The Brauer-Manin Obstruction of Del Pezzo Surfaces of Degree 4

The Brauer-Manin Obstruction of del Pezzo Surfaces of Degree 4

Manar Riman August 26, 2017

Abstract Let k be a global field, and X a del Pezzo surface of degree 4. Under the assumption of existence of a proper regular model, we prove in particular cases that if the Brauer-Manin set is empty over k then it is empty over L where L is an odd degree extension of k. This problem provides evidence for a conjecture of Colliot-Thélèneand Sansuc about the sufficiency of the Brauer-Manin obstruction to existence of rational points on a rational surface.

Contents

1 Introduction 2

2 Rational Points 3 2.1 Brauer Group of a Field ...... 3 2.1.1 Cyclic Algebras ...... 5 2.1.2 The Brauer Group of a Local Field ...... 6 2.1.3 The Brauer Group of a Global Field ...... 6 2.2 The Brauer Group of a Scheme ...... 6 2.2.1 Hochschild-Serre Spectral Sequence ...... 7 2.2.2 Residue Maps ...... 8 2.2.3 The Brauer-Manin Obstruction ...... 9 2.3 Torsors ...... 9

3 Del Pezzo Surfaces 10 3.1 Geometry ...... 10 3.1.1 Picard Group ...... 13 3.2 Arithmetic ...... 14

4 The Brauer-Manin Set of a Del Pezzo surface of degree 4 14 4.1 The Main Problem ...... 14 4.2 Geometry and the Evaluation Map ...... 16 4.2.1 Purity: Gysin Sequence ...... 16 4.2.2 Constant Evaluation ...... 16

1 1 Introduction

Let X be a smooth, projective, geometrically integral variety over a global field k. We define the 1 adèlering to be the restricted product pk , q where Ω is the set of places of k. The vPΩk v Ov k existence of a k-point implies the existence of a kv-point for every place v of k, i.e., XpAkq ‰ H. We are interested in passing from local pointsś to global points. A class of varieties over k satisfies the Hasse principle if existence of local points implies existence of global points for every variety in the class. Quadrics are an example of varities satisfying the Hasse principle. Br Manin showed that the Brauer group can be used to construct a set XpAkq , that contains the k-points Xpkq and is contained in the adelic points XpAkq. If a variety X has XpAkq ‰ H and Br XpAkq “H, we say that there is a Brauer-Manin obstruction to the Hasse principle. Colliot-Thélèneand Sansuc in 1980 conjectured that the Brauer-Manin obstruction is sufficient for geometrically rational surfaces. Conjecture 1.1. (Colliot-Thélèneand Sansuc, [CTS80]) For geometrically rational surfaces over a number field, the Brauer-Manin obstruction is the only obstruction to the Hasse principle. Assuming Schinzel’s hypothesis and the finiteness of Tate-Shafarevich groups for elliptic curves, Wittenberg proved the conjecture for general del Pezzo surfaces of degree 4[Wit07]. We will study a consequence of the conjecture for a del Pezzo surface X of degree 4 which can be realized as the intersection of two quadrics [Section 3]. Quadrics satisfy the following theorems. Theorem 1.2. Springer Theorem [Lam05, thm 2.7] Let Q be a quadric over k and L{k an odd degree extension. Let X be the vanishing set of Q. If X has an L-point then X has a k-point. Theorem 1.3. Amer-Brumer Theorem [Lam05] Let Q1, and Q2 be two quadrics over k. Let X “ V pQ1,Q2q over k, and Xλ “ V pQ1 ` λQ2q over kpλq where λ is an indeterminant. Then

Xpkq ‰ H ðñ Xλpkpλqq ‰ H. We deduce that for a del Pezzo surface X of degree 4 over k, and L{k an odd degree extension, the existence of an L-point implies the existence of a k-point. Assuming Colliot-Th´el`eneand Sansuc’s conjecture, we have the following proposition.

Proposition 1.4. Let X be a Del Pezzo surface of degree 4 over k such that XpAkq ‰ H, and L{k an odd degree extension. Then

Br Br XpAkq “ H ùñ XpALq “H. In this paper we prove Proposition 1.4 under the assumption of the existence of a proper regular model when Br X{ Br k » Z{2Z, the special fiber of the model is geometrically irreducible, and the residue field is sufficiently big.

Outline

In section 2 we define the Brauer group and the Brauer-Manin obstruction. In section 3 we provide equivalent definitions of del Pezzo surfaces and prove some of its geometric and arithmetic properties. Finally in the the last section we characterize our problem in terms of valuations and prove it under the assumption of the existence of a proper regular model when Br X{ Br k » Z{2Z, the special fiber of the model is geometrically irreducible, and the residue field is sufficiently big.

2 Notation k: a field ks: the separable closure of k k: an algebraic closure of k Ωk: set of places of k Ak: adèlering of k Xpkq: k-points of X XpAkq: adèlicpoints of X

By a nice surface we mean a smooth projective, and geometrically integral surface. On such a surface there exists a pairing of the divisors p., .q: Pic X ˆ Pic X Ñ Z [Har77, V.3]. For a nice variety the Picard group Pic X coincides with the group of Weil divisors modulo linear equivalence. We denote by KX the class of the canonical sheaf ωX in Pic X. An exceptional curve on a smooth projective k-surface is an irreducible curve C in Xk such that pC,Cq “ pKX ,Cq “ ´1.

2 Rational Points

2.1 Brauer Group of a Field In this section we deﬁne the Brauer group of a ﬁeld in terms of Azumaya algebras. Then we deduce a cohomological characterization of the Brauer group.

Deﬁnition 2.1. An Azumaya algebra A over k is a k-algebra such that A bk ks is isomorphic, as a ks-algebra, to Mnpksq for some n ě 1. Deﬁnition 2.2. A quaternion algebra over k is a 4-dimensional Azumaya algebra over k.

Example 2.3. Hamilton’s ring of quaternions H is a 4 dimensional R algebra generated by i and j 2 2 satisfying i “ j “ ´1 and ji “ ´ij. It is a quaterion algebra because H b C » M2pCq. Example 2.4. Let pa, bq where a, b P k˚ be the k-algebra generated by i, j satisfying i2 “ a, j2 “ b and ij “ ´ji. The algebra pa, bq is a quaternion algebra.

