∈ Div_{v} for some non-constant function ϕ ∈ k̄(Z)^{∗}, then the restriction to Z_{K} shows, since Z_{K} is proper and does not intersect with (ϕ), that ϕ ∈ k̄(X)^{∗}. But this is impossible because then (ϕ) has a non-trivial intersection with \overline{Y}.

The proof we have given explains where the map T̂ → Div_{\overline{Z}\\overline{Y}} \overline{Z} comes from.

1.2. The pull-back of the Brauer group to a torsor

In connection with arithmetic applications of torsors under tori we are interested in describing the cokernel of the map f^{∗} : Br X → Br Y, especially when Y is a universal torsor. Arguably a more essential question is the computation of the cokernel of the map f_{nr}^{∗} : Br_{nr} (k(X)/k) → Br_{nr} (k(Y)/k). Colliot-Thélène and Sansuc pointed out that when k is algebraically closed the map f_{nr}^{∗} is an isomorphism. (This follows from the fact that in this situation Y is birationally equivalent to X ×_{k} P_{k}^{r}.)

Our first result concerns the cokernel of f^{∗} : Br X → Br Y over an algebraically closed field.

Theorem 1.5 Let k be an algebraically closed field of characteristic 0. Let X be a smooth and geometrically integral variety over k such that k[X]^{∗} = k^{∗} and Pic X is a free abelian group of finite type. Let f : Y → X be a torsor under a k-torus such that k[Y]^{∗} = k^{∗} and Pic Y is a free abelian group of finite type (for example, a universal torsor). Then the map f^{∗} : Br X → Br Y is an isomorphism.

Note that after [CS87a] the injectivity of f^{∗} : Br X → Br Y is known to be true for any torsor under a torus, but the surjectivity does not hold in general. A well known example is the torsor G_{m,k}^{2} → G_{m,k} under G_{m,k}. Here Br G_{m,k} = 0, but Br G_{m,k}^{2} = Q/Z.

Proof of Theorem 1.5. Let X be a smooth and geometrically connected variety over an algebraically closed field k, such that k[X]^{∗} = k^{∗} and Pic X is torsion-free. Let f : Y → X be a torsor under T. We denote its type by τ_{Y}. The assumptions of Theorem 1.5 are equivalent to the condition that τ_{Y} is injective and τ_{Y} (T̂) is a primitive sublattice of Pic X.

Let us choose a Z-basis in T̂ ≃ Z^{n}, and factor f : Y → X into a composition of torsors under G_{m}:

Y = Y_{0} → Y_{1} → … → Y_{n} = X,

where each Y_{i} is a k-variety. Then we have H^{0}(Y_{i}, G_{m}) = k^{∗}. It is also clear that Pic Y_{i} is a torsion free abelian group of finite type. It is therefore enough to prove Theorem 1.5 when T = G_{m}. We assume this from now on.

The spectral sequence (2) and Lemma 1.1 imply that Br Y = H^{2}(X, f_{∗}G_{m}). Now the exact sequence (3) together with the remark that follows it, and the fact that Br Y is a torsion group lead to an exact sequence

0 → Br X → Br Y → H^{2}(X, Z)_{tors}.  (10)

The Kummer exact sequence gives rise to the exact sequence

0 → k[X]^{∗}/k[X]^{∗n} → H^{1}(X, Z/n) → (Pic X)[n] → 0.

Now our assumptions imply H^{1}(X, Z/n) = 0. Using the exact sequence

0 → Z ⟶^{×n} Z → Z/n → 0

we see that H^{2}(X, Z)[n] = 0 for any n > 0, hence H^{2}(X, Z)_{tors} = 0. It now follows from (10) that the map f^{∗} : Br X → Br Y is an isomorphism. Theorem 1.5 is proved. QED

Over general fields the situation is more complicated.

Theorem 1.6 Let k be a field of characteristic 0 such that H^{3}(k, k̄^{∗}) = 0. Let X be a smooth and geometrically integral k-variety such that k̄[X]^{∗} = k̄^{∗} and Pic \overline{X} is a free abelian group of finite type.

(a) Let f : Y → X be a torsor under a k-torus T, such that k̄[Y]^{∗} = k̄^{∗} and Pic \overline{Y} is a free abelian group of finite type (for example, a universal torsor). Let τ : T̂ → Pic \overline{X} be the type of f : Y → X. Let P_{τ} be the intersection of the image of (Br \overline{X})^{Γ} → H^{2}(k, Pic \overline{X}) with τ_{∗}(H^{2}(k, T̂)) in H^{2}(k, Pic \overline{X}). Then there is an exact sequence of abelian groups

0 → Br X/Br_{1} X → Br Y/Br_{1} Y → P_{τ} → 0.

(b) Let f : Y → X be a universal torsor. Then there is a natural isomorphism Br Y/Br k = (Br \overline{X})^{Γ} such that the canonical map Br X → (Br \overline{X})^{Γ} identifies with the map Br X → Br Y/Br k defined by f^{∗}.

Note that, in the assumptions of this theorem, the surjectivity of the map f^{∗} : Br X → Br Y for a universal torsor f : Y → X is equivalent to the surjectivity of the map Br X → (Br \overline{X})^{Γ}.

The following particular case of Theorem 1.6 (b) is valid over any field of characteristic 0.

Corollary 1.7 Let k be a field of characteristic 0. Let X be a smooth and geometrically integral k-variety such that k̄[X]^{∗} = k̄^{∗}, Pic \overline{X} is a free abelian group of finite type, and (Br \overline{X})^{Γ} = 0. If f : Y → X is a universal torsor, then Br Y = Br k.

This corollary follows from Theorem 1.5 and the exact sequence (1) applied to Y. It improves an earlier similar result of Colliot-Thélène and Sansuc valid for smooth compactifications of universal torsors over projective varieties.

Proof of Theorem 1.6. The natural map of Γ-modules f^{∗} : Br \overline{X} → Br \overline{Y} is an isomorphism by Theorem 1.5. Hence (Br \overline{X})^{Γ} → (Br \overline{Y})^{Γ} is also an isomorphism.

We now exploit the functoriality of the Hochschild–Serre spectral sequence

H^{p}(Γ, H^{q}(\overline{X}, G_{m})) ⇒ H^{p+q}(X, G_{m})

with respect to the morphism f : Y → X. In view of our assumption H^{3}(k, \overline{k}^{∗}) = 0 this spectral sequence gives rise to the following commutative diagram with exact rows:

0 → Br X/Br_{1} X → (Br \overline{X})^{Γ} → H^{2}(k, Pic \overline{X})
↓ || ↓
0 → Br Y/Br_{1} Y → (Br \overline{Y})^{Γ} → H^{2}(k, Pic \overline{Y})

(11)

The statement (a) now follows from this diagram combined with exact sequence (4). Taking into account that Pic \overline{Y} = 0 for a universal X-torsor Y one obtains (b). Theorem 1.6 is proved. QED

The following theorem, in which the torsor is replaced by a relative smooth compactification, seems to be more useful in applications.

