ON THE BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES

JÖRG JAHNEL AND DAMARIS SCHINDLER

Abstract. We show that, for every integer 1 ≤ d ≤ 4 and every finite set S of places, there exists a degree d del Pezzo surface X over ℚ such that Br(X)/ Br(ℚ) ≅ ℤ/2ℤ and the non-trivial Brauer class has a nonconstant local evaluation exactly at the places in S. For d = 4, we prove that in all cases, with the exception of S = {∞}, this surface may be chosen diagonalisably over ℚ.

1. Introduction

A del Pezzo surface is a smooth, proper algebraic surface X over a field K with ample anti-canonical sheaf 𝒦 ^{−1}. Over an algebraically closed field, every del Pezzo surface of degree d ≤ 7 is isomorphic to P^{2}, blown up in (9 − d) points in general position [Man, Theorem 24.4.iii)].

According to the adjunction formula, a smooth complete intersection of two quadrics in P^{4} is del Pezzo. The converse is true, as well. For every del Pezzo surface of degree four, its anticanonical image is the complete intersection of two quadrics in P^{4} [Do, Theorem 8.6.2].

For an arbitrary proper variety X over ℚ, the Brauer-Manin obstruction is a phenomenon that can explain failures of weak approximation or even the Hasse principle. Its mechanism works as follows.

Let p be any prime number. The Grothendieck-Brauer group is a contravariant functor from the category of schemes to the category of abelian groups. In particular, for an arbitrary scheme X and a ℚ_{p}-rational point x: Spec ℚ_{p} → X on it, there is a restriction homomorphism x^{∗}: Br(X) → Br(ℚ_{p}) ≅ ℚ/ℤ. For a Brauer class α ∈ Br(X), we call

ev_{α,p}: X(ℚ_{p}) ⟶ ℚ/ℤ , x ↦ x^{∗}(α) ,

the local evaluation map, associated to α. Analogously, for the real place, there is the local evaluation map ev_{α,∞}: X(ℝ) → \frac{1}{2}ℤ/ℤ.

Date: August 6, 2016.

2010 Mathematics Subject Classification. Primary 14F22; Secondary 14G25, 14J26, 11G35.

Key words and phrases. Degree four del Pezzo surface, Brauer-Manin obstruction.

2 J. JAHNEL AND D. SCHINDLER

Let us use the notation Ω for the set of all places of ℚ, i.e. for the union of all finite primes together with ∞. The local evaluation maps are continuous with respect to the p-adic, respectively real, topologies on X(ℚ_{ν}), for ν ∈ Ω. Moreover, it is well-known that ev_{α,ν} is constant for all but finitely many places. Thus, only adelic points x = (x_{ν})_{ν∈Ω} ∈ X(𝔸_{ℚ} ) satisfying

(1.1)  ∑_{ν∈Ω} ev_{α,ν}(x_{ν}) = 0 ∈ ℚ/ℤ

may possibly be approximated by ℚ-rational points.

We say that a Brauer class α ∈ Br(X) works at a place ν if the local evaluation map ev_{α,ν}: X(ℚ_{ν}) → ℚ/ℤ is non-constant. This is in fact a property of the residue class of α in Br(X)/ Br(ℚ).

Observe that if X(𝔸_{ℚ} ) ≠ ∅ and there exists a Brauer class that works at least at a single place then weak approximation is violated on X. On the other hand, if Br(X)/ Br(ℚ) ≅ ℤ/2ℤ and a nontrivial class works at least at one place then there exist adelic points fulfilling (1.1). That is, the Brauer-Manin obstruction cannot explain a violation of the Hasse principle.

The goal of this paper is to investigate which subsets of Ω may occur as the set of places at which a nontrivial Brauer class works, in the situation of a degree four del Pezzo surface. Our first main result is as follows.

Theorem 1.1. Let S ⊂ Ω be any finite subset. Then there exists a degree four del Pezzo surface X over ℚ having a ℚ-rational point such that Br(X)/ Br(ℚ) ≅ ℤ/2ℤ and the nontrivial Brauer class works exactly at the places in S.

For the construction, we make use of surfaces given as the intersection of two quadrics of the form

−A_{1}(T_{0} − T_{1})(T_{0} + T_{1}) = T_{3}^{2} − DT_{4}^{2} ,
−A_{2}(T_{0} − T_{2})(T_{0} + T_{2}) = T_{3}^{2} − B^{2}DT_{4}^{2} ,

with A_{1}, A_{2}, D, B ∈ ℚ. This family is inspired by a surface studied in work of Birch and Swinnerton-Dyer [BSD] and has the advantage that there is a standard way to write down a Brauer class. Moreover, we obtain the following example, which is different in nature.

Example 1.2. Let X ⊂ P_{ℚ}^{4} be the degree four del Pezzo surface that is given by the equations

T_{0}T_{1} = T_{2}^{2} + 7T_{3}^{2} ,
(T_{0} − 4T_{1})(T_{0} − 6T_{1}) = T_{2}^{2} + 7T_{4}^{2} .

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 3

Then X has a ℚ-rational point, Br(X)/ Br(ℚ) ≅ ℤ/2ℤ, and the nontrivial Brauer class works exactly at the infinite place. In particular, the surface X(ℝ) has two connected components and ℚ-rational points on only one of them.

The particular case that S = {∞} is perhaps the most interesting one. In this case, the set of ℚ-rational points of the surface X is dense in the real component of X that has ℚ-rational points. This is in line with Mazur’s conjecture [Maz, Conjecture 1] stating that the closure of X(ℚ) with respect to the real topology is equal to a union of connected components.

We give more details about Example 1.2 in section 2. Similar examples for other kinds of surfaces are available in the literature, including singular cubic surfaces [SD1, §3], conic bundles with five singular fibres [Maz, §3], and others.

Recall that over an algebraically closed field two quadratic forms are always simultaneously diagonalisable. We say that a degree four del Pezzo surface is diagonalisable over ℚ if the defining quadratic forms are diagonalisable over ℚ.

The surface from Example 1.2 is not diagonalisable over ℚ but only over ℚ(\sqrt{6}), as is easily seen using Fact 2.1.b.iii). Somewhat surprisingly, such a behaviour is necessary at this point. There is the following result.

Theorem 1.3. Let X be a degree four del Pezzo surface over ℚ having an adelic point and α ∈ Br(X) a Brauer class that works exactly at the infinite place. Then X is not diagonalisable over ℚ.

Our method of proof uses the fact that diagonal degree four del Pezzo surfaces have nontrivial automorphisms that are defined over the ground field. By functoriality, these operate on Br(X), but the induced operation on Br(X)/ Br(ℚ) turns out to be trivial automatically. Therefore, every α ∈ Br(X)/ Br(ℚ) induces a homomorphism i_{α}: Aut′(X) → Br(ℚ). Cf. Construction 3.2 for more details.

Moreover, we prove that if α ∈ Br(X) works at ∞ then there is an automorphism σ ∈ Aut(X) witnessing this, i.e. such that i_{α}(σ) has a nontrivial component at ∞. From this, the claim easily follows.