Let Azk be the category of Azumaya algebras over k with k-algebra homomorphisms as morphisms. The opposite algebra Aopp is a k-algebra with the same k-vector space structure as A but multiplication deﬁned as a ¨ b “ ba.

opp Proposition 2.5. If A and B are in Azk then so are A and A bk B. Furthermore if L is a ﬁeld extension of k then A bk L P AzL.

Proof. By deﬁnition of an Azumaya algebra we have A bk ks » Mnpksq, and B bk ks » Mmpksq opp opp opp for some positive integers m, n. We have A bk ks » Mnpksq » Mnpksq so A P Azk. Also

A bk B bk ks » A bk ks bks B bk ks » Mnpksq bks Mmpksq » Mn`mpksq so A bk B P Azk. Finally

A bk L bk Ls » A bk ks bks Ls » Mnpksq bks Ls » MnpLsq so A bk L P AzL.

We deﬁne an equivallence relation on the set Azk as follows. Two Azumaya algebras A and B are similar A „ B if either one of the equivalent following conditions is satisﬁed:

(a) There exists m, n ě 1 and a division algebra D P Azk such that A » MnpDq and B » MmpDq as k-algebras.

3 (b) There exists m, n ě 1 such that MnpAq » MmpBq as k-algebras. Deﬁnition 2.6. The Brauer group of a ﬁeld k, which we denote by Br k is, the set of equivalence classes of Azumaya algebras by the above relation, i.e.,

Br k :“ Azk{„

It is a group under tensor product and inverse deﬁned by Aopp.

If L is a finite extension of k then there is a group homomorphism Br k Ñ Br L defined by A ÞÑ A bk L. In fact Br is a covariant functor from fields to abelian groups. In the remainder of this section we derive the cohomological characterization of Br k.Let G “ Galpks{kq. Consider A P Azk, and σ P G. By definition of an Azumaya algebra there is an isomorphism Φ : A bk ks Ñ Mrpksq for some r ą 0. We want to define the action of σ on Φ. Let σ act on A b ks by acting on the second factor, and act on Mrpksq entrywise. We define the action of σ σ on the isomorphism Φ : A bk ks Ñ Mrpksq to be Φ satisfying the commutative diagram:

A b k M pk q k s Φ r s σ σ A b k M pk q. k s σΦ r s Proposition 2.7. There is a natural injection from the isomorphism classes of Azumaya algebras 1 to H pG, PGLrpksqq;

2 tAzumaya algebras over k of dimension r u 1 ãÑ H pG, PGLrpksqq. „ Proof. Let A be an Azumaya algebra of dimension r2. Then there is a k-algebra isomorphism s s Φ: Mrpk q Ñ A bk ks. We deﬁne the image cocycle to be f : G Ñ PGLrpk q by

´1 σ s s fσ “ Φ p Φq P Autk´algebraspMrpk qq “ PGLrpk q.

It is indeed a cocycle because

´1 στ ´1 σ σ ´1 στ ´1 σ σ ´1τ σ fστ “ Φ p Φq “ Φ p Φqp Φ qp Φq “ Φ p Φqp pΦ Φq “ fσ. fτ

s If we compose Φ with an automorphism of Mrpk q, i.e., the algebra corresponds to an isomorphic algebra, we get a cohomologous cocycle. In fact, the dimension is ﬁxed for the class of algebras we are considering so equivalence class is the same as isomorphism class of the algebras. Hence, the map passes to the quotient.

In fact, the injection deﬁned in Proposition 2.7 is a bijection as we explain now.

1 1 Deﬁnition 2.8. Let k {k be a Galois extension and X a k-variety. A k {k twist of X is a k-variety „ Y such that Xk1 ÝÑ Yk1 . Theorem 2.9. [Ser97, III.1 Proposition 1] There is a bijection

1 tk {k ´ twists of Xu „ 1 1 ÝÑ H pGalpk {kq, AutpX 1 qq. k ´ isomorphisms k Corollary 2.10. The injection deﬁned in Proposition 2.7 is a bijection.

4 2 s Proof. An Azumaya algebra A of dimension r is a k {k twist of Mrpkq by deﬁnition. Furthermore, 1 AutpXk1 q “ PGLrpk q. We have the natural short exact sequence

˚ 1 Ñ ks Ñ GLrpksq Ñ PGLrpksq Ñ 1 which gives rise to a long exact sequence of cohomology. One of the boundary maps gives a group homomorphism 1 2 ˚ H pGalpks{kq, PGLrpksqq Ñ H pGalpks{kq, ks q 2 ˚ that associates to each Azumaya algebra A a class rAs in H pGalpks{kq, ks q. We also denote 2 ˚ 2 H pGalpks{kq, ks q by H pk, Gmq. The homomorphism

2 Br k Ñ H pk, Gmq is an isomorphism as explained in [Ser79, Chapter X,Section 5].

2.1.1 Cyclic Algebras Definition 2.11. Let L{k be a cyclic extension of degree n. Fix σ a generator of GalpL{kq, and fix a P k˚. We define the cyclic algebra pσ, aq to be

Lrxsσ pσ, aq :“ xxn ´ ay

σ where Lrxsσ is the twisted polynomial ring with multiplication deﬁned as x` “ ` x.

One can prove that pσ, aq is a central simple algebra [GS06, Section 2.5]. Furthermore, a finitely dimensional central simple algebra is an azumaya algebra [GS06, Theorem 2.2.1]. Hence pσ, aq P Br k. We associate to a character χ: GalpL{kq Ñ Z{nZ the cyclic algebra pτ, aq where a P k˚ is fixed and τ “ χ´1p1q; this algebra is also refered to by pχ, bq. We say that A P Br k is split by L{k if it lies in the kernel of the map Br k Ñ Br L. This kernel, which is denoted by BrpL{kq, is isomorphic to H2pGalpL{kq,L˚q by the inflation-restriction sequence below [Poo, Proposition 1.5.13]:

2 ˚ inf 2 res 2 0 Ñ H pGalpL{kq,L q ÝÝÑ H pk, Gmq ÝÝÑ H pL, Gmq.

Theorem 2.12. [GS06, corollary 4.7.4] We have

k˚ „ Br L k ˚ ÝÑ p { q NL{kpL q deﬁned by a ÞÑ pσ, aq.