Theorem 1.8 We keep the notation and conditions of Theorem 1.6 (a). Let Q_{τ} be the intersection of the image of (Br \overline{X})^{Γ} → H^{2}(k, Pic \overline{X}) with τ_{∗}(�_{ω}^{2}(k, T̂)) in H^{2}(k, Pic \overline{X}). Then for any relative smooth compactification Z → X of Y → X the following sequence of abelian groups is exact:

0 → Br X/Br_{1} X → Br Z/Br_{1} Z → Q_{τ} → 0.

Proof. The beginning of the proof is similar to the proof of Theorem 1.6. Let g : Z → X be any relative smooth compactification of f : Y → X. This time we use the functoriality of the Hochschild–Serre spectral sequence

H^{p}(Γ, H^{q}(\overline{X}, G_{m})) ⇒ H^{p+q}(X, G_{m})

with respect to the morphism g : Z → X.

The natural map g^{∗} : (Br \overline{X})^{Γ} → (Br \overline{Z})^{Γ} is an isomorphism. Indeed, by [G] the map i^{∗} : Br \overline{Z} → Br \overline{Y} is injective (since Y ⊂ Z is an open and dense embedding of regular varieties). The composed map i^{∗}g^{∗} = f^{∗} : Br \overline{X} → Br \overline{Y} is an isomorphism by Theorem 1.5. Thus g^{∗} : Br \overline{X} → Br \overline{Z} is also an isomorphism. All these maps are homomorphisms of Γ-modules, hence g^{∗} : (Br \overline{X})^{Γ} → (Br \overline{Z})^{Γ} is an isomorphism.

It is clear that H^{0}(\overline{Z}, G_{m}) = \overline{k}^{∗} since the same property is true for Y. In view of our assumption that H^{3}(k, \overline{k}^{∗}) = 0 the Hochschild–Serre spectral sequence now gives rise to the following commutative diagram with exact rows:

0 → Br X/Br_{1} X → (Br \overline{X})^{Γ} → H^{2}(k, Pic \overline{X})
↓ || ↓
0 → Br Z/Br_{1} Z → (Br \overline{Z})^{Γ} → H^{2}(k, Pic \overline{Z})

(12)

The proof is now completed by an easy diagram chase and an application of Lemma 1.3. QED

For a proper variety X we obtain, as a corollary, a result about the cokernel of the map f_{nr}^{∗} : Br X = Br_{nr} (k(X)/k) → Br_{nr} (k(Y)/k).

Corollary 1.9 In the assumptions of Theorem 1.6 (a) assume further that X is proper. Then we have an exact sequence

0 → Br X/Br_{1} X → Br_{nr} (k(Y)/k)/Br_{nr,1}(Y) → Q_{τ} → 0,

where

Br_{nr,1}(Y) := Ker[Br_{nr} (k(Y)/k) → Br_{nr} (k̄(Y)/k̄)].

The corollary follows from Theorem 1.8 because in this case Z is a smooth and proper model of Y, hence Br Z = Br_{nr} (k(Y)/k).

Corollary 1.10 In the assumptions of Theorem 1.6 (a) assume further that X is proper and Y is a universal torsor. Then Br_{nr} (k(Y)/k) = f_{nr}^{∗}(Br_{nr} (k(X)/k)) whenever (Br \overline{X})^{Γ} = 0 or �_{ω}^{2}(k, Pic \overline{X}) = 0.

More precisely, in the case when X is a proper variety over a number field k the unramified Brauer group of a universal torsor f : Y → X can be described as follows:

Br_{nr} (k(Y)/k)/f_{nr}^{∗}(Br_{nr} (k(X)/k)) = ⋂ Ker[(Br \overline{X})^{Γ}/Br X → (Br \overline{X_{v}})^{Γ_{v}}/Br X_{v}].

Here Γ_{v} = Gal (\overline{k}_{v}/k_{v}), X_{v} = X ×_{k} k_{v}, \overline{X_{v}} = X ×_{k} \overline{k_{v}}, and the intersection is taken over the places v of k such that the Γ-module Pic \overline{X} is unramified at v (the inertia group at v acts trivially). Such are almost all the places of k. It would be interesting to give an example of a proper and smooth variety X satisfying the assumptions of Theorem 1.6 (a) such that the group described in the last displayed formula is non-zero.

Remark 1.11 If Y (k) ≠ ∅, then the existence of a section of the map H^{3}(k, k̄^{∗}) → H^{3}(Y, G_{m}) makes the assumption H^{3}(k, k̄^{∗}) = 0 in Theorem 1.6 redundant. The same is true for Theorem 1.8 if Z(k) ≠ ∅.

2. Arithmetic of vertical torsors

In this section we cross the fibration method of [H94] and [H97] with the descent method of [S90], [S96] and [CS00].

2.1. Main theorem

We shall consider a geometrically integral variety X equipped with a surjective morphism f to P_{k}^{1}. Let η be the generic point of P_{k}^{1}, K = k(η) = k(P_{k}^{1}). Let X_{K} := X ×_{P_{k}^{1}} K be the generic fibre of f, X_{k̄(η)} = X ×_{K} k̄(η), and let X\overline{K} := X ×_{K} \overline{K} be the generic geometric fibre.

Recall that a k-scheme V is called split if it contains a non-empty geometrically integral open k-subscheme (which need not be dense). By a further restriction such a subscheme can be assumed to be smooth.

The rank of f, denoted by rk f, is defined as the sum of the degrees of the closed points P ∈ P_{k}^{1} such that the fibre X_{P} = f^{−1}(P) is not split over the residue field k(P) of P. Let U ⊂ X be a dense open subset such that

(a) the restriction of f to U is a smooth surjective morphism with integral closed fibres;

(b) for any closed point P ∈ P_{k}^{1} the fibre U_{P} is geometrically integral if and only if X_{P} is split;

(c) the generic fibre U_{K} coincides with X_{K}. Note that such a dense open subset exists if and only if f : X → P_{k}^{1} locally in étale topology has a section at every closed point P ∈ P_{k}^{1}. Equivalently, every closed fibre of f contains an irreducible component of multiplicity 1. We define

M_{U} = Ker [Pic \overline{U} → Pic X_{k̄(η)}].

A Γ-module M such that M = M_{U} for a dense open subset U ⊂ X satisfying (a),

(b) and (c) will be called a vertical module associated to f. A U-torsor of type M_{U} ↪ Pic \overline{U} is called vertical. It is easy to see that M_{U} is a torsion-free abelian group ([S96], Prop. 3.2.3).

Theorem 2.1 Let X be a smooth, projective and geometrically integral variety over a number field k, and let f : X → P_{k}^{1} be a dominant morphism with geometrically integral generic fibre. Suppose that the following conditions hold:

(1) rk f ≤ 2;

(2) Pic X\overline{K} is torsion-free, Br X\overline{K} is finite, and X_{K} has a k̄(η)-point;

(3) either Br X_{k̄(η)} = 0, or there exists a vertical module M associated to f such that �_{ω}^{2}(k, M) = 0.