Our third main result asserts that, for diagonalisable degree four del Pezzo surfaces, the subset {∞} is the only exception of this kind.

Theorem 1.4. Let S ⊂ Ω be a finite subset, different from {∞}. Then there exists a diagonalisable degree four del Pezzo surface X over ℚ having a ℚ-rational point such that Br(X)/ Br(ℚ) ≅ ℤ/2ℤ and the nontrivial Brauer class works exactly at the places in S.

Conjecturally, for degree four del Pezzo surfaces, all failures of weak approximation are due to the Brauer-Manin obstruction. More precisely, it is conjectured that X(ℚ) is dense in

X(𝔸_{ℚ} )^{Br} := ⋂_{α∈Br(X)}X(𝔸_{ℚ} )^{α},

for X(𝔸_{ℚ} )^{α} ⊆ X(𝔸_{ℚ} ) the subset defined by condition (1.1) and X(𝔸_{ℚ} ) endowed with the product topology induced by the ν-adic topologies on X(ℚ_{ν}).

Due to work of P. Salberger and A. N. Skorobogatov [SSk, Theorem 0.1], this conjecture is proven under the assumption that X has a ℚ-rational point. The weaker statement that if X has a ℚ-rational point then the ℚ-rational points on X are Zariski dense already follows from the fact that del Pezzo surfaces of degree four that have a rational point are unirational (see Theorem 29.4 and Theorem 30.1 in [Man]).

Recall that each of the surfaces provided by Theorem 1.1 has a ℚ-rational point. We may thus blow up ℚ-rational points in general position to obtain del Pezzo surfaces of low degree. It actually requires some thought to see that, on every del Pezzo surface X of degree ≥2, there exists a non-empty Zariski open subset U ⊂ X of admissible blow-up points. Admissible meaning here that the blow-up is indeed del Pezzo, not just the desingularisation of a weak del Pezzo surface. We omit the details on this issue here. Moreover, Brauer groups do not change under blowup and the local evaluation maps are compatible in the sense that ev_{α,ν}(π(x)) = ev_{π^{∗}α,ν}(x). Hence we obtain the following corollary.

Corollary 1.5. Let S ⊂ Ω be an arbitrary finite subset and d ≤ 4 a positive integer. Then there exists a del Pezzo surface X of degree d over ℚ having a ℚ-rational point such that Br(X)/ Br(ℚ) ≅ ℤ/2ℤ and the nontrivial Brauer class works exactly at the places in S.

It is well-known that every del Pezzo surface X of degree at least five has Br(X)/ Br(ℚ) = 0. One way to see this is to systematically inspect all possible Galois operations on the exceptional curves in a way analogous to [Ja, Chapter III, 8.21–8.23] and to apply [Man, Proposition 31.3]. Thus, Corollary 1.5 cannot have an analogue for del Pezzo surfaces of higher degree.

At least for d = 5 and 7, as well as for d = 6 under the additional assumption that X has an adelic point, there is also a geometric argument.

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 5

Indeed, these surfaces are birationally equivalent to P_{ℚ}^{2} [VA, Theorem 2.1], cf. [Man, Theorem 29.4].

It would be interesting to produce examples as in Corollary 1.5 with the additional restriction that the surface is minimal.

Remark 1.6. For a degree four del Pezzo surface, the group Br(X)/ Br(ℚ) may be isomorphic to either 0, ℤ/2ℤ, or (ℤ/2ℤ)^{2}. In the cases of degree 3, 2, or 1, there are even more options [Co, Theorem 4.1], [Man, Section 31, Table 3] and [SD2]. We do not know whether the analogue of Corollary 1.5 is true for a prescribed Brauer group.

Acknowledgements. We would like to thank the anonymous referee for carefully reading an earlier version of this paper and for many helpful remarks that led to an improved exposition of the material.

2. Brauer classes on degree four del Pezzo surfaces

The goal of this section is to gather some facts about degree four del Pezzo surfaces that are necessary for what follows. This includes some results on their Brauer groups and finally leads us to a proof of the assertions made in Example 1.2. In other words, we show Theorem 1.1 under the assumption of Theorem 1.4. Unless a specific choice is made, we work in this section over an arbitrary base field K of characteristic ≠2. Let us denote by \overline{K} an algebraic closure of K.

A del Pezzo surface X ⊂ P_{K}^{4} of degree four is the base locus of a pencil (μQ^{(1)} + νQ^{(2)})_{(μ:ν)∈P^{1}} of quadratic forms in five variables with coefficients in the field K. The generic member of the pencil must be of rank five, as otherwise X would be a cone. The condition that det(μQ^{(1)} + νQ^{(2)}) = 0 therefore defines a finite subscheme 𝒮_{X} ⊂ P_{K}^{1} of degree 5.

Choosing a different basis of the pencil yields another embedding of 𝒮_{X} into the projective line. Thus, one may consider the subscheme 𝒮_{X} ⊂ P_{K}^{1} as an invariant of the surface X itself. Moreover, the definition extends to arbitrary intersections of two quadrics in P^{4} that are not cones.

Facts 2.1. a) X is nonsingular if and only if the scheme 𝒮_{X} is reduced.

b) Let X ⊂ ℙ^{4} be a smooth intersection of two quadrics. Then the following statements hold.

i) If \{s_{0}, …, s_{4}\} = 𝒮_{X}(\overline{K}) then the quadratic forms Q_{s_{0}}, …, Q_{s_{4}} are exactly of rank 4. ii) The cusps of the cones defined by Q_{s_{i}} = 0, for i = 0, …, 4, are in general linear position in P^{4}, i.e. not contained in any hyperplane.

6 J. JAHNEL AND D. SCHINDLER

iii) X is diagonalisable over K if and only if 𝒮_{X} is split over K.

Proof. These statements are rather well-known. Proofs may be found, for example, in [Wi]. Concretely, parts a) and b.i) are implied by [Wi, Proposition 3.26]. Furthermore, part b.ii) is [Wi, Corollaire 3.29], while part b.iii) is [Wi, Corollaire 3.30]. □

Let X be a degree four del Pezzo surface over a field K and assume that there is a K-rational point s ∈ 𝒮_{X}(K) as well as that the corresponding degenerate quadric Q_{s} has a K-rational point, different from the cusp. Then there exist four linearly independent linear forms l_{1}, …, l_{4} such that

cQ_{s} = l_{1}l_{2} − (l_{3}^{2} − Dl_{4}^{2}) ,

for some constant c, see [VAV, Lemma 2.1]. Furthermore, D is the discriminant of cQ_{s}, considered as a quadratic form in four variables.

The case most interesting for us is when there are two distinct K-rational points s_{1}, s_{2} ∈ 𝒮_{X}(K) and the corresponding degenerate quadrics Q_{s_{1}}, Q_{s_{2}} have the same discriminant. Then X may be given by a system of equations in the form

(2.1)  l_{11}l_{12} = l_{13}^{2} − Dl_{14}^{2} ,
(2.2)  l_{21}l_{22} = l_{23}^{2} − Dl_{24}^{2} .