˚ Corollary 2.13. The cyclic algebra pσ, aq is split by L if and only if a P NL{kpL q. Notation 2.14. We will abuse notation and denote the cyclic algebra by pL{k, aq when the generater of GalpL{kq is not essential for the argument. If k contains the nth roots of unity then by Kummer ? ? theory any cyclic extension of k is of the form kp n aq. In this case we refer to pkp n aq{k, bq by pa, bqn.

5 2.1.2 The Brauer Group of a Local Field Let k be a nonarchimedian local field. As defined by Serre [Ser79, Introduction, section 1] there is an isomorphism „ inv: Br k ÝÑ Q{Z. This map is known as the invariant map. It satisfies the following properties.

Proposition 2.15. [Mil, Example 4.2, Proposition 4.3] Let L{k be an unramified cyclic extension of a nonarchimedian local field and consider pσ, bq P Br k where σ P GalpL{kq. Then vpbq invppσ, bqq “ . rL : ks Proposition 2.16. [Ser79, Theorem 3] If L is a finite extension of k then the following diagram commutes: Br k { inv Q Z rL:ks Br L { . inv Q Z Proposition 2.17. Every element of Br k is a cyclic algebra.

Proof. For every m{n P Q{Z, consider the cyclic algebra pσ, aq corresponding to the unramiﬁed ˚ extension Lw{kv of degree n, and a P kv of valuation m. By Proposition 2.15, invppσ, aqq “ m{n. This characterizes all elements of Br kv.

2.1.3 The Brauer Group of a Global Field

Let k be a global ﬁeld and let Ωk be its set of places. Denote by kv the completion of k at v P Ωk, and invv : Br kv Ñ Q{Z the corresponding invariant map. The fundamental exact sequence of global class ﬁeld theory completely characterizes Br k:

v invv 0 Ñ Br k Ñ‘vPΩk Br kv ÝÝÝÝÝÑ Q{Z. ř Proposition 2.18. If L{k is a ﬁnite extension then the following diagram commutes, where the rows are exact: 0 Br k ‘v Br kv Q{Z 0 v invv ř rL:ks

0 Br L ‘v ‘w Br Lw Q{Z 0. w invw Proof. The left square commutes by functoriality of Br.ř The right square commutes by Proposition

2.16, and the identity w{vrLw : kvs “ rL : ks for a finite extension L{k. ř 2.2 The Brauer Group of a Scheme 2 We have seen that Br k » H pSpec k, Gmq is the cohomological definition of the Brauer group of a field. Let us generalize this definition to a scheme X. Definition 2.19. For a scheme X, we define the cohomological Brauer group as

2 Br X :“ H´etpX, Gmq.

6 It follows by functoriality of cohomology that Br is a contravariant functor from the category of schemes to the category of abelian groups. Theorem 2.20. Let X be a regular, integral, noetherian scheme. The map Br X Ñ Br KpXq is injective where KpXq is the function ﬁeld of X. So Br X is a torsion abelian group. Proof. See [Gro68a, Corollary 1.10] to prove injectivity. Furthermore, Br KpXq is deﬁned by Galois cohomology hence it is torsion. So Br X is also torsion.

In the following section we present two main methods that have been used in the literature to calculate the Brauer group. One uses the Hochschild-Serre spectral sequence and the other uses the residue maps.

2.2.1 Hochschild-Serre Spectral Sequence Recall that a spectral sequence p,q p`q E2 ñ L . gives rise to a long exact sequence [Poo, Section 6.7]:

1,0 1 0,1 2,0 2 0,2 1,1 3,0 0 Ñ E2 Ñ L Ñ E2 Ñ E2 Ñ kerpL Ñ E2 q Ñ E2 Ñ E2 . Let K 1 be a ﬁnite Galois extension of k with Galois group G. The Hochschild-Serre spectral sequence

p,q p q p`q p`q E2 :“ H pG, H´etpXK , Gmqq ñ H´et pX, Gmq “: L . will be used in the Brauer group computation. Theorem 2.21. Let X be everywhere locally soluble scheme over k then

„ 1 Br1 X{ Br k ÝÑ H pGalpks{kq, Pic Xks q where Br1 X :“ kerpBr X Ñ Br Xks q, known as the algebraic Brauer group. Proof. Let K be a finite Galois extension of k. We have H1pG, K˚q “ 0 by Hilbert 90. Also, 1 2 H pX, Gmq “ Pic X [Poo, Proposition 6.6.1], and HétpX, Gmq “ Br X by definition. Then the Hochschild-Serre spectral sequence in this case gives rise to the long exact sequence

G 2 ˚ 1 3 ˚ 0 Ñ 0 Ñ Pic X Ñ pPic XK q Ñ H pG, K q Ñ kerpBr X Ñ Br XK q Ñ H pG, Pic XK q Ñ H pG, K q.

Taking the limit over all ﬁnite extensions of k we get the same sequence over ks. By a result due 3 ˚ to Tate [NSW08, Theorem 8.3.11(iv)], we have that H pGalpks{kq, ks q “ 0. Furthermore, local solubility of X induces a morphism Spec kv Ñ X which splits the structure map πv : Xkv Ñ Spec kv. ˚ Thus by functoriality of the Brauer group we deduce that the induced map πv : Br kv Ñ Br Xkv ˚ also splits. Because πv splits and Br k Ñ‘v Br kv is injective, the natural map Br k Ñ Br X is injective. Hence the exact sequence now becomes a short exact sequence

1 0 Ñ Br k Ñ Br1 X Ñ H pGalpks{kq, Pic Xks q Ñ 0.

In general, if X is not locally soluble then there is an isomorphism

„ 1 Br1 X{ Br0 X ÝÑ H pGalpks{kq, Pic Xks q where Br0 X :“ ImpBr k Ñ Br Xq.

7 2.2.2 Residue Maps Let X be regular, integral, noetherian scheme, and D a Weil divisor. We associate to D the discrete valuation ring R with valuation v and ‘quotient field K “ KpXq. Let F be the residue field of R with respect to v, let p be the characteristic of F , and let Knr be the maximal unramified extension 1 of K with respect to v. If A is an abelian group then we denote by App q the prime to p subgroup of A. The valuation v extends uniquely to Knr and induces a map

2 ˚ 1 2 1 f1 : H pGalpKnr{Kq,Knrqpp q Ñ H pGalpKnr{Kq, Zqpp q.