Then if the Manin obstruction is the only obstruction to the Hasse principle or to weak approximation for the k-fibres of f in a dense open subset of P_{k}^{1}, then the same property holds for X.

This theorem should be compared with Thm. A of [CS00]. Under additional cohomological assumptions we replace the arithmetic hypothesis that the k-fibres of f satisfy the Hasse principle or weak approximation by a weaker condition.

Let us describe the idea of proof of Theorem 2.1. As in [S96] and [CS00] we can lift an adelic point {P_{v}} of U(A_{k})^{Br} to an adelic point on some vertical torsor Y → U. The difficulty, pointed out in [H97], 4.5, is that in our situation there is no reason why Y should satisfy the Hasse principle: the hypothesis that the Manin obstruction to the Hasse principle is the only one for the k-fibres of f is too weak for this. However, using Proposition 2.3 below and the results of Section 1 we show that our condition (3) implies that {P_{v}} lifts to an adelic point {Q_{v}} ∈ Y (A_{k})^{Br Z}, where Z is a relative smooth compactification of Y. On the other hand, we show that on Z the Manin obstruction to the Hasse principle is the only one: this is a consequence of the hypothesis rk f ≤ 2 and the results of [H97] (which replace Thm. 2.1 of [S96]). Therefore the assumption U(A_{k})^{Br} ≠ ∅ implies that Y (k) is not empty, hence U(k) is non-empty. As U(A_{k})^{Br} is dense in X(A_{k})^{Br} (by [CS00], Prop. 1.1) the statement regarding the Hasse principle follows. The argument for weak approximation is similar.

Remark 2.2 If we just want to deal with the case Br X_{k̄(η)} = 0, the results of Section 1 are not needed. Indeed, this assumption implies Br \overline{X} = 0 and we can apply [H97], 4.5.

The key observation for the proof of Theorem 2.1 is the following statement.

Proposition 2.3 Let X be a smooth and geometrically integral variety over a number field k, such that H^{0}(\overline{X}, G_{m}) = \overline{k}^{∗} and Pic \overline{X} is an abelian group of finite type. Let T be a k-torus, and τ be an injection of Γ-modules T̂ → Pic \overline{X}. Suppose that {M_{v}} ∈ X(𝔸_{k})^{Br_{1} X}. Then there exists a torsor f : Y → X under T of type τ, and a point {P_{v}} ∈ Y (𝔸_{k})^{Br_{1} Y} such that M_{v} = f(P_{v}) for all places v.

Proof. Let S be the k-group of multiplicative type dual to the Γ-module Pic \overline{X}, S → T be the surjective k-homomorphism dual to τ, and S_{1} be its kernel. We denote by Id the canonical isomorphism Ŝ → Pic \overline{X}. Under our assumptions the main theorem of the descent theory ([CS87a], [S01], 6.1.2) states that an adelic point on X, which is orthogonal to Br_{1} X with respect to the Brauer–Manin pairing, can be lifted to an adelic point {N_{v}} on some universal torsor Z → X of type Id. Then the quotient Y = Z/S_{1} is an X-torsor under T of type τ (this trivially follows from the functoriality of type with respect to the structure group).

Let us prove that Z → Y is a universal torsor. Since Z is a universal X-torsor and H^{0}(\overline{X}, G_{m}) = \overline{k}^{∗}, we see from (4) that H^{0}(\overline{Z}, G_{m}) = \overline{k}^{∗} and Pic \overline{Z} = 0. The same exact sequence (4) considered for the torsor Z → Y now shows that T̂ → Pic \overline{Y} is an isomorphism, hence Z is a universal Y -torsor.

Now let {P_{v}} be the image of {N_{v}} on Y. The inverse main theorem of the descent theory states that the image of an adelic point on a universal torsor is orthogonal to Br_{1} Y. This is exactly what we wanted to prove. QED

Note that this statement is non-trivial in the following sense: in general the natural map Br_{1} X → Br_{1} Y is not surjective.

We shall need the following small refinement of [H97], Thm. 3.2.1:

Proposition 2.4 Let F be a closed subset of A_{k}^{n} of codimension at least 2, L = k(A_{k}^{n}). Let V be a quasi-projective k-variety equipped with a surjective morphism p : V → A_{k}^{n} \ F with split fibres and geometrically integral generic fibre V_{L}. Let X_{L} be a smooth and projective model of V_{L}. Assume the following conditions:

(a) The restriction of p to a general line in An ; k̄ has a section.

(b) Br X\overline{L} is finite, and Pic (X\overline{L}) torsion-free, where \overline{L} is an algebraic closure of L, and X\overline{L} = X_{L} ×_{L} \overline{L}.

(c) There exists a non-empty open subset Ω of A_{k}^{n} such that for each θ ∈ Ω(k), the Manin obstruction to the Hasse principle (resp. to weak approximation) is the only one for smooth and projective models of the fibre V_{θ}.

Then we have the following statements:

(i) The Manin obstruction to the Hasse principle (resp. to weak approximation) is the only one for a smooth and projective model X of V.

(ii) Assume further that X(A_{k})^{Br} ≠ ∅. Let Ω′ be any non-empty open subset of A_{k}^{n}. Then there exists a k-point θ of Ω′ such that a smooth and projective model of the fibre V_{θ} has a k-point.

In particular, if V_{L} is smooth and projective, and X(A_{k})^{Br} ≠ ∅, then p(V (k)) is Zariski dense in A_{k}^{n}.

Proof. (i) is precisely [Ha97], Thm. 3.2.1. We prove (ii) using the same induction argument. We can find a k-variety V′ equipped with a k-morphism p′ : V′ → A_{k}^{n} \F such that p′ is projective and its generic fibre is smooth and L-birational to V_{L}. We can also suppose that p′ satisfies the same assumptions as p (see [H97], proof of Thm. 3.2.1). In the case n = 1 it follows immediately from [H97], proof of Prop. 3.1.1 (see also [H94], proof of Thm. 4.2.1) that if X(A_{k})^{Br} ≠ ∅, then the image of V′(k) by p′ is Zariski dense in A_{k}^{1}. This proves (ii) for n = 1 via Nishimura’s lemma.

Assume that (ii) is proved for any integer less than n. As in [H97], proof of Thm. 3.2.1, we can find a k-point M_{0} in A_{k}^{n} such that the k-morphism π : V′ \ p′^{−1}(M_{0}) → P_{k}^{n−1} obtained by composing p′ with the map A_{k}^{n} \ {M_{0}} → P_{k}^{n−1}, has the property that the restriction of π to π^{−1}(A_{k}^{n−1}) satisfies the assumptions of the proposition. (Recall that V′ is k-birationally equivalent to V.) If X(A_{k})^{Br} ≠ ∅, then the induction hypothesis implies that for any non-empty Zariski open subset U ⊂ P_{k}^{n−1} there exists a k-point m of U (corresponding to an affine line D passing through M_{0}) such that a smooth and projective model Z of π^{−1}(m) = p′^{−1}(D) has a k-point. This implies that Z(A_{k})^{Br} ≠ ∅. Applying the case n = 1 to the restriction of p′ to p′^{−1}(D) (this is possible by [H97], Lemma 3.2.2), we obtain that each non-empty open subset of D contains a k-rational point θ such that a smooth and projective model of the fibre p′^{−1}(θ) has a k-point. This proves (ii) for p′, hence for p by Nishimura’s lemma. QED

Proof of Theorem 2.1. The case rk f ≤ 1 is known ([H97], Prop. 3.1.1, here condition (3) is not required). We now assume that rk f = 2.