For such surfaces, there is a standard way to write down a Brauer class, which goes back, at least to B. Birch and Sir Peter Swinnerton-Dyer [BSD].

Proposition 2.2. Let X be the degree four del Pezzo surface over a field K, given by the equations (2.1,2.2). Assume that D is a non-square in K and put L := K(\sqrt{D}).

a) Then the quaternion algebra (see [Pi, Section 15.1] for the notation)

𝒜 := (L(X), τ, \frac{l_{11}}{l_{21}} )

over the function field K(X) extends to an Azumaya algebra over the whole of X. Here, by τ ∈ Gal(L(X)/K(X)), we denote the nontrivial element.

b) In the case that K = ℚ, denote by α ∈ Br(X) the Brauer class, defined by the extension of 𝒜. Let ν be any (archimedean or non-archimedean) place of ℚ.

i) Let x ∈ X(ℚ_{ν}) be a point and assume that, for some i, j ∈ {1, 2}, one has l_{1i}(x) ≠ 0 and l_{2j}(x) ≠ 0. Denote the corresponding quotient l_{1i}(x)/l_{2j}(x) by q. Then

ev_{α,ν}(x) = \{ \begin{matrix}0 & if (q, D)_{ν} = 1 , \\ \frac{1}{2} & if (q, D)_{ν} = −1 , \end{matrix}

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 7

for (q, D)_{ν} the Hilbert symbol. ii) If ν is split in L then the local evaluation map ev_{α,ν} is constantly zero.

Proof. a) This is a consequence of [VAV, Lemma 3.2]. b.i) The quotients

\frac{l_{11}}{l_{21}} /\frac{l_{11}}{l_{22}} = \frac{l_{23}^{2}−Dl_{24}^{2}}{l_{21}^{2}} , \frac{l_{12}}{l_{21}} /\frac{l_{12}}{l_{22}} = \frac{l_{23}^{2}−Dl_{24}^{2}}{l_{21}^{2}} , and \frac{l_{11}}{l_{21}} /\frac{l_{12}}{l_{21}} = \frac{l_{13}^{2}−Dl_{14}^{2}}{l_{12}^{2}}

are norms of rational functions from L(X). Therefore, they define the trivial element of H^{2}(⟨σ⟩, K(X_{L})^{∗}) ⊆ Br K(X), and hence in Br X. In particular, the four expressions l_{1i}/l_{2j} define the same Brauer class.

The general description of the evaluation map, given in [Man, Paragraph 45.2], shows that ev_{α,ν}(x) is equal to 0 or \frac{1}{2} depending on whether q is in the image of the norm map N_{L_{𝔫}/ℚ_{ν}} : L_{𝔫}^{∗} → ℚ_{ν}^{∗}, or not, for 𝔫 a place of L lying above ν. This is exactly what is tested by the Hilbert symbol (q, D)_{ν}. ii) If ν is split in L then the norm map

N_{K(X_{L𝔫})/K(X_{ℚν} )} : K(X_{L_{𝔫}})^{∗} → K(X_{ℚ_{ν}} )^{∗}

is surjective. In particular, l_{11}/l_{21} ∈ K(X_{ℚ_{ν}} )^{∗} is the norm of a certain rational function on X_{L_{𝔫}}. Therefore, it defines the zero class in H^{2}(⟨σ⟩, K(X_{L_{𝔫}})^{∗}) ⊆ Br K(X_{ℚ_{ν}} ), and thus in Br X_{ℚ_{ν}} . To complete the argument, we note that every ℚ_{ν}-rational point x: Spec ℚ_{ν} → X factors via X_{ℚ_{ν}} . □

In the following, we will make heavy use of the two facts below. The first one recalls the explicit description of the situation when the Brauer group of X is isomorphic to (ℤ/2ℤ)^{2}. It is convenient to introduce the following assumption on X defined over a local or global field K.

Assumption A. In the local field case, assume that X(K) ≠ ∅, and in the global field case that X has an adelic point.

Fact 2.3. Let X be a degree four del Pezzo surface over a local or global field K satisfying Assumption A. Then Br(X)/ Br(K) ≅ (ℤ/2ℤ)^{2} if and only if 𝒮_{X} has three distinct points s_{0}, s_{1}, s_{2} ∈ 𝒮_{X}(K) so that all three discriminants D_{s_{0}}, D_{s_{1}}, D_{s_{2}} are nonsquares in K and coincide up to square factors. In this case, representatives of the three nontrivial classes may be obtained as follows. Choose a subset {s_{i}, s_{j}} ⊂ {s_{0}, s_{1}, s_{2}} of size two. Write X accordingly in the form (2.1,2.2) and take the corresponding Azumaya algebra as described in Proposition 2.2.

8 J. JAHNEL AND D. SCHINDLER

Proof. This is a well-known fact and a proof may be found, for example, in [VAV, Theorem 3.4]. Note that the assumption on X implies that, for every closed point s ∈ 𝒮_{X}, the corresponding rank-4 quadric has a regular point over the residue field of s [VAV, Lemma 5.1]. □

Fact 2.4. Let X be a degree four del Pezzo surface over a local or global field K satisfying Assumption A. Assume X to be diagonalisable over K. Let D_{i} ∈ K, for 0 ≤ i ≤ 4, be the five rank-4 discriminants and assume that D_{0} = D_{1} =: D.

a) Let D be a non-square in K. Then the Brauer class α ∈ Br(X) described in Proposition 2.2 is trivial, i.e. α ∈ Br(K), if and only if D_{2}, D_{3}, and D_{4} are all squares in K.

b) If the conditions in a) hold or all five discriminants D_{i} are squares in K, then one has Br(X)/ Br(K) ≅ 0.

Proof. a) This equivalence statement is established in [VAV, Proposition 3.3].

b) Fact 2.3 above proves that Br(X)/ Br(K) is at most of order two. If it were of order exactly two then, by [VAV, Theorem 3.4], the nontrivial class could be obtained as described in Proposition 2.2. In particular, only the case that D is a non-square remains to be considered. However, as the other three discriminants are squares, this is exactly the situation in which part a) proves that the Brauer class is trivial. □

Remark 2.5. Under the assumptions of Fact 2.4, there is an isomorphism

Br(X)/ Br(K) ⟵^{≅} ker(o: (ℤ/2ℤ)^{5} → K^{∗}/(K^{∗})^{2})/T ,

where the homomorphism o is given by o: (a_{0}, …, a_{4}) ↦ (D_{0}^{a_{0}}·. . .·D_{4}^{a_{4}} mod (K^{∗})^{2}) and T is generated by the vector (1, …, 1) and the standard vectors e_{i}, for those i ∈ {0, …, 4} for which D_{i} is a perfect square. Note that D_{0}· . . . · D_{4} is a perfect square in K, cf. Fact 3.4.

Once one has an explicit description of the Brauer classes, one needs criteria to understand whether or not they evaluate constantly at a given place. For this, the following result turns out to be very useful.

Criterion 2.6 (A. Várilly-Alvarado and B. Viray). Let X be the degree four del Pezzo surface over ℚ given by the equations (2.1,2.2). Assume that D is a non-square and let α ∈ Br(X) be the Brauer class described in Proposition 2.2.