Furthermore, the exact sequence

0 Ñ Z Ñ Q Ñ Q{Z Ñ 0 induces an isomorphism

1 f2 2 H pGalpKnr{Kq, Q{Zq “ HompGalpKnr{Kq, Q{Zq ÝÑ H pGalpKnr{Kq, Zq.

1 2 1 ´1 We deﬁne the residue map Bv : pBr Kqpp q Ñ H pGalpKnr{Kq, Q{Zqpp q as the composite f2 ˝ f1. Proposition 2.22. [Sal99] Let pχ, aq be a cyclic algebra corresponding to the cyclic extension L{k ´1 of degree m. The map Bvpχ, aq factors through GalpL{kq and sends τ “ χ p1q to 1{m ` Z. Theorem 2.23. [Sal99, Theorem 10.3] The following sequence is exact

1 1 0 Ñ pBr Rqpp q Ñ pBr Kqpp q Ñ HompGalpKnr{Kq, Q{Zq Ñ 0.

In general if we consider all codimension 1 points of a regular integral noetherian scheme X we get the following result.

Theorem 2.24. Let X be a regular integral noetherian scheme. Then we have the exact sequence

‘Bx 1 0 Ñ Br X Ñ Br KpXq ÝÝÑ H pkpxq, Q{Zq xPXp1q à where Xp1q is the set of codimension 1 points of X and kpxq is the residue ﬁeld corresponding to x. However this is true with the caveat of excluding the p-primary part if dim X ď 1 and some kpxq is imperfect of characteristic p, or dim X ě 2 and some kpxq is of characteristic p.

Proof. This follows from [Gro68b, Proposition 2.1], and Grothendiecks absolute cohomological purity theorem [Fuj02].

To deﬁne the residue maps Bx it is enough to deﬁne them on cyclic algebras as before and then apply Merkurjev-Suslin theorem [GS06, Theorem 4.6.6].

Corollary 2.25. If L{KpXq is a cyclic extension with prime degree q that is unramiﬁed at x P Xp1q then pσ, bq P ker Bx if and only if vxpbq “ 0 pmod qq.

Proof. By deﬁnition of the residue map if vpbq “ 0 pmod qq then Bxppσ, bqq “ 0. If vpbq ‰ 0 pmod qq then Bxppσ, bqq “ 0 if and only if the degree of L{KpXq is 1 which can not occur because q is prime.

8 2.2.3 The Brauer-Manin Obstruction

Let k be a global ﬁeld with set of places Ωk. Let X a proper k-scheme and consider A P Br X.A point P P Xpkq can be interpreted as a section P : Spec k Ñ X. We deﬁne

˚ evA pP q :“ P A .

There is a finite set of places S Ă Ωk such that for every v R S, X can be extended to a SpecpOvq scheme, and A to an element of Br Xv. For any point Pv P X pOvq, evA pPvq P Br Ov which is trivial. Hence for pPvqvPΩ P XpAkq we have invvpevA pPvqq “ 0 for almost all places v of k. So the sum vPΩ invvpevA pPvqq is well defined. Definitionř 2.26. Let A P Br X. We define

A XpAkq :“ tpPvq P XpAkq : invvpevA pPvqq “ 0u. vPΩ ÿ And the Brauer-Manin set Br A XpAkq :“ XpAkq . PBr X A č Br Proposition 2.27. We have the inclusions Xpkq Ă XpAkq Ă XpAkq. Proof. The ﬁrst inclusion follows from the fundamental exact sequence of class ﬁeld theory

Xpkq XpAkq

evA evA

v invv 0 Br k ‘v Br kv Q{Z. ř The second inclusion follows from the deﬁnition of the Brauer-Manin set.

We say that there is a Brauer-Manin obstruction to the Hasse principle or local-global Br principle for X if XpAkq ‰ H and XpAkq “H.

2.3 Torsors Later in Section 6.3.2 we will use the language of torsors; we brieﬂy recall this now. For a reference see [Sko01].

Deﬁnition 2.28. Let G Ñ S be an fppf group scheme over a general base S.A G-torsor over S is an fppf S-scheme X endowed with a right G-action X ˆS G Ñ X such that it satisﬁes one of the following equivalent conditions:

1 1. XS1 » GS1 for some fppf base change S Ñ S.

2. The map X ˆS G Ñ X ˆS X given by px, gq ÞÑ px, xgq is an isomorphism. Theorem 2.29. [Poo, Theorem 6.5.10] Let G be an fppf group scheme over a locally noetherian scheme S. Assume that G Ñ S is an abelian scheme, and G is locally factorial. Then we have

tG ´ torsorsu ÝÑ„ H1pS, Gq. isomorphisms

9 Proposition 2.30. Let X be a G-torsor as above. The following are equivalent

1. X is isomorphic to the trivial torsor G.

2. XpSq ‰ H

3. The class of rXs in H1pS, Gq is the neutral element.

Proof. The implication p1q ñ p2q follows by the existence of the identity element in GpSq. The reverse implication p2q ñ p1q follows by base extending XX » GX over the section S Ñ X to get the isomorphism XS » GS. Finally p1q ðñ p3q by Theorem 2.29.

3 Del Pezzo Surfaces

3.1 Geometry b´1 Deﬁnition 3.1. A del Pezzo surface X is a nice k-surface with ample anticanonical sheaf ωX . The degree of X is the intersection number d “ pKX ,KX q.

Following the exposition in [VA13, Section 1] to prove that, we will prove that over ks, the 2 separable closure of k, a del Pezzo surface of degree d is isomorphic to the blowup of P at 9 ´ d 1 1 points or to P ˆ P . In the latter case the degree is 8. To prove this we will need the following propositions from [Coo88] and the lemma.

Proposition 3.2. [Coo88, Proposition 1.7] Let X and Y are smooth projective surfaces over a separably closed ﬁeld k and f : X Ñ Y be any birational morphism. Then f factors as

X “ X0 Ñ X1 Ñ ¨ ¨ ¨ Ñ Xr “ Y, where each map Xi Ñ Xi`1 is a blowup of a k-closed point of Xi`1. The minimal model program gives the factorization of f over k. The idea of the proof of Proposition 3.2 is to show that the blow-up at a closed point whose residue ﬁeld is a nontrivial purely inseparable extension of k cannot give rise to a smooth surface. Therefore f factors over a separably closed ﬁeld as a sequence of blowups of closed points.