If Br X_{k̄(η)} = 0, then we take U = X. In this case Br \overline{U} is trivial since it is a subgroup of Br X_{k̄(η)} by Grothendieck’s theorem.

If Br X_{k̄(η)} ≠ 0 we choose a vertical module M associated to f with �_{ω}^{2}(k, M) = 0, which is possible by condition (3). In this case we define U to be the dense open subset of X such that M = M_{U}.

We observe that k̄[U]^{∗} = k̄^{∗}. Indeed, the generic fibre U_{k̄(η)} = X_{k̄(η)} of \overline{U} → P1 ; k̄ is proper and geometrically integral, hence every regular function on U_{k̄(η)} comes from a rational function on P1 ; k̄. However, no such function can be regular on \overline{U} because of surjectivity of the morphism \overline{U} → P1 ; k̄.

We now claim that condition (2) implies the finiteness of Br U_{k̄(η)}. Indeed, Br k̄(η) = 0 by Tsen’s Theorem; H^{1}(k̄(η), Pic X\overline{K}) is finite because Pic X\overline{K} is torsion-free; and Br X\overline{K} is finite. The exact sequence (1) now shows that Br X_{k̄(η)} = Br U_{k̄(η)} is finite. Since Br \overline{U} injects into Br U_{k̄(η)} we conclude that Br \overline{U} is also finite.

We also note that Pic \overline{U} is torsion-free, because Pic U_{k̄(η)} and M = Ker [Pic \overline{U} → Pic U_{k̄(η)}] are torsion-free.

Let {Q_{v}} be an adelic point of X(A_{k})^{Br} , and Σ be a finite set of places of k containing all the places where we want to approximate. By [CS00], Prop. 1.1, there exists an adelic point {M_{v}} in U(A_{k})^{Br} . If we deal with weak approximation, then the points M_{v} for v ∈ Σ can be chosen to be as close as we wish to the corresponding points Q_{v}.

Let τ be the natural injection of Γ-modules M ↪ Pic \overline{U}. By Proposition 2.3 there exist a vertical torsor Y → U and an adelic point {P_{v}} ∈ Y (A_{k})^{Br_{1} Y} that is mapped to {M_{v}} by the structure map Y → U. Let g : Z → U be a relative smooth compactification of Y → U. Recall that Br \overline{U} injects into Br X_{k̄(η)}, thus condition (3) implies that Q_{τ} = 0 in the notation of Theorem 1.8 (with X replaced by U). Therefore by Theorem 1.8 the map Br U → Br Z/Br_{1} Z is surjective. Let α be an arbitrary element of Br Z. Then we can write α = β + γ, where β ∈ Im [Br U → Br Z] and γ ∈ Br_{1} Z. Since {M_{v}} ∈ U(A_{k})^{Br} , the projection formula gives that ∑_{v∈Ω} j_{v}(β(P_{v})) = 0. On the other hand, ∑_{v∈Ω} j_{v}(γ(P_{v})) = 0 since {P_{v}} ∈ Y (A_{k})^{Br_{1} Y} and Br_{1} Z ⊂ Br_{1} Y. This proves that {P_{v}} ∈ Y (A_{k})^{Br Z}.

The assumption rk f = 2 allows us to apply [S96], Cor. 1.4 (see also [CS00]). This yields an open subset W ⊂ A_{k}^{n} with codim (A_{k}^{n}\W) ≥ 2, equipped with a morphism W → P_{k}^{1} such that Y is k-birationally equivalent to Y′ := U ×_{P_{k}^{1}} W. Moreover, the following properties hold: the projection p : Y′ → W is surjective, its generic fibre is projective, and all fibres of p are split. Furthermore, the restriction of p to a general line in An ; k̄ has a section because we assumed that the generic fibre of f had a k̄(η)-point. Let Y ′ ; \overline{L} be the generic geometric fibre of p over the algebraic closure \overline{L} of L = k(A_{k}^{n}). Then the assumptions that Pic X\overline{K} is torsion-free and Br X\overline{K} is finite imply that Pic Y ′ ; \overline{L} is torsion-free and Br Y ′ ; \overline{L} is finite. Let V be a smooth and projective compactification of Y. By [H97], Thm. 3.2.1, on V the Brauer–Manin obstruction to the Hasse principle (or to weak approximation if we assumed the same property for the k-fibres of f) is the only one. Since {P_{v}} ∈ Y (A_{k})^{Br Z} ⊂ V (A_{k})^{Br} we can use Proposition 2.4 (ii). Thus p(Y′(k)) is Zariski dense in W, which implies U(k) ≠ ∅. If we deal with weak approximation, then there exists a k-point P on Y that is arbitrary close to P_{v} for v ∈ Σ. Projecting the point P to U, we obtain a k-point on X, which is as close to Q_{v} for v ∈ Σ in the k_{v}-topology as we wish. QED

There is a natural way to construct an open set U with properties (a), (b) and (c). For any closed point P of P_{k}^{1} the fibre X_{P} contains an irreducible component of multiplicity 1. (Indeed, X_{K} has a k̄(η)-point and X is proper, thus f has a k̄-section. Since X is regular, the intersection point of this section with X_{P} must be regular on X_{P} ([S96], Lemma 1.1 (b)). In particular, this point belongs to an irreducible component of multiplicity 1.) For every closed point P choose an irreducible component E_{P} ⊂ X_{P} of multiplicity 1. Whenever there is a geometrically irreducible component we choose it as E_{P}. Let F_{P} be the union of all the remaining components of X_{P}, and let G_{P} be the singular locus of E_{P}. Define U ⊂ X as the complement to the (finite) union of the F_{P} and the G_{P} for all the closed points P ∈ P_{k}^{1}. This dense open subset U ⊂ X satisfies properties (a), (b) and (c). It is clearly enough to consider so constructed open subsets to get all possible vertical modules.

Let Δ ⊂ P_{k}^{1} be the union of the closed points P such that the fibre X_{P} is not split (equivalently, the fibre U_{P} is not geometrically integral). For each P ∈ Δ we denote by k_{P} the integral closure of k(P) in the function field k(E_{P}) of E_{P}.

The inclusion Pic P1 ; k̄ ↪ M = M_{U} gives rise to the exact sequence

0 → Z → M → N → 0.  (13)

Here the Γ-module N is isomorphic to the module of characters of the torus

∏_{P ∈Δ} R_{k(P)/k}(R_{k_{P} /k(P)}^{1}G_{m}),

where R_{k(P)/k} is the Weil restriction of scalars from k(P) to k, and R_{k_{P} /k(P)}^{1}G_{m} is the kernel of the norm map R_{k_{P} /k(P)}G_{m} → G_{m}. For any finite extension k′/k we write Z[k′/k] for the induced Γ-module Z[Γ/Γ_{k′}], where Γ_{k′} = Gal (k̄/k′). Then the Γ-module N is the direct sum of the natural ‘diagonal’ quotients of Z[k_{P} /k] by Z[k(P)/k], for P ∈ Δ.