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 9

Then, for any place ν ≠ 2, ∞ such that the reductions modulo ν of the quadratic forms in (2.1) and (2.2) both have rank 4, the local evaluation map ev_{α,ν} is constant.

Proof. This is [VAV, Proposition 5.4]. □

Proof of Theorem 1.1 assuming Theorem 1.4. Theorem 1.4 solves the problem for every subset S ≠ {∞}. Thus, in order to establish Theorem 1.1, it suffices to verify the assertions made in Example 1.2.

For this, one first checks that 𝒮_{X} has exactly three ℚ-rational points, corresponding to the quadratic forms independent of the variable T_{2}, T_{3}, and T_{4}, respectively, and a point of degree two that splits over the quadratic field ℚ(\sqrt{6}) . In particular, X is nonsingular.

The discriminants of the three ℚ-rational quadratic forms of rank 4 are, up to square factors, 1, (−7), and (−7). Therefore, Fact 2.3 shows that Br(X)/ Br(ℚ) is at most of order 2. On the other hand, by Proposition 2.2, we have a Brauer class α ∈ Br(X) that is given over the function field ℚ(X) as the quaternion algebra (ℚ(\sqrt{−7})(X), τ, φ) for φ := \frac{T_{0}−4T_{1}}{T_{1}} .

Next, we observe that X has no real points such that x_{0} = x_{1} = 0. Moreover, for an arbitrary real point x ∈ X(ℝ) such that x_{1} ≠ 0, the equations imply x_{0}/x_{1} ≥ 0 and (\frac{x_{0}}{x_{1}} − 4)(\frac{x_{0}}{x_{1}} − 6) ≥ 0, hence

x_{0}/x_{1} ∈ [0, 4] or x_{0}/x_{1} ≥ 6 .

There exist real points of both kinds, for example (1 : 1 : 1 : 0 : \sqrt{2}) and (8:1:1:1:1). Since (−7) < 0, we have that (q, −7)_{∞} is the sign of q. Thus, ev_{α,∞} distinguishes the two kinds of real points. In particular, Br(X)/ Br(ℚ) is indeed of order two and the nontrivial element works at the infinite place.

It remains to show that it does not work at any other place. Criterion 2.6 shows constancy of the evaluation map ev_{α,ν} for all finite places ν ≠ 2, 7. Furthermore, ev_{α,2} is constant by Proposition 2.2.b.ii), as the prime 2 splits in ℚ(\sqrt{−7}) .

Finally, for the prime 7, we argue as follows. Let x ∈ X(ℚ_{7}) be any 7-adic point on X. Normalise the coordinates x_{0}, …, x_{4} such that each is a 7-adic integer and at least one is a unit. If 7|x_{0} and 7|x_{1} then the equations imply that all coordinates must be divisible by 7, a contradiction. Hence, at least one of x_{0} and x_{1} is a unit. Modulo 7, we have (\overline{x}_{0} − 4\overline{x}_{1})(\overline{x}_{0} − 6\overline{x}_{1}) = \overline{x}_{0}\overline{x}_{1} (since both expressions are equal to \overline{x}_{2}^{2}), and this equation has the solutions \overline{x}_{0}/\overline{x}_{1} = 1, 3 in ℤ/7ℤ. However, the solution \overline{x}_{0}/\overline{x}_{1} = 3 is contradictory, as then \overline{x}_{0}\overline{x}_{1} would be a non-square. Consequently, both x_{0} and x_{1} must be

10 J. JAHNEL AND D. SCHINDLER

units and

\frac{x_{0}−4x_{1}}{x_{1}} ≡ −3 (mod 7) ,

which implies that \frac{x_{0}−4x_{1}}{x_{1}} is a square in ℚ_{7}. This shows (\frac{x_{0}−4x_{1}}{x_{1}} , −7)_{7} = 1 and ev_{α,7}(x) = 0. □

Remark 2.7. In Example 1.2, weak approximation is disturbed in a rather astonishing way. The smooth manifold X(ℝ) has two connected components. There are two kinds of real points x ∈ X(ℝ), those with x_{0}/x_{1} ∈ [0, 4] and those such that x_{0}/x_{1} ∈ [6, ∞]. However, for every ℚ-rational point x ∈ X(ℚ), one has x_{0}/x_{1} > 6.

A naively implemented point search shows that there are exactly 792 ℚ-rational points of naive height up to 1000 on X. The smallest value of the quotient x_{0}/x_{1} is 319/53 ≈ 6.019.

3. Diagonal degree four del Pezzo surfaces

The goal of this section is to collect some facts about diagonal degree four del Pezzo surfaces. These will lead us to a proof of Theorem 1.3.

Let X be a diagonal degree four del Pezzo surface over a base field K, i.e. one that is given by equations of the form

(3.1)  a_{0}T_{0}^{2} + … + a_{4}T_{4}^{2} = 0 ,
(3.2)  b_{0}T_{0}^{2} + … + b_{4}T_{4}^{2} = 0

with coefficients in K. Then, for every (i_{0}, …, i_{4}) ∈ {0, 1}^{5}, the map

(T_{0} : …: T_{4}) ↦ ((−1)^{i_{0}}T_{0} : …: (−1)^{i_{4}}T_{4})

defines a K-automorphism of X. Thus, we explicitly described a subgroup Aut′(X) ⊆ Aut_{K}(X) that is isomorphic to (ℤ/2ℤ)^{4}.

It is known that the automorphism group of a degree four del Pezzo surface over an algebraically closed field is generically isomorphic to (ℤ/2ℤ)^{4} and that there are particular cases, where the automorphism group is larger [Do, Theorem 8.6.8].

Lemma 3.1. Let X be a diagonal degree four del Pezzo surface over a local or global field K satisfying Assumption A. Then the natural operation of Aut′(X) on Br(X) induces the trivial operation on Br(X)/ Br(K).

Proof. This is trivially true if Br(X)/ Br(K) ≅ 0 or ℤ/2ℤ. Otherwise, it follows from the description of the representatives given in Fact 2.3. □

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 11

Construction 3.2. Let X be a diagonal degree four del Pezzo surface over a local or global field K satisfying Assumption A.

By functoriality, the operation of Aut′(X) on X induces an operation on Br(X), which is necessarily trivial on Br(X)/ Br(K). Thus, for every α ∈ Br(X), there is a natural homomorphism

i_{α}: Aut′(X) ⟶ Br(K) ,

given by the condition that σ^{∗}α = α + i_{α}(σ) for σ ∈ Aut′(X).

Moreover, the homomorphism i_{α} depends only on the class of α in Br(K)/ Br(K).

Definition 3.3. Let K = ℚ and assume that, for certain α ∈ Br(K)/ Br(ℚ) and σ ∈ Aut′(X), the Brauer class i_{α}(σ) has a nontrivial image in Br(K_{ν}) at the place ν. Then, as

ev_{α,ν}(σ(x)) = ev_{σ^{∗}α,ν}(x) = ev_{α,ν}(x) + i(σ)_{ν} ,

the Brauer class certainly works at ν. We say in this situation that σ is a witness for the non-constancy of the local evaluation map at ν.