Proposition 3.3. [Coo88, Theorem 1.3] The minimal smooth projective rational surfaces over a 2 separably closed ﬁeld k are and the Hirzebruch surfaces n :“ pO 1 ‘ O 1 pnqq for n “ 0 and Pk F P Pk Pk n ě 2.

Lemma 3.4. Let X be del Pezzo surface of degree d over an algebraically closed ﬁeld k. Then every irreducible curve with negative self intersection is exceptional.

Proof. The adjunction formula applied to C is

2papCq ´ 2 “ pC,Cq ´ pC, ´KX q.

The arithmetic genus of C satisﬁes papCq ě 0 since C is irreducible. Also pC,Cq ă 0 and pC, ´KX q ą 0 since ´KX is ample. Then the only way we have equality is when papCq “ 0 and pC,Cq “ pC,KX q “ ´1. Hence C is exceptional.

2 We say that r points in P are in general position if all of the following conditions hold:

10 • no three points lie on a line,

• no six points lie on a conic, and

• no eight points lie on a cubic with one point on a singularity.

Theorem 3.5. A scheme X is a del Pezzo surface of degree 1 ď d ď 9 over ks if and only if it is 2 1 1 isomorphic to the blowup of s at 9 d points in general position, or to . In the latter Pk ´ Pks ˆ Pks case d “ 8.

Proof. We consider Y a minimal model of X and the corresponding birational morphism f : X Ñ Y . By Proposition 3.2, f can be factored as

X “ X0 Ñ X1 Ñ ¨ ¨ ¨ Ñ Xr “ Y.

By Proposition 3.3 we only need to consider the following cases:

2 Y “ P : By Lemma 3.4 there are no irreducible curves on X of degree less than ´1. Hence no point that is blown up in one step lies on the exceptional curve of a previous exceptional curve. 2 r So X is the blow up of P at r distinct points. The canonical divisor is KX “ ´3l ` 1 ei 2 [Har77, V.6.6] where l is the line in P not passing through any of the r points. Therefore the 2 ř degree is d “ KX “ p´3l, ´3lqq ´ r “ 9 ´ r. It remains to show that the points are in general ˚ ˚ 3 3 2 position. If three points lie on a line L then pL, Lq “ pπ L, π Lq ´ 2 1 ripL, eiq ` 1 ri by [Har77, V 3.6] and the linearity of the intersection pairing where L is the strict transform ř ř of L and ri is the multiplicity of Pi on L. Sincer r pL, eiq “ ri by [Har77, Vr 3.7], it follows ˚ ˚ 3 2 that pL, Lq “ pπ L, π Lq ´ 1 ri ď 1 ´ 3 “ ´2 ă ´1 by [Har77r, V 3.2(a)] and [Har77, V 3.5.2] which contradicts Lemma 3.4. Similarly,r when 6 of the points lie on a conic C ˚ ˚ ř 3 2 then pC,r rCq “ pπ C, π Cq ´ 1 ri ď 4 ´ 6 “ ´2 ă ´1 which contradicts Lemma 3.4. If 8 points lie on a singular cubic E with one point Pj at the singularity, we get pE, Eq “ ˚ ˚ 3 2 ř pπ E, πr Erq ´ 1 ri ď 9 ´ 7 ´ 4 “ ´2 ă ´1 because rj ě 2 by [Har77, V 3.5.2]. This also contradicts Lemma 3.4. r r ř Y “ F0: If X “ Y then X is a del Pezzo surface of degree pKX ,KX q “ 2pOp´2q, Op´2qq “ 8. If 1 1 X ‰ Y then Xr´1 is the blow-up of P ˆ P at a point which is isomorphic to the blowp-up of 2 P at two points. If we blow down the two exceptional curves corresponding to the blow-up of 2 2 P at two points we get a birational map to P . Hence we constructed a birational map from 2 2 Xr´1 to P and so from X to P . Therefore we reduce to case 1. Y “ : If n ě 2 then there is a curve on Y that satisﬁes pC,Cq ă ´1. Its strict transform f ´1pCq Fn k also has intersection multiplicity ă ´1, which contradicts Lemma 3.4.

2 Now we prove the converse, i.e., P blown up at 0 ď r ď 8 points in general position is a 2 del Pezzo surface of degree d “ 9 ´ r. The blowup of P is a nice variety. It remains to prove is that the anti canonical sheaf is ample; for this we use the Kleiman condition. We need to prove that ´KX . ´ KX ą 0 and ´KX .C ą 0 for any irreducible curve on X. Let te1, . . . , eru be 2 the classes of the exceptional divisors and l is the pullback of a line in P not passing through r any of the blown up points. The canonical divisor is ´KX “ 3l ´ 1 ei [Har77, V.6.6]. So 2 2 KX “ 9 ´ r ą 0 where 0 ď r ď 8. Now let π : X Ñ P be the monoidal transformation associated ř 2 to the blowup. Let D “ πpCq and P1,...Pr be the points that are blown up in P . We have

´KX .C “ 3l.C ´ ei.C “ 3 deg D ´ multPi pDq where the last equality follows by [Har77,V ř ř 11 2 3.2(a)]. Let Q be a point on D distinct from P1,...,P5 in P . Consider a conic passing through 4 points P1,...,P4 and a line through P5 and Q; this is possible because the points P1,...,P5 are in general position. Let E be the cubic deﬁned by the union of the conic and the line. We have D.E “ r 3 deg D “ pPDXE ipD, E, pq “ pPDXE multppDqmultppEq “ pPDXE multppDq ą 1 multPi pDq. So ´KX .C ą 0 for any irreducible curve C on X. By the Kleiman condition, the anti canonical ř ř 2 ř ř sheaf is ample. Hence the blowup of P at r points in general position is a del Pezzo surface. Theorem 3.6. A scheme X is a del Pezzo surface of degree 4 if and only if it is isomorphic to an 4 integral smooth complete intersection of two quadrics in P .