2.2. Variants and applications

We now give a sufficient condition for the vanishing of �_{ω}^{2}(k, M) in the situation of Theorem 2.1. For any finite field extension k′/k we write H^{1}(k′/k, Q/Z) for the kernel of the restriction map H^{1}(k, Q/Z) → H^{1}(k′, Q/Z). If k′ and k′′ are subfields of k̄, then k′k′′ is the smallest subfield of k̄ containing k′ and k′′.

Proposition 2.5 Let X be a smooth, projective, and geometrically integral variety over a number field k, and let f : X → P_{k}^{1} be a dominant morphism with geometrically integral generic fibre. Assume that two k-fibres of f, say X_{P} and X_{Q}, are not split, and all the other closed fibres are split. Assume also that the fibre X_{P} (resp. X_{Q}) contains a multiplicity 1 component E_{P} (resp. E_{Q}) such that if k_{P} (resp. k_{Q}) is the integral closure of k in k(E_{P}) (resp. in k(E_{Q})), then

(i) �_{ω}^{2}(k, Z[k_{P} /k]/Z) = �_{ω}^{2}(k, Z[k_{Q}/k]/Z) = 0, and

(ii) for some k-embeddings k_{P} ⊂ k̄, k_{Q} ⊂ k̄, the following natural map is surjective:

H^{1}(k_{P} /k, Q/Z) ⊕ H^{1}(k_{Q}/k, Q/Z) ⟶ H^{1}(k_{P} k_{Q}/k, Q/Z).

Then �_{ω}^{2}(k, M) = 0 for the vertical module M constructed from E_{P} and E_{Q}. Moreover, (i) and (ii) hold as long as k_{P} and k_{Q} are cyclic extensions of k.

Proof. From the exact sequence (13) we obtain a commutative diagram with exact rows

H^{1}(k, N) ⟶ H^{1}(k, Q/Z) ⟶ H^{2}(k, M) ⟶ H^{2}(k, N)
↓  ↓  ↓  ↓
⨁_{v∈Ω_{k}} H^{1}(k_{v}, N) ⟶ ⨁_{v∈Ω_{k}} H^{1}(k_{v}, Q/Z) ⟶ ⨁_{v∈Ω_{k}} H^{2}(k_{v}, M) ⟶ ⨁_{v∈Ω_{k}} H^{2}(k_{v}, N)

From the description of N given above it is clear that N = Z[k_{P} /k]/Z ⊕ Z[k_{Q}/k]/Z. Thus �_{ω}^{2}(k, N) = 0 by condition (i). Hence any non-zero α ∈ �_{ω}^{2}(k, M) is the image of an element β ∈ H^{1}(k, Q/Z) = Hom(Γ, Q/Z). We have

H^{1}(k, N) = H^{1}(k_{P} /k, Q/Z) ⊕ H^{1}(k_{Q}/k, Q/Z).

The restriction of the differential H^{1}(k, N) → H^{2}(k, Z) = H^{1}(k, Q/Z) from (13) to the subgroup H^{1}(k_{P} /k, Q/Z) ⊂ H^{1}(k, N) is the natural inclusion of H^{1}(k_{P} /k, Q/Z) into H^{1}(k, Q/Z) (and the same for Q). Indeed, the exact sequence of Γ-modules

0 → Z → Z[k_{P} /k] → Z[k_{P} /k]/Z → 0,

where the second arrow is the diagonal embedding, naturally maps to the exact sequence (13). The desired fact now follows from the commutative diagram where the left hand arrow is the natural inclusion:

H^{1}(k, N)  → H^{2}(k, Z)
↑  ||
H^{1}(k, Z[k_{P} /k]/Z) ↪ H^{2}(k, Z)

Because of the assumption (ii) a character of Γ belongs to the image of H^{1}(k, N) if and only if its restriction to Γ′ := Gal (k̄/k_{P} k_{Q}) = Gal (k̄/k_{P}) ∩ Gal (k̄/k_{Q}) is trivial. Since α ≠ 0 the restriction of β to Γ′ is non-trivial. By Tchebotarev’s density theorem (cf. [H94], Prop. 2.2.1), there exist infinitely many places v of k such that the image β_{v} of β in H^{1}(k_{v}, Q/Z) is not zero, but k_{v} is a direct summand of k_{P} k_{Q} ⊗_{k} k_{v}. For such a place v we have H^{1}(k_{v}, N) = 0, hence β_{v} does not come from H^{1}(k_{v}, N). This contradicts the fact that α_{v} = 0 for almost all v.

Now suppose that k_{P} /k and k_{Q}/k are cyclic. Then �_{ω}^{2}(k, N) = 0 by [CS77], Prop. 2 and Prop. 6, [CS87b], Prop. 9.1. Since Γ/Γ′ is a subgroup of the finite abelian group Gal (k_{P} /k) × Gal (k_{Q}/k), any character of Γ/Γ′ extends to this group. Therefore the condition (ii) is also satisfied. QED

Remark 2.6 Condition (i) holds as long as the degrees [k_{P} : k] and [k_{Q} : k] are prime numbers. It also holds if the Galois closures of k_{P} /k and k_{Q}/k are metacyclic (loc. cit.).

Here is a particular case of Theorem 2.1 with conditions that are easy to check.

Theorem 2.7 Let X be a smooth, projective, and geometrically integral variety over a number field k, and let f : X → P_{k}^{1} be a dominant morphism with geometrically integral generic fibre. Suppose that the following conditions hold:

(1) two k-fibres of f, say X_{P} and X_{Q}, are not split, and all the other closed fibres are split;

(2) Pic X\overline{K} is torsion-free, Br X\overline{K} is finite, and X_{K} has a k̄(η)-point;

(3) the fibre X_{P} (resp. X_{Q}) contains a multiplicity 1 component E_{P} (resp. E_{Q}) such that if k_{P} (resp. k_{Q}) is the integral closure of k in k(E_{P}) (resp. in k(E_{Q})), then [k_{P} : k] (resp. [k_{Q} : k]) is a prime number.

Then if the Manin obstruction is the only obstruction to the Hasse principle or to weak approximation for the k-fibres of f in a dense open subset of P_{k}^{1}, then the same property holds for X.

Proof. In view of the previous remark we only need to check condition (ii) of Proposition 2.5. We fix k-embeddings of k_{P} and k_{Q} into k̄. Let p = [k_{P} : k], q = [k_{Q} : k]. If k_{P} and k_{Q} are Galois (hence cyclic) extensions of k, we can apply Proposition 2.5. If k_{Q} is not Galois and p ≠ q, then a non-trivial cyclic subextension of k_{P} k_{Q}/k is either k_{P}, or is linearly disjoint from k_{P} and k_{Q}. The latter case is impossible because neither p^{2} nor q^{2} divides [k_{P} k_{Q} : k] = pq. Thus Proposition 2.5 still applies in this case. From now on we assume that p = q. We need the following lemma:

Lemma 2.8 Let D be a subgroup of the symmetric group S_{p} not isomorphic to Z/p. Then Hom(D, Z/p) = 0.