Fact 3.4. Let X be a diagonal degree four del Pezzo surface over a field K and D_{0}, …, D_{4} be the discriminants of the five associated quadratic forms of rank 4. Then D_{0}· . . . · D_{4} is a square in K.

Proof. This is a direct calculation. □

Lemma 3.5. Let X be a diagonal degree four del Pezzo surface over ℝ that has a real point. Assume that Br(X)/ Br(ℝ) ≠ 0. Then X(ℝ) has exactly two connected components. Moreover, there is an element σ ∈ Aut′(X) that interchanges these components.

Proof. By Fact 3.4, there are three cases. The number of negative discriminants among the five rank-4 discriminants is either zero, or two or four. Facts 2.4.b) and 2.3 show that Br(X)/ Br(ℝ) ≠ 0 only in the last case.

Then the pencil of quadrics in P^{4} associated with X contains four rank-4 quadrics of negative discriminant. We may write each of them in the shape

−c_{0}T_{i_{0}}^{2} + c_{1}T_{i_{1}}^{2} + c_{2}T_{i_{2}}^{2} + c_{3}T_{i_{3}}^{2} = 0 ,

for c_{0}, …, c_{3} > 0, and say that the variable T_{i_{0}} is distinguished by the form considered.

We claim that not all four forms may distinguish the same variable. Indeed, if that would be the case then we would also have the form −c′_{0}T_{i_{0}}^{2} + c′_{1}T_{i_{1}}^{2} + c′_{2}T_{i_{2}}^{2} + c′_{4}T_{i_{4}}^{2} = 0, which shows that the form in the pencil

12 J. JAHNEL AND D. SCHINDLER

that does not involve T_{i_{0}} has opposite signs at T_{i_{3}}^{2} and T_{i_{4}}^{2}. The same argument for all combinations of two of the four quadratic forms enforces six opposite signs among the four coefficients of T_{i_{1}}^{2}, …, T_{i_{4}}^{2}, a contradiction.

Thus, X may be given by two equations of the form

−c_{0}T_{i_{0}}^{2} + c_{1}T_{i_{1}}^{2} + c_{2}T_{i_{2}}^{2} + c_{3}T_{i_{3}}^{2} = 0 ,
−d_{0}T_{j_{0}}^{2} + d_{1}T_{j_{1}}^{2} + d_{2}T_{j_{2}}^{2} + d_{3}T_{j_{3}}^{2} = 0 ,

for c_{k}, d_{k} > 0, i_{0} ≠ j_{0}, and {i_{0}, …, i_{3}} ∪ {j_{0}, …, j_{3}} = {0, …, 4}. These equations imply x_{i_{0}} ≠ 0 and x_{j_{0}} ≠ 0 for every real point x ∈ X(ℝ). In particular, X(ℝ) has at least two connected components, given by the two possible signs of x_{i_{0}}/x_{j_{0}}. Clearly, these two components are interchanged under the operation of Aut′(X).

We finally note that a real degree four del Pezzo surface cannot have more than two connected components [Silh, Chapter III, Theorem 3.3]. □

Remark 3.6. The stronger statement that if X(ℝ) splits into two connected components then the operation of Aut′(X) interchanges them is true, as well.

Indeed, by blowing up a real point not lying on any exceptional curve, one obtains a real cubic surface that has two connected components. According to L. Schläfli [Sch, pp. 114f.], there are exactly five real types of real cubic surfaces and those correspond in modern language to the four conjugacy classes of order-2 subgroups in W (E_{6}) together with the trivial group. Only for one of these five cases, the Brauer group is nontrivial [Ja, Appendix, Table 2], it is isomorphic to (ℤ/2ℤ)^{2} then, and that is the single case in which the surface is disconnected. We will not make use of this observation.

Proof of Theorem 1.3. Let X be a diagonalisable degree four del Pezzo surface over ℚ that has an adelic point and a Brauer class α ∈ Br(X) working at the infinite place. We note that, since X has an adelic point, it clearly has a real point. The local evaluation ev_{α,∞}(x) for x ∈ X(ℝ) is defined using the restriction homomorphism x^{∗}: Br(X) → Br(ℝ), which factors via Br(X_{ℝ} ). Hence, non-constancy of ev_{α,∞} implies that the restriction α_{ℝ} ∈ Br(X_{ℝ} )/ Br(ℝ) is a nonzero class.

In this case, Lemma 3.5 shows that X(ℝ) splits into two connected components. Moreover, there exists an automorphism σ ∈ Aut′(X_{ℝ} ) ≅ Aut′(X) interchanging these. Since ev_{α,∞} is locally constant, this implies that σ is a witness for the non-constancy of the local evaluation map at ∞. In other

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 13

words, the natural homomorphism i_{α}: Aut′(X) → Br(ℚ) has in its image a Brauer class i_{α}(σ) ∈ Br(ℚ) with a nonzero component at infinity.

According to global class field theory [Ta, Section 10, Theorem B], i_{α}(σ) necessarily has a nonzero component at a second place ν ≠ ∞. Consequently, α works at the place ν, too, which implies the claim. □

4. Surfaces with a Brauer class working at a prescribed set of places

The goal of this section is to prove Theorem 1.4. We distinguish between the cases #S > 1, #S = 1, and S = ∅. The family below will serve us in all cases.

4.1. A family of degree four del Pezzo surfaces. For D, A_{1}, A_{2}, B ∈ ℚ, let S := S^{(D;A_{1},A_{2},B)} ⊂ P_{ℚ}^{4} be given by the system of equations

(4.1)  −A_{1}(T_{0} − T_{1})(T_{0} + T_{1}) = T_{3}^{2} − DT_{4}^{2} ,
(4.2)  −A_{2}(T_{0} − T_{2})(T_{0} + T_{2}) = T_{3}^{2} − B^{2}DT_{4}^{2} .

Theorem 4.1. Let D, A_{1}, A_{2}, B be nonzero rational numbers. A.a) Then S is not a cone. The degree-5 scheme 𝒮_{X} has a point at infinity and four others, which are the roots of a completely reducible polynomial of degree four having discriminant

Δ := A_{1}^{2}(A_{1} − A_{2})^{2}(A_{1}B^{2} − A_{2})^{2}B^{4}(B − 1)^{2}(B + 1)^{2}/A_{2}^{6}B^{12} .

b) S has the ℚ-rational point (1:1:1:0:0) ∈ X(ℚ).

c) If Δ ≠ 0 then the five rank-4 discriminants are, up to factors being perfect squares, given by D, D, as well as DA_{1}A_{2}(A_{1} − A_{2})(B^{2} − 1), A_{1}A_{2}(A_{1}B^{2} − A_{2})(B^{2} − 1), and D(A_{1} − A_{2})(A_{1}B^{2} − A_{2}). B.a) There is a Brauer class α ∈ Br(X) extending that of the quaternion algebra (ℚ(\sqrt{D})(X), τ, \frac{T_{0}+T_{1}}{T_{0}+T_{2}} ) over the function field ℚ(X).

b) Moreover, one has ev_{α,ν}(x) = 0 for x = (1:1:1:0:0) and every ν ∈ Ω.

c) At an arbitrary place ν ∈ Ω, the local evaluation map ev_{α,ν} is constant if one of the following conditions holds.