Lemma 3.7. Let X be a del Pezzo surface of degree 4. The linear system | ´ KX | separates points and tangent vectors. 2 Proof. By Theorem 3.5, X is the blowup of P at 5 points P1,P2,...,P5. Denote the blowdown 2 by π : X Ñ P . The idea is to prove the divisors in | ´ KP2 | “ |3l| that pull back to a divisor r in | ´ KX | “ |3l ´ 1 ei| separate the image under π of points and tangent vectors in X. Let Q1 and Q2 be two distinct points on X. First we consider the case when Q1 and Q2 don’t lie on the ř 2 same exceptional curve. Consider the points tP1,...,Pr, πpQ1q, πpQ2qu in P . There exists a conic passing through 4 points but not πpQ1q. Consider this conic union the line passing through the remaining 2 points to be the cubic that passes through πpQ1q but not πpQ2q. The blowup of this cubic is a divisor in the linear system | ´ KX | separating Q1 and Q2 in X. Now we consider the case when the points Q1 and Q2 are on the same exceptional curve. In this case consider the line in 2 P passing through πpQ1q of a ﬁxed slope and one of the points P1,...,Pr union the conic passing through the remaining blown up points and blow up to get a divisor separating the points Q1 and Q2 in X. Hence | ´ KX | separates points. We now prove that | ´ KX | separates tangent vectors. Consider a point Q and a tangent direction tQ at Q in X. If the tangent direction is not along 2 an exceptional curve then we consider a line in P passing through πpQq and some point Pi such that πptQq is not along this line union the conic passing through the remaining points Pj where 2 j ‰ i. The line and the conic form a cubic in P whose blowup separates Q and tQ. If tQ is along an exceptional curve then consider the line of a ﬁxed slope through πpQq and a point Pi union 2 the conic through the remaining points Pj, j ‰ i in P . The blowup of the constructed cubic is a divisor in | ´ KX | that separates Q and tQ. Therefore the linear system | ´ KX | separates points and tangent vectors.

4 Proof of Theorem 4.7. First let’s prove that we can embed X in P by the linear system of | ´ ωX | where ´ωX is ample. Apply Riemann Roch to the divisor ´mωX where m ą 0 as follows: 0 1 0 h pX, ´mωX q ´ h pX, ´mωX q ` h pX, p1 ` mqωX q “ 1{2p´mωX , p´m ´ 1qωX q ` χpX, OX q. (1) 1 By Kodaira vanishing it follows that h pX, ωX q “ 0. So the Euler characteristic is 0 χpX, OX q “ h pX, OX q “ 1. 1 The second term h pX, ´mωX q of (1) vanishes by Kodaira vanishing theorem. The third term 0 h pX, p1 ` mqωX q of (1) vanishes because ´ωX is ample and so its inverse doesn’t have global sections. Hence 0 h pX, ´mωX q “ 1{2mpm ` 1qp4q ` 1. 0 4 4 For m “ 1, h pX, ´ωX q “ 5. Let i : X Ñ P be the morphism of X to P given by | ´ ωX |. By Lemma 3.5, the linear system |´ωX | separates points and tangent vectors. Hence, i is an embedding 4 of X into P . We have the exact sequence

0 Ñ IX Ñ OP4 Ñ i˚OX Ñ 0.

12 Twisting we get the exact sequence

0 Ñ IX p2q Ñ OP4 p2q Ñ i˚OX p´2ωX q Ñ 0. This yields the long exact sequence of cohomology

0 4 0 4 0 0 Ñ H pP , IX p2qq Ñ H pP , OP4 p2qq Ñ H pX, ´2ωX q.

0 4 6 0 We have h pP , OP4 p2qq “ 2 “ 15. To calculate h pX, ´2ωX q we apply Riemann Roch again. Recall that ` ˘ 0 h pX, mωX q “ 1{2mpm ` 1qp4q ` 1. So we deduce that 0 h pX, ´2ωX q “ 1{2p2qp3qp4q ` 1 “ 13. 0 Hence by the exact sequence of cohomology above, h pX, IX p2qq is at least 2-dimensional. So X is contained in the intersection of two quadrics Q1 and Q1 of linearly independent deﬁning equations. 4 By [Har77, Exercise 8.4(d)], Q1 X Q2 is a complete intersection. Hence it is of codimension 2 in P . The del Pezzo surface X is of dimension 2 and degree 4 and X Ă Q1 X Q2 of same dimension and degree. Hence X “ Q1 X Q2 Conversely let Q1 X Q2 be a smooth integral intersection of two quadrics Q1 and Q2. Let 4 i: Q1 X Q2 Ñ Q1 and j : Q1 Ñ P . By the adjuction formula we have

˚ ˚ ωQ1XQ2 “ pj ˝ iq pωP 4 b OP4 pQ1q b OP4 pQ2qq “ pj ˝ iq OP4 p´5 ` 2 ` 2q “ OQ1XQ2 p´1q.

Hence ´ωQ1XQ2 is ample. So Q1 X Q2 is a del Pezzo surface. The intersection Q1 X Q2 is a complete intersection so by Bezout’s theorem its degree is 2 ˆ 2 “ 4.

3.1.1 Picard Group Theorem 3.8. Let X be a del Pezzo surface of degree d over a separably closed ﬁeld not isomorphic 1 1 2 10 d to P ˆ P , i.e., isomorphic to the blowup of P at 9 ´ d points. Then Pic X » Z ´ and ´KX “ 3l ´ ei.

2 Proof. Let Xiřbe the total transform of P after blowing up the i-th point. For every blow up map at the point Pi we have a projection πi : Xi`1 Ñ Xi. We notice that PicpXiq » PicpXi ´ Piq by [Har77, II 6.5] and PicpXi ´ Piq » PicpXi`1 ´ Eiq because Xi ´ Pi » Xi`1 ´ Ei. Each Ei is of codimension 1 so we get the exact sequence

Z Ñ Pic Xi`1 Ñ Pic Xi Ñ 0

2 2 where the first map maps 1 to Ei. Moreover, for every n ‰ 0, pnEq “ ń ‰ 0 and so the first ˚ map is injective. It is also split by πi : Pic Xi Ñ Pic Xi`1, and hence

Pic Xi`1 “ Pic Xi ‘ Z.