Proof. Let φ : D → Z/p be a non-zero homomorphism. If σ ∈ D is not a p-cycle, then the order of σ is prime to p, hence φ(σ) = 0. In particular, D contains a p-cycle c such that φ(c) ≠ 0. Suppose that D contains an element σ which does not belong to the cyclic group < c >. There exists an integer m, 0 < m < p, such that σc^{m} has a fixed point. Therefore σc^{m} is not a p-cycle, hence φ(σc^{m}) = 0. If σ is not a p-cycle, then φ(σ) = 0 and we have a contradiction with φ(c) ≠ 0. Thus any non-trivial element of D is a p-cycle. This leads to a contradiction because if σ is a p-cycle not in < c >, then σc is not a p-cycle. QED.

We resume the proof of Theorem 2.7. If k_{P} is Galois and k_{Q} is not Galois, then the Galois closure k′ of k_{Q} has a non-abelian Galois group D ⊂ S_{p}. Since Hom(D, Z/p) = 0 by Lemma 2.8, k_{P} and k′ are linearly disjoint, that is, Gal (k_{P} k′/k) = Z/p × D. Thus the only normal subgroup of Gal (k_{P} k′/k) of index p is D. This means that the only cyclic subextension of k_{P} k_{Q}/k is k_{P}, and we conclude by applying Proposition 2.5.

Finally, if neither k_{P} nor k_{Q} is Galois, the assumption (ii) of Proposition 2.5 fails only if k_{P} and k_{Q} are linearly disjoint, and k_{P} k_{Q} contains a cyclic extension E/k of degree p. Then Ek_{P} = Ek_{Q} = k_{P} k_{Q}, and Gal (Ek′/k) = Z/p × D. Define H_{P} = Gal (Ek′/k_{P}), H_{Q} = Gal (Ek′/k_{Q}). Then Gal (Ek′/E) = D, hence H_{P} ∩ H_{Q} ⊂ D. The projection H_{P} → D is an isomorphism (H_{P} does not contain Z/p since k_{P} is not contained in k′). Therefore, Hom(H_{P} , Z/p) = 0 by Lemma 2.8, so that the projection H_{P} → Z/p is zero. Hence we obtain H_{P} = D, which is a contradiction because H_{P} is not normal in Gal (Ek′/k) (recall that k_{P} /k is not Galois). Theorem 2.7 is proved. QED

We have the following variant of Theorem 2.1 applicable in some cases when rk f = 3.

Theorem 2.9 Let X be a smooth, projective and geometrically integral variety over a number field k, and let X → P_{k}^{1} be a dominant morphism with geometrically integral generic fibre. Assume that rk f ≤ 3 and every non-split fibre X_{P} contains a multiplicity 1 component E_{P} such that the integral closure k_{P} of k(P) in k(E_{P}) is a quadratic extension of k(P). Assume also the hypothesis (2) of Theorem 2.1. Then we have the same conclusion as in Theorem 2.1 unless rk f = 3 and there exists a biquadratic field extension k(\sqrt{a}, \sqrt{b}) and non-split closed fibres X_{P_{1}} and X_{P_{2}} with k(P_{1}) = k, k(P_{2}) = k(\sqrt{a}), k_{P_{1}} = k(\sqrt{a}) and k_{P_{2}} = k(\sqrt{a}, \sqrt{b}).

Proof. The beginning of the proof is similar to that of Theorem 2.1, so we only indicate the places where this proof is different. We construct U ⊂ X from the components E_{P}, then the variety W is the punctured affine cone over the complement to a closed subset of codimension 2 in a smooth projective quadric of dimension 4 (see [S90] or [CS00]). We have a vertical torsor Y such that Y (A_{k}) ≠ ∅ and Y is k-birational to Y′ := U ×_{P_{k}^{1}} W. Thus W (k) ≠ ∅ by the Minkowski–Hasse Theorem. Hence W is k-isomorphic to an open subset of the affine space whose complementary set is of codimension at least 2.

To conclude the proof it is sufficient to check that �_{ω}^{2}(k, M) = 0 for the vertical module M = M_{U}. Set n = rk f and denote by l_{i}, l′_{i}, 1 ≤ i ≤ n, the two conjugate components corresponding to the E_{P} ’s. The Weyl group W (B_{n}) of the root system B_{n} is the semi-direct product of (Z/2)^{n} (permutations interchanging l_{i} with l′_{i}) and S_{n} (permutations of the indices of the l_{i} and l′_{i}). This action gives M and N the structure of a W (B_{n})-module. In [KST], Def. 1.2, the W (R)-modules Q(R) and M(R) are defined for root systems R = B_{n} and C_{n}. Using the exact sequence (13) we see that N = Q(B_{n}) and M = M(C_{n}) as W -modules (see [KST], p. 29).

The action of Γ on the set of the l_{i} and l′_{i} defines a homomorphism Γ → W (B_{n}). Let G be its image. Thm. 1.22 of [KST] describes �_{ω}^{2}(k, M(C_{n})) = �_{ω}^{2}(G, M(C_{n})) in terms of the subgroup G ⊂ W (B_{n}). (Note that by [KST], Prop. 1.19, we have �_{ω}^{2}(G, M(C_{n})) = �_{ω}^{2}(G, Q(C_{n})).) It follows from Cor. 4.14 of [KST] that the assumption n ≤ 3 implies that �_{ω}^{2}(k, M(C_{n})) ≠ 0 only in the exceptional case described in the statement of the theorem. QED

2.3. Examples: conic bundles over the plane

Let V be the conic bundle over A_{k}^{2} defined in P_{k}^{2} × A_{k}^{2} by the equation

a_{0}(x, t)X_{0}^{2} + a_{1}(x, t)X_{1}^{2} + a_{2}(x, t)X_{2}^{2} = 0, (X_{0}, X_{1}, X_{2}) ∈ P_{k}^{2}, (x, t) ∈ A_{k}^{2}, (14)

where the a_{i}(x, t) are non-zero polynomials in x and t. By a simple change of variables one can arrange that the a_{i}(x, t) are pairwise coprime; we assume this from now on. Then for any given value of t at most one a_{i}(x, t_{0}) vanishes. Let us now introduce some notation.

For i = 0, 1, 2 let d_{i} := deg_{t} a_{i}(x, t), and let b_{i}(x) be the leading coefficient of a_{i}(x, t) as a polynomial in t. Then we can write

a_{i}(x, t) = b_{i}(x)t^{d_{i}} + terms of lower degree in t.

Let Δ_{l} be the union of closed points P ∈ A_{k}^{1} such that

a_{l}(x, P) = 0 and a_{m}(x, P)a_{n}(x, P) ∈ \overline{k(P)}^{∗2} \ k(P)^{∗2},

where {l, m, n} = {0, 1, 2}. Let Δ = Δ_{0} ∪ Δ_{1} ∪ Δ_{2}, and let N be the cardinality of Δ(k̄).