• ν = p is a finite place, p ≠ 2, and p divides neither D, nor A_{1}, nor A_{2}, nor B.

• ν = p splits in ℚ(\sqrt{D}) , or ν = ∞ and D > 0.

• D is square-free, ν = p is a finite place, p|D, p ≠ 2, gcd(B, D) = 1, (\frac{−A_{1}}{p}) = 1, and A_{1} ≡ A_{2} (mod p).

14 J. JAHNEL AND D. SCHINDLER

d) At a place ν, the local evaluation map ev_{α,ν} cannot be constant if (−A_{1}, D)_{ν} = −1 or (−A_{2}, D)_{ν} = −1. Proof. A.a) and c) are standard calculations, while b) is directly checked. Moreover, B.a) is a direct application of Proposition 2.2.a) and the assertion of b) follows from the fact that \frac{x_{0}+x_{1}}{x_{0}+x_{2}} = 1 for x = (1:1:1:0:0). B.c) The sufficiency of the first condition is Criterion 2.6, while that of the second was shown in Proposition 2.2.b). In order to establish the sufficiency of the third, we argue as follows.

First of all, the prime p ramifies in ℚ(\sqrt{D}) . A p-adic unit u ∈ ℚ_{p} is a local norm from ℚ(\sqrt{D}) if and only if (u mod p) ∈ 𝔽_{p}^{∗} is a square. Moreover, we note that (\frac{−A_{1}}{p}) = (\frac{−A_{2}}{p}) = 1 implies that each of the four rational functions \frac{T_{0}±T_{1}}{T_{0}±T_{2}} may be used to evaluate the Brauer class α at the place p.

Let now x ∈ X(ℚ_{p}) be any p-adic point. Normalise the coordinates x_{0}, …, x_{4} such that each is a p-adic integer and at least one is a unit. If p|x_{0} and p|x_{1} or p|x_{0} and p|x_{2} then the equations imply that all coordinates must be divisible by p, a contradiction. Modulo p, we have \overline{x}_{0}^{2} − \overline{x}_{1}^{2} = \overline{x}_{0}^{2} − \overline{x}_{2}^{2}, hence \overline{x}_{1} = ±\overline{x}_{2}, which implies that one of the four quotients \frac{x_{0}±x_{1}}{x_{0}±x_{2}} is congruent to 1 modulo p, and therefore a norm. B.d) We note first that X has X(ℚ_{ν})-rational points such that x_{0} ≠ ±x_{1} and x_{0} ≠ −x_{2}. Indeed, putting x_{0} := 1 and choosing x_{3} and x_{4} sufficiently close to 0 in the ν-adic topology, we see that (4.1) and (4.2) become soluble when viewed as equations for x_{1} and x_{2}, respectively.

Now, let us assume without loss of generality that (−A_{1}, D)_{ν} = −1. Then the automorphism σ: (T_{0} : …: T_{4}) ↦ (T_{0} :(−T_{1}):T_{2} :T_{3} :T_{4}) changes the rational function \frac{T_{0}+T_{1}}{T_{0}+T_{2}} to

\frac{T_{0}−T_{1}}{T_{0}+T_{1}} = −\frac{1}{A_{1}} \frac{T_{3}^{2}−DT_{4}^{2}}{(T_{0}+T_{1})^{2}} ,

which takes only ν-adic non-norms from ℚ(\sqrt{D}) , since (−A_{1}, D)_{ν} = −1. This shows that i_{α}(σ) has a nonzero component at ν, i.e. that the automorphism σ witnesses the non-constancy of the local evaluation map ev_{α,ν}.

□

4.2. More than one place. Let S ⊂ Ω be a given set that consists of at least two places. We incorporate the notation {p_{1}, …, p_{r}} = S\{2, ∞}.

To construct a diagonalisable degree four del Pezzo surface such that a nontrivial Brauer class works exactly at the places in S, we first choose a square-free integer D ≠ 0 satisfying the following conditions.

• D > 0 if and only if ∞ ∉ S.

• D ≡ 3 (mod 4) in the case that 2 ∈ S and D ≡ 1 (mod 8) when 2 ∉ S.

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 15

• D is divisible by p_{1}, …, p_{r} and has exactly one further prime divisor, which we call q. That such a choice of D is possible follows immediately from the fact that there are infinitely many primes in every odd residue class modulo 8.

Now write S = S_{1} ∪ S_{2} as a union of two not necessarily disjoint subsets of even size. This is possible, because #S ≥ 2. In addition, we may put 2 into both subsets in case it occurs as an element of S, and the same for ∞.

Next, we choose primes A_{1} ≠ A_{2} not dividing D such that, for i = 1, 2,

(4.3)  (−A_{i}, D)_{ν} = −1 ⟺ ν ∈ S_{i} .

To see that this may be achieved, we observe at first that one has (−A_{1}, D)_{ν} = (−A_{2}, D)_{ν} = 1 for all places ν ≠ 2, ∞; p_{1}, …, p_{r}, q, A_{1}, and A_{2}. The requirement at ν = 2 may be realised by choosing A_{i} ≡ 1 (mod 4), the condition at ν = ∞ is fulfilled as the A_{i} are positive. Furthermore, what we require in (4.3) is

(\frac{−A_{i}}{p_{j}} ) = \{ \begin{matrix}−1 & if p_{j} ∈ S_{i} , \\ 1 & otherwise , \end{matrix}

for i = 1, 2, and (\frac{−A_{1}}{q}) = (\frac{−A_{2}}{q}) = 1. Let us impose, in addition, the condition that

(4.4)  A_{2} ≡ A_{1} (mod q) .

All these are congruence conditions modulo distinct odd primes. Therefore, the existence of a prime A_{1} satisfying (4.3) for all places except, possibly, A_{1} itself, is implied by Dirichlet’s Theorem on primes in arithmetic progressions. Moreover, as #S_{1} is even, we have (−A_{1}, D)_{A_{1}} = 1 by the Hilbert reciprocity law [Ne, Chapter VI, Theorem 8.1].

In a completely analogous manner, Dirichlet’s Theorem and the Hilbert reciprocity law imply the existence of a prime A_{2} ≠ A_{1} fulfilling (4.3) and (4.4).

We may now formulate the main result of this paragraph.

Theorem 4.2. Let the integers D, A_{1}, and A_{2} be chosen as above.

a) Then, for every integer B ≥ 2, the surface X ⊂ P_{ℚ}^{4} given by

−A_{1}(T_{0} − T_{1})(T_{0} + T_{1}) = T_{3}^{2} − DT_{4}^{2} ,
−A_{2}(T_{0} − T_{2})(T_{0} + T_{2}) = T_{3}^{2} − B^{2}DT_{4}^{2}

is nonsingular and has a ℚ-rational point.

b) There is a Brauer class α ∈ Br(X) extending that of the quaternion algebra (ℚ(\sqrt{D})(X), τ, \frac{T_{0}+T_{1}}{T_{0}+T_{2}} ) over the function field ℚ(X).