2 9´d 10´d Carrying this calculation over all blowups we get Pic X “ Pic X9´d “ Pic P ‘ Z » Z . Let te1, . . . , er, lu be the generators of Pic X where te1, . . . , eru are the classes of the exceptional divisors 2 and l is the pullback of a line in P not passing through the blown up points. ˚ It follows from [Har77, V.6.6] that KX “ π pKP2 q ` ei. Since KP2 » Op´3q, we get KX “ ´3l ` ei. Hence ´KX “ 3l ´ ei. ř ř ř 13 To study the action of the Galois group on Pic X we need to ﬁnd the exceptional curves. The intersection multiplicities are as follows pei, eiq “ ´δi,j, pei, lq “ 0,and pl, lq “ 1. Let C “ r al ´ i“1 biei be an exceptional curve. Then pC,Cq “ pC,KX q “ ´1 gives the two equations r ř 2 2 a ´ bi “ ´1 i“1 ÿ r 3a ´ bi “ 1 i“1 ÿ which depend on r “ 9 ´ d. For d “ 4, we get 16 exceptional curves see [VA13, Section 1.5]. We will study the action of the Galois group on the exceptional curves. 1 Consider σ P Galpks{kq and σ : Spec ks Ñ Spec ks the corresponding morphism on schemes. After base changing to ks we get 1 idX ˆ σ : Xks “ X ˆk Spec ks Ñ Xks .

This morphism induces an automorphism on Pic Xks . Hence we obtain an action Galpks{kq Ñ

AutpPic Xks q. This action fixes the canonical class and the intersection multiplicity [Man74, Theorem 23.8]. We define the splitting field of X to be the smallest extension of k where the exceptional divisors are defined. Then the Galois group GalpK{kq acts on the coefficients of the exceptional curves resulting in an action on Pic XK .

3.2 Arithmetic The ﬁrst step in computing the Brauer-Manin set is to compute the Brauer group Br X{ Br k. Recall 1 that this is isomorphic to H pGalpks{kq, Pic Xks q by Theorem 2.21. Theorem 3.9. [Man74] Let X be a del Pezzo surface of degree 4. Then Br X{ Br k is isomorphic to the trivial group, Z{2Z, or Z{2Z ˆ Z{2Z. Remark 3.10. Some algorithms for calculating the Brauer classes and the corresponding Magma scripts can be found in the papers [BBFL07] and [VAV14]. Br We will be interested in calculating XpAkq for a degree 4 del Pezzo surface X.

4 The Brauer-Manin Set of a Del Pezzo surface of degree 4

4.1 The Main Problem Recal the problem: Problem 4.1. Let k be a number ﬁeld and L{k be an odd degree extension. Let X be a Del Pezzo surface of degree 4 over k such that XpAkq ‰ H. Prove that Br Br XpAkq “ H ùñ XpALq “H.

Br Xk Br XL For the rest of the paper we will consider the case when Br k “ xA y » Z{2Z, and Br L “ Br B xA y » Z{2Z for all odd degree ﬁeld extensions L{k. So then we have XpAkq “ BPBr X XpAkq “ A Br XpAkq . The same holds for XpALq . Thus the problem reduces to the statement that Ş A A XpAkq “ H ùñ XpALq “H. We will give an equivalent characterization of this statement in terms of valuations after the following lemma.

14 Lemma 4.2. Let k be a number ﬁeld. Let X be a variety over k with XpAkq ‰ H, and let A P Br X. If there exists a place v such that inv : X k 1 is surjective then X A . 0 v0 A p v0 q Ñ ord A Z{Z pAkq ‰H 1 Proof. Fix pPvqvPΩ P XpAkq. By the surjectivity of invv0 A there exists a Pv0 P Xpkv0 q such that inv P 1 inv P . Consider the adelic point Q where v0 A p v0 q “ ´ v‰v0 vA p vq p vqvPΩ ř Pv v ‰ v0 Qv “ 1 #Pv0 v “ v0

1 A A By deﬁnition of Pv0 we have v invvA pQvq “ 0. Hence pQvqvPΩ P XpAkq and so XpAkq ‰H.

The lemma provides a necessaryř condition for XpAkq “ H. Proposition 4.3. Let X be variety over a number ﬁeld k. Let A P Br X be an algebra of order 2. The following are equivalent:

A (i) XpAkq “H. 1 (ii) For all places v P Ωk, invvA : Xpkvq Ñ 2 Z{Z is constant, and number of places v such that invvA pXpkvqq “ t1{2u is odd. Proof. The implication piiq ùñ piq follows from the deﬁnition of

A XpAkq “ tpPvq P XpAkq : invvA pPvq “ 0u. v ÿ Now we consider the reverse implication. Lemma 4.2 implies that invvA is not surjective for all 1 places v. Since A is of order 2 in Br X{ Br k then invvA : Xpkvq Ñ 2 Z{Z is constant for all places Br v. Since XpAkq “H then we must have v invvA pPvq ‰ 0 for every pPvq P XpAkq. Hence the number of places v such that invvA pXpkvqq “ t1{2u must be odd. ř Corollary 4.4. Let X be variety over a number ﬁeld k, and let A P Br X be an algebra of order 1 2. Let L{k be an odd degree extension. Assume that invvA : Xpkvq Ñ 2 Z{Z is constant for every 1 place v on k, and invwA : XpLwq Ñ 2 Z{Z is constant for every place w on L. If the number of places v on k such that invvA pXpkvqq “ t1{2u is odd then the number of places w on L such that invwA pXpLwqq “ t1{2u is odd. Furthermore,

A A XpAkq “ H ùñ XpALq “H. Proof. Theorem 2.16 and the functoriality of the Brauer group yields the commutative diagram:

evA invv Xpkvq ÝÝÝÝÑ Br kv ÝÝÝÝÑ Q{Z

rLw:kvs

§ evA § invw § Xp§Lwq ÝÝÝÝÑ Br§Lw ÝÝÝÝÑ Q§{Z. đ đ đ If invvA pXpkvqq “ t0u or rLw : kvs is even, then there exists a point Qw P XpLwq such that invwA pQwq “ 0. Since invwA is constant, invwA pXpLwqq “ t0u. Moreover if invvA pXpkvqq “ t1{2u and rLw : kvs is odd, then there exists a point Qw P XpLwq such that invwA pQwq “ 1{2. Since invwA is constant, invwA pXpLwqq “ t1{2u. Hence the number of places w P ΩL such that invwA pXpLwqq “ t1{2u is the same as the number of places w P ΩL such that rLw : kvs is odd and invvA pXpkvqq “ t1{2u. Since rL : ks “ w{vrLw : kvs is odd then for any place v, there exists an ř 15 odd number of places w|v such that rLw : kvs is odd. Hence, if #tv P Ωk : invvA pXpkvqq “ t1{2uu is odd then the number of places w P ΩL such that invwA pXpLwqq “ t1{2u is odd. A Now if XpAkq “H then Proposition 4.3 implies that the number of places v on k such that invvA pXpkvqq “ t1{2u is odd. Hence the number of places w on L such that invwA pXpLwqq “ t1{2u A is odd. Proposition 4.3 applied again implies that XpALq “H.