Proposition 2.10 Let V be the subvariety of P_{k}^{2} × A_{k}^{2} defined by equation (14). In the above notation assume the following conditions:

(i) deg_{x} a_{0}(x, t) + deg_{x} a_{1}(x, t) + deg_{x} a_{2}(x, t) ≤ 4;

(ii) N = 1; or N = 2 and we are not in the exceptional case E_{2}; or N = 3, d_{0} ≡ d_{1} ≡ d_{2} (mod 2), and we are not in the exceptional case E_{3}.

Then the Manin obstruction to the Hasse principle and weak approximation is the only one for any smooth and projective model of V.

The exceptional cases are defined as follows:

E_{2} : d_{l} ≢ d_{m} ≡ d_{n} (mod 2), where {l, m, n} = {0, 1, 2}, and there exists a biquadratic extension k(\sqrt{a}, \sqrt{b}) and a closed point P ∈ A_{k}^{1} with residue field k(P) = k(\sqrt{a}) such that we have

(1) −ab_{m}(x)b_{n}(x) ∈ k(x)^{∗2}, and

(2) a_{p}(x, P) = 0, −ba_{q}(x, P)a_{r}(x, P) ∈ k(P)(x)^{∗2}, where {p, q, r} = {0, 1, 2};

E_{3} : there exists a biquadratic extension k(\sqrt{a}, \sqrt{b}) and closed points P_{1}, P_{2} ∈ A_{k}^{1} with residue fields k(P_{1}) = k, k(P_{2}) = k(\sqrt{a}), such that we have

(1) a_{l}(x, P_{1}) = 0, −aa_{m}(x, P_{1})a_{n}(x, P_{1}) ∈ k(x)^{∗2}, where {l, m, n} = {0, 1, 2}, and (2) a_{p}(x, P_{2}) = 0, −ba_{q}(x, P_{2})a_{r}(x, P_{2}) ∈ k(P_{2})(x)^{∗2}, where {p, q, r} = {0, 1, 2}.

Proof. Consider the morphism V → A_{k}^{1} given by the projection to the coordinate t. This morphism naturally extends to a surjective morphism V′ → P_{k}^{1}, where the fibre at ∞ is given in P_{k}^{2} × A_{k}^{1} by the equation:

b_{0}(x)X_{0}^{2} + b_{1}(x)X_{1}^{2} + b_{2}(x)X_{2}^{2} = 0

if the d_{i} have the same parity. In the opposite case we can assume that d_{p} ≢ d_{q} ≡ d_{r} (mod 2) for some {p, q, r} = {0, 1, 2}. Then the fibre at ∞ is given by

b_{q}(x)X_{q}^{2} + b_{r}(x)X_{r}^{2} = 0.

Let P ∈ Δ, say P ∈ Δ_{0}. Then (14) shows that the fibre V_{P} contains an irreducible component of multiplicity 1 defined over the quadratic extension of k(P) given by the square root of a_{1}(x, P)a_{2}(x, P). The closed fibres V_{Q} at the closed points Q ∉ Δ ∪ {∞} are geometrically integral.

The restricted morphism V′_{smooth} → P_{k}^{1} is still surjective. It extends to a morphism f : X → P_{k}^{1}, where X is a smooth and projective model of V (by Hironaka’s theorem). Split fibres of V′_{1} → P_{k}^{1} give rise to split fibres of X → P_{k}^{1}.

The generic fibre X_{k(t)} of f : X → P_{k}^{1} is a conic bundle surface over P_{k(t)}^{1}. In particular, X_{k(t)} is a rational surface, defined over K = k(t), hence its geometric Brauer group Br X\overline{K} is zero and its geometric Picard group Pic X\overline{K} is torsion-free. Moreover, X_{k(t)} has a k̄(t)-point since any smooth conic over k̄(t) has a k̄(t)-point by Tsen’s theorem. In view of condition (i) each smooth k-fibre X_{m} of f is a conic bundle over P_{k}^{1} with at most 5 degenerate fibres. Then by the results of [CSS87],

[CT] and [SS] the Manin obstruction to the Hasse principle and weak approximation is the only one for X_{m}.

To apply Theorem 2.9 it remains to check that for N = 2 or N = 3 we are not in the exceptional case of that theorem. This follows from the description of the fibres at the points of Δ ∪ {∞} given in the beginning of this proof. QED.

Remark 2.11 One can give a variant of the previous proposition, where a_{0}(x, t) and a_{1}(x, t) are in k[t], and a_{2}(x, t) is a product of two irreducible polynomials of degrees 2 and 4 in x. The arithmetic condition on the k-fibres is ensured by a theorem of Swinnerton-Dyer [SD], see [S01], Thm. 7.4.1. However, to satisfy the irreducibility condition one would need a slightly stronger version of Theorem 2.1, namely, a version only requiring that the k-fibres in a Hilbertian subset of P_{k}^{1}(k) have the property that the Manin obstruction is the only one. Such a version can be easily proved along the same lines as Theorem 2.1 (cf. [H97], p. 158).

2.4. Examples: equations of norm type

Let C be a geometrically integral, smooth and proper curve, and let π : C → P_{k}^{1} be a finite morphism of degree n. Write π^{−1}(A_{k}^{1}) = Spec A, then A is a finitely generated k[t]-module. Since π is flat, A is projective, and hence is free because k[t] is principal. Thus we can choose a basis ω_{1}, …, ω_{n} of the k[t]-module A. Let L = k(C); it is also the field of fractions of A. Let N_{t}(z_{1}, …, z_{n}) be the norm of z_{1}ω_{1} + … + z_{n}ω_{n} for the finite field extension L(z_{1}, …, z_{n})/k(t)(z_{1}, …, z_{n}). It is clear that N_{t}(z_{1}, …, z_{n}) is homogeneous of degree n in variables z_{1}, …, z_{n}.

Let g(t) ∈ k[t] be a polynomial with leading coefficient c ≠ 0. Consider the affine k-variety V defined by the equation in variables z_{1}, …, z_{n}, t:

N_{t}(z_{1}, …, z_{n}) = g(t).  (15)

Set formally g(∞) = 0 if n does not divide deg g, and g(∞) = c otherwise. For a closed point P ∈ P_{k}^{1} we denote by m_{P} the highest common factor of the ramification indices of of π at closed geometric points in the fibre C_{P} (for example, m_{P} = 1 if π is unramified at some point over P). Define Δ as the union of closed points P ∈ P_{k}^{1} such that g(P) = 0, and there is no k(P)-point Q ∈ C_{P} such that π is unramified at Q.

Proposition 2.12 Let V be an affine variety defined by equation (15). In the above notation assume that

(i) Δ consists of at most two k-points of P_{k}^{1};

(ii) for any P ∈ Δ there is a closed point Q ∈ C_{P} such that π is unramified at Q, and the residue field k(Q) has prime degree over k(P) = k; (iii) for any closed point P ∈ P_{k}^{1} we have m_{P} = 1. Then the Manin obstruction to the Hasse principle and weak approximation is the

only one for a smooth and projective model of V.