16 J. JAHNEL AND D. SCHINDLER

c) The Brauer class α works at every place ν ∈ S. If B is a prime number that splits in ℚ(\sqrt{D}) then α does not work at any other place.

d) There are infinitely many prime numbers B splitting in ℚ(\sqrt{D}) such that Br(X)/ Br(ℚ) ≅ ℤ/2ℤ.

Proof. a) follows from Theorem 4.1.A.a) and b). b) is Theorem 4.1.B.a).

c) Our choices of A_{1}, A_{2}, and D guarantee that Theorem 4.1.B.d) applies to every ν ∈ S. On the other hand, as B is a prime that splits in ℚ(\sqrt{D}) , Theorem 4.1.B.c) shows constancy of the evaluation map ev_{α,ν} at all other places.

d) In order to exclude the option Br(X)/ Br(ℚ) ≅ (ℤ/2ℤ)^{2}, according to Fact 2.3 we have to choose the parameter B such that none of the terms

A_{1}A_{2}(A_{1}−A_{2})(B^{2}−1), DA_{1}A_{2}(A_{1}B^{2}−A_{2})(B^{2}−1), and (A_{1}−A_{2})(A_{1}B^{2}−A_{2})

is a perfect square. By Siegel’s Theorem on integral points on elliptic curves [Silv, Theorem IX.4.3], the term in the middle is a square only finitely many times. The two others lead to Pell-like equations, the integral solutions of which are known to have exponential growth, cf. for example [Ch, Chapter XXXIII, §§15–18], and hence are much sparser than the set of primes that split in ℚ(\sqrt{D}) . The assertion follows. □

4.3. No place. Let X ⊂ P_{ℚ}^{4} be the surface given by

−(T_{0} − T_{1})(T_{0} + T_{1}) = T_{3}^{2} − 17T_{4}^{2} ,
−103(T_{0} − T_{2})(T_{0} + T_{2}) = T_{3}^{2} − 68T_{4}^{2} .

Then X is nonsingular and X(ℚ) ≠ ∅. Moreover, Br(X)/ Br(ℚ) ≅ ℤ/2ℤ but the nontrivial Brauer class works at no place.

Proof. The first two assertions follow from Theorem 4.1.A.b) and a). The discriminants of the five rank-4 forms are, up to square factors, 17, 17, 66, 206, and 3399 such that, by Facts 2.4.a) and 2.3, we have Br(X)/ Br(ℚ) ≅ ℤ/2ℤ.

Let α ∈ Br(X) be nontrivial. By Theorem 4.1.B.c), the local evaluation map is constant at all places ν ≠ 2, 17, 103, and ∞. Moreover, it is constant at ν = ∞ as the field ℚ(\sqrt{17}) is real-quadratic. Constancy at ν = 2 and 103 is clear, too, since these primes split in ℚ(\sqrt{17}) . Finally, ev_{α,17} is constant as (\frac{−1}{17}) = 1 and 103 ≡ 1 (mod 17). □

4.4. Exactly one place. The examples here are necessarily a bit different, as the 16 automorphisms must not witness the non-constancy of the evaluation map. We may nonetheless work with the family from Theorem 4.1.

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 17

Example 4.3. Let l be a prime number such that l ≡ 3 (mod 4). Choose a prime D ≡ 1 (mod 8) such that (\frac{D}{l}) = −1 and another prime A > l such that A ≡ 1 (mod D) and (A^{2} − 1)(A^{2} − l^{2}) is a non-square. Then the surface X ⊂ P_{ℚ}^{4} given by

−(T_{0} − T_{1})(T_{0} + T_{1}) = T_{3}^{2} − DT_{4}^{2} ,
−A^{2}(T_{0} − T_{2})(T_{0} + T_{2}) = T_{3}^{2} − l^{2}DT_{4}^{2}

is nonsingular and has a ℚ-rational point. Moreover, Br(X)/ Br(ℚ) ≅ ℤ/2ℤ and the nontrivial class works exactly at the place l.

Proof. We first note that the restrictions on D and A are easy to fulfil due to Dirichlet’s and Siegel’s theorems. Furthermore, the first three assertions follow directly from Theorem 4.1.A.a) and b), as well as Facts 2.4.a) and 2.3. The nontrivial Brauer class α ∈ Br(X) may be understood as an extension of the quaternion algebra (ℚ(\sqrt{D})(X), τ, \frac{T_{0}+T_{1}}{T_{0}+T_{2}} ) over ℚ(X) to the whole scheme X. Theorem 4.1.B.c) implies that the local evaluation map is constant at all places ν ≠ l. Non-constancy of ev_{α,l}: Note that l is an inert prime, since (\frac{D}{l}) = −1. An element u ∈ ℚ_{l}^{∗} is a local norm from ℚ(\sqrt{D}) if and only if ν_{l}(u) is even.

For \underline{x} = (1:1:1:0:0), we have ev_{α,l}(\underline{x}) = 0 by Theorem 4.1.B.b). On the other hand, the substitutions T_{0} = lT′_{0}, T_{1} = T′_{1}, T_{2} = lT′_{2}, T_{3} = lT′_{3}, and T_{4} = T′_{4} yield a different model X′ of X that is given by

−(lT′_{0} − T′_{1})(lT′_{0} + T′_{1}) = l^{2}T′_{3}^{2} − DT′_{4}^{2} ,
−A^{2}( T′_{0} − T′_{2})( T′_{0} + T′_{2}) = T′_{3}^{2} − DT′_{4}^{2} .

Moreover, \frac{T_{0}+T_{1}}{T_{0}+T_{2}} = \frac{lT′_{0}+T′_{1}}{lT′_{0}+lT′_{2}} = \frac{1}{l} \frac{lT′_{0}+T′_{1}}{T′_{0}+T′_{2}} . It suffices to find a ℚ_{l}-rational point on X′ so that \frac{lT′_{0}+T′_{1}}{T′_{0}+T′_{2}} is a l-adic unit.

The reduction of X′ modulo l is given by

T′_{1}^{2} =  −\overline{D}T′_{4}^{2} ,
−\overline{A}^{2}(T′_{0} − T′_{2})(T′_{0} + T′_{2}) = T′_{3}^{2} − \overline{D}T′_{4}^{2} .

We observe that the first equation has a nontrivial solution, as (\frac{D}{l}) = −1 and l ≡ 3 (mod 4) together imply that (−\overline{D}) ∈ 𝔽_{l}^{∗} is a square. Let ρ ∈ 𝔽_{l}^{∗} be one of its square roots.

The 𝔽_{l}-rational point

x = (\frac{\overline{A}^{2}+\overline{D}}{2\overline{A}^{2}} : ρ : \frac{\overline{A}^{2}−\overline{D}}{2\overline{A}^{2}} : 0 : 1) ∈ X′(𝔽_{l})

is non-singular, as is easily checked using the Jacobian criterion. The quotient \frac{lT′_{0}+T′_{1}}{T′_{0}+T′_{2}} is an l-adic unit for any l-adic lift of x, as required. The assertion follows. □

In order to provide a corresponding example in the case l ≡ 1 (mod 4), we need the following lemma.

Lemma 4.4. Let 𝔽_{l} be a finite field of characteristic ≠2. Then there exists an element κ ∈ 𝔽_{l} such that 2(1 + κ^{2}) is a non-square in 𝔽_{l}.

Proof. If l = 3 then put κ := 0. Otherwise, i.e. for l ≥ 5, let c ∈ 𝔽_{l} be any non-square. The equation cT_{0}^{2} = 2(T_{1}^{2} + T_{2}^{2}) defines a conic over 𝔽_{l}, which has exactly (l +1) 𝔽_{l}-rational points. Among them, at most four have x_{1} = 0 or x_{0} = 0. For the others, κ := x_{2}/x_{1} fulfils the required condition.

□

Example 4.5. Let l be a prime number that is either l ≡ 1 (mod 4) or l = 2. If l = 2 then choose B := 2, otherwise let B be an odd prime number that is split in ℚ(\sqrt{l}) and such that neither l(l − 1)(B^{2} − 1), nor (lB^{2} − 1)(B^{2} − 1), nor (l − 1)(lB^{2} − 1) is a perfect square.

Then the surface X ⊂ P_{ℚ}^{4} given by

(4.5)  −l(T_{0} − T_{1})(T_{0} + T_{1}) = T_{3}^{2} − lT_{4}^{2} ,
(4.6)  −(T_{0} − T_{2})(T_{0} + T_{2}) = T_{3}^{2} − B^{2}lT_{4}^{2}

is nonsingular and has a ℚ-rational point. Moreover, Br(X)/ Br(ℚ) ≅ ℤ/2ℤ and the nontrivial class works exactly at the place l.

Proof. Once again, the restrictions on B are easy to fulfil due to Dirichlet’s and Siegel’s theorems. Furthermore, the first three assertions follow directly from Theorem 4.1.A.a) and b) as well as Facts 2.4.a) and Fact 2.3. There is a Brauer class α ∈ Br(X), which may be understood as an extension of the quaternion algebra (ℚ(\sqrt{l})(X), τ, \frac{T_{0}+T_{1}}{T_{0}+T_{2}} ) over ℚ(X) to the whole scheme X. Moreover, Theorem 4.1.B.c) implies that the local evaluation map is constant at all places ν ≠ 2, l. Thus, all which remains to be shown is that ev_{α,l} is non-constant and that ev_{α,2} is constant in the case l ≡ 1 (mod 4).

Non-constancy of ev_{α,2} for l = 2: For ℚ(\sqrt{2}) , the prime 2 is ramified. A 2-adic unit u is a local norm from ℚ(\sqrt{2}) if and only if u ≡ ±1 (mod 8).

For \underline{x} = (1 : 1 : 1 : 0 : 0), we have ev_{α,2}(\underline{x}) = 0 by Theorem 4.1.B.b). On the other hand, there is the 2-adic point x = (1:0:\sqrt{−7}:0:1) ∈ X(ℚ_{2}) . Observe that (−7) ≡ 1 (mod 8) implies that (−7) is a square in ℚ_{2}. Moreover, we may choose \sqrt{−7} ∈ 5 + 16ℤ_{2} since 5^{2} ≡ −7 (mod 32). Then \frac{x_{0}+x_{1}}{x_{0}+x_{2}} = 1/(1 + \sqrt{−7}), which is in the residue class \frac{1}{2} ·(3 mod 8). Consequently, ev_{α,2}(x) = \frac{1}{2}.

BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 19

Non-constancy of ev_{α,l} for l ≡ 1 (mod 4): For ℚ(\sqrt{l}) , the prime l is ramified. A l-adic unit u is a local norm from ℚ(\sqrt{l}) if and only if (u mod l) ∈ 𝔽_{l}^{∗} is a square.

For \underline{x} = (1 : 1 : 1 : 0 : 0), we have ev_{α,l}(\underline{x}) = 0. On the other hand, the substitution T_{3} = lT′_{3} yields a different model X′ of X that is given by

(T_{0} − T_{1})(T_{0} + T_{1}) = T_{4}^{2} − lT′_{3}^{2} ,
(T_{0} − T_{2})(T_{0} + T_{2}) = l(B^{2}T_{4}^{2} − lT′_{3}^{2}) .

The 𝔽_{l}-rational point

x = ((1 + κ^{2}):2σ:(1 + κ^{2}):0:(1 − κ^{2})) ∈ X(𝔽_{l})

is non-singular for any κ ∈ 𝔽_{l} such that κ^{2} ≠ −1.

If, moreover, κ is chosen as in Lemma 4.4 then \frac{x_{0}+x_{1}}{x_{0}+x_{2}} = \frac{(1+κ)^{2}}{2(1+κ^{2})}, which is a non-square. Then, for every l-adic point that lifts x, the local evaluation map has value \frac{1}{2}. Constancy of ev_{α,2} for l ≡ 1 (mod 4): If l ≡ 1 (mod 8) then the prime p = 2 is split in ℚ(\sqrt{l}) and there is nothing to prove.

On the other hand, assume that l ≡ 5 (mod 8), in which case 2 is an inert prime. Then an element u ∈ ℚ_{2}^{∗} is a local norm from ℚ(\sqrt{l}) if and only if ν_{2}(u) is even. In particular, (−1) and (−l) are local norms, such that any of the quotients \frac{T_{0}±T_{1}}{T_{0}±T_{2}} may be used to evaluate α at the place 2.

Furthermore, any 2-adic point x ∈ X(ℚ_{2}) may be represented by coordinates x_{0}, …, x_{4} that are 2-adic integers, at least one of which is a unit. It is now a routine matter to determine all quintuples of residues modulo 8 that do not entirely consist of even ones and satisfy the system (4.5,4.6) of equations modulo 8. From the list obtained, one already sees that ev_{α,2}(x) = 0 in each case. □

Acknowledgements. The authors would like to thank the Hausdorff Center for Mathematics at Bonn for the excellent working conditions.

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BRAUER-MANIN OBSTRUCTION FOR DEGREE FOUR DEL PEZZO SURFACES 21

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[VA] Várilly-Alvarado, A.: Arithmetic of del Pezzo surfaces, Notes of lectures given at the Lorentz Center, Leiden, in October 2010, available

at: http://math.rice.edu/~av15/Files/LeidenLectures.pdf

[VAV] Várilly-Alvarado, A. and Viray, B.: Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups, Adv. Math. 255 (2014), 153– 181

[Wi] Wittenberg, O.: Intersections de deux quadriques et pinceaux de courbes de genre 1, Lecture Notes in Mathematics 1901, Springer, Berlin 2007

Département Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany

E-mail address: ~jjahnel@mathematik.uni-siegen.de
URL: http://wwwuser.gwdg.de/~jjahnel

Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, NL-3584 CD Utrecht, The Netherlands

E-mail address: d.schindler@uu.nl
URL: http://www.math.uni-bonn.de/people/dschindl