Proof. The morphism V_{smooth} → A_{k}^{1} given by the coordinate t extends to a morphism f : X → P_{k}^{1}, where X is a smooth and projective model of V. Let K = k(t). The generic fibre X_{K} of f is a smooth compactification of a principal homogeneous space V_{K} of the K-torus T_{K} = R_{L/K}^{1}G_{m}. In particular, X_{K} is a rational variety (over \overline{K}), and V_{K} has a k̄(t)-point because H^{1}(k̄(t), T_{K}) = 0 by Steinberg’s Theorem (recall that k̄(t) is a C_{1}-field). Thus condition (2) of Theorem 2.7 is satisfied.

Let P be a closed point of P_{k}^{1} such that g(P) ≠ 0. Condition (iii) implies that the fibre V_{P} is geometrically integral. Indeed, V_{P} ×_{k(P)} \overline{k(P)} is the product of an affine space and an affine variety given by

u_{1}^{m_{1}} … u_{r}^{m_{r}} = a,

where a ≠ 0. By condition (iii) the highest common factor of the multiplicities m_{i} is 1. Hence the \overline{k(P)}-variety given by the displayed equation is integral.

If P is a closed point of P_{k}^{1} not in Δ but such that g(P) = 0, then the fibre V_{P} is split. Indeed, it follows from the definition of Δ that in this case the k(P)-variety V_{P} contains a geometrically irreducible component of multiplicity 1. Now (i) shows that condition (1) of Theorem 2.7 holds.

By (ii) the fibre X_{P} for P ∈ Δ contains an irreducible component E_{P} of multiplicity 1 such that the integral closure k_{P} of k in k(E_{P}) is of prime degree over k. We use these components to construct a vertical module M. It is a theorem of Colliot-Thélène and Sansuc that (smooth compactifications of) the principal homogeneous spaces of k-tori have the property that the Manin obstruction to the Hasse principle and weak approximation is the only one (see, e.g. [S01], Thm. 6.3.1). For m ∈ k such that g(m) ≠ 0 and π : C → P_{k}^{1} is not ramified at m, the fibre V_{m} is a principal homogeneous space of a k-torus. We can now conclude by applying Theorem 2.7. QED

Remark 2.13 The assumption (iii) of Proposition 2.12 can be satisfied only if the degree n of π is at least 3.

Explicit example. Let C be the smooth and proper model of the curve given by x^{4} + p(t)x + a = 0, where the degree of the non-constant polynomial p(t) is divisible by 3, and a ∈ k^{∗}. Then π : C → P_{k}^{1} is unramified at a k-point over ∞. We apply Proposition 2.12 to study weak approximation on the variety V given by

N_{t}(z_{1}, …, z_{4}) = ct(t − 1),

where c ∈ k^{∗}. (V has a smooth k-point t = z_{i} = 0.) One checks easily that conditions (i) and (iii) of Proposition 2.12 are satisfied. Condition (ii) is satisfied as long as we assume that none of the polynomials x^{4} + p(0)x + a = 0 and x^{4} + p(1)x + a = 0 is irreducible. (The weirdness of these conditions is partly due to the fact that we need an example not covered by the methods already described in literature. For example, we could replace 4 by 3, and require the degree of p(x) to be even, but then the smooth k-fibres satisfy the Hasse principle and weak approximation, so that we would be able to conclude using previously known results.)

Acknowledgements. The second named author thanks the University of Strasbourg and the Max-Planck-Institut für Mathematik in Bonn for hospitality. Both authors are grateful to the organizers of the conference “Higher dimensional varieties and rational points” in Alfréd Rényi Institute in Budapest.

REFERENCES

[CF] J.W.S. Cassels, A. Fröhlich (ed.). Algebraic number theory, Academic Press, London and New-York, 1967.

[CT] J.-L. Colliot-Thélène. Surfaces rationnelles fibrées en coniques de degré 4. In: Séminaire de Thorie des Nombres, Paris 1988–1989, Progr. Math. 91, Birkhäuser, Boston 1990, 43–55.

[CS77] J.-L. Colliot-Thélène et J.-J. Sansuc. La R-équivalence sur les tores. Ann. Sci. École Norm. Sup. 10 (1977) 175–230.

[CS87a] J.-L. Colliot-Thélène et J.-J. Sansuc. La descente sur les variétés rationnelles, II. Duke Math. J. 54 (1987) 375–492.

[CS87b] J.-L. Colliot-Thélène and J.-J. Sansuc. Principal homogeneous spaces under flasque tori; applications. J. Algebra 106 (1987) 148–205.

[CSS87] J.-L. Colliot-Thélène, J.-J. Sansuc and Sir Peter Swinnerton-Dyer. Intersections of two quadrics and Châtelet surfaces. J. reine angew. Math. 373 (1987) 37–107; 374 (1987) 72–168.

[CS00] J.-L. Colliot-Thélène and A. N. Skorobogatov. Descent on fibrations over P_{k}^{1} revisited. Math. Proc. Camb. Phil. Soc. 128 (2000) 3 383-393.

[EGA4] A. Grothendieck. Élements de géométrie algébrique. Étude locale des schémas et des morphismes de schémas (EGA IV), Publ. Math. IHES, 1964-1967.

[G] A. Grothendieck. Le groupe de Brauer, III : exemples et compléments. In: Dix exposés sur la cohomologie des schémas, Masson-North-Holland, Amsterdam 1968.

[H94] D. Harari. Méthode des fibrations et obstruction de Manin. Duke Math. J. 75 (1994) 221–260.

[H97] D. Harari. Flèches de spécialisation en cohomologie étale et applications arithmétiques. Bull. Soc. math. France 125 (1997) 143–166.

[KST] B. Kunyavskiǐ, A.N. Skorobogatov and M.A. Tsfasman. Del Pezzo surfaces of degree four. Mém. Soc. math. France 37 (1989).

[M] J.-S. Milne. Étale Cohomology, Princeton University Press 33, Princeton 1980.

[SS] P. Salberger and A.N. Skorobogatov. Weak approximation for surfaces defined by two quadratic forms. Duke Math. J. 63 (1991) 517–536.

[S90] A. N. Skorobogatov. Arithmetic on certain quadric bundles of relative dimension 2, I. J. reine angew. Math. 407 (1990) 57–74.

[S96] A. N. Skorobogatov. Descent on fibrations over the projective line. Amer. J. Math. 118 (1996) 905–923.

[S01] A. N. Skorobogatov. Torsors and rational points, Cambridge Univ. Press, Cambridge, 2001.

[SD] Sir Peter Swinnerton-Dyer. Rational points on some pencils of conics with 6 singular fibres. Ann. Fac. Sci. Toulouse 8 (1999) 331–341.

[V] V. E. Voskresenskiǐ. Algebraic groups and their birational invariants, Amer. Math. Soc., Providence, 1998.

D. Harari: DMA, École Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France. e-mail: harari@dma.ens.fr

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