4.2 Geometry and the Evaluation Map In [Bri15], Bright studied the relation between the Brauer group of a smooth variety and the geometry of the special fiber of its regular model using purity theorems. He described the relation between constant evaluation maps and the existence of points on the fibers of torsors over the special fiber of a regular model. We will use his results in this section.

4.2.1 Purity: Gysin Sequence Theorem 4.5 (Gysin sequence [Mil80, Corollary 16.2]). Let X be a regular scheme, let Z be a regular closed subscheme of X of codimension c, and let F be a finitely generated locally free sheaf of Z{n-modules on X. Then we have a long exact sequence of Z{n-modules: 0 Ñ H2c´1pX, F q Ñ H2c´1pXzZ, F q Ñ H0pZ, F pćqq Ñ H2cpX, F q Ñ H2cpXzZ, F q Ñ H1pZ, F pćqq Ñ ... that is functorial in the pair pX,Zq. Corollary 4.6 ([Bri15, Corollary 2.6]). Let X be a regular excellent scheme, and Z any reduced, closed subscheme of codimension 1. Let X˝, and Z˝ be the non-singular locus of X, and Z respectively. Then there is an exact sequence

1 1 1 ˝ 1 3 ˝ 1 3 1 0 Ñ Br Xpp q Ñ BrpXzZqpp q Ñ H pZ , Ql{Zlqpp q Ñ H pX , Gmqpp q Ñ H pXzZ, Gmqpp q.

4.2.2 Constant Evaluation

Let k denote a finite extension of Qp with ring of integers O and residue field F. We let X be a scheme over O that is regular, faithfully flat, separated, and of finite type. We further assume that the generic fiber X “ X Ô Spec k is smooth and geometrically irreducible over k. We denote the ˝ ˝ special fiber by X0 “ X Ô Spec F, and the non-singular locus of the special fiber by X0 . Let X0 denote the base change of the special fiber to an algebraic closure of k. For n a positive ineteger and p a prime, if A is an abelian group then Arns denotes the n-torsion subgroup where n is a positive 1 integer, and App q denotes the prime to p subgroup where p is a prime number. In what follows we assume that X pOq ‰ H. Corollary 4.6 applied to the pair pX , X0q yields the exact sequence:

1 1 B 1 ˝ 1 0 Ñ Br X pp q Ñ Br Xpp q ÝÑ H pX0 , Q{Zqpp q. Lemma 4.7. Let P be a k-point of X lying in the image of X pOq in Xpkq, and reducing to a point ˝ P0 of X0 . Then the following diagram commutes:

1 1 Br Xpp q H1pX ˝, { qpp q B 0 Q Z ˚ ˚ P P0

1 1 1 Br kpp q » H pF, Q{Zqpp q.

16 Proof. This follows by functoriality of the Gysin sequence applied to the morphism P : Spec O Ñ X .

We will consider algebras A P Br X such that there exists a point P P X pOq where A pP q “ 0; such algebras are known as normalized algebras.

˝ Theorem 4.8. Assume that the nonsingular locus of the special ﬁber X0 is geometrically irreducible, and assume that p ‰ 2. Let L{k be a ﬁnite extension. Consider A P Br X a normalized algebra of order 2. Then there exists a positive constant B such that when |F| ą B the following holds. If the evaluation map evA : X pOkq Ñ Br k is constant then so is evA : X pOLq Ñ Br L. Denote by B the composite of the maps:

1 B 1 ˝ 1 1 ˝ 1 Br Xpp q ÝÑ H pX0 , Q{Zqpp q Ñ H pX0 , Q{Zqpp q.

1 Lemma 4.9 ([Bri15, Lemma 5.11]). Let A be a normalized algebra of Br Xpp q. If BA has order 1 ˝ 1 ˝ n in H pX0 , Q{Zq, then BA has order n in H pX0 , Q{Zq. Moreover, the evaluation map takes values in Br krns.

Proof. The algebra BA has order exactly n so n divides the order of BA . If BpnA q “ 0 then it follows by the exact sequence

1 1 1 ˝ 1 1 ˝ 1 H pF, Q{Zqpp q Ñ H pX0 , Q{Zqpp q Ñ H pX0 , Q{Zqpp q

1 1 that BpnA q is in the image of an element in H pF, Q{Zqpp q. Hence BpnA q is constant. Furthermore, A is normalized so BpnA q “ 0. Therefore, BA has order dividing n. Hence BA has order n in 1 ˝ H pX0 , Q{Zq, and the evaluation map takes values in Br krns. Lemma 4.10 ([Bri15, Lemma 5.15]). Suppose that S is a normal connected scheme. Then the 1 connected Z{nZ-torsors over S are those representing classes of order n in H pS, Z{nZq. Proof. Because S is normal, it follows by [Gro71, Exposition I, Section 10] that there is an equivalence of the category of connected finite ètalecovers of S and the category of ètalealgebras of K “ KpSq. Hence it suffices to consider the case S “ Spec k for some field k. There is an isomorphism between torsors over Spec k and HompGalpk{kq,Gq defined by: for a fixed geometric point x of T Ñ Spec k we associate Φ P HompGalpk{kq,Gq such that Φpσq “ g where g is the unique elemet in G such that σx “ gx. This isomrphism is independent of the chosen geometric point x as Tk is the trivial torsor. The homomorphism Φ is surjective if and only if the Galois action of Galpk{kq is transitive on the geometric points of T . This is equivalent to transitivity of the monodromy action of πpSpec kq “ Galpk{kq on the fiber T which means T is connected. In particular the Z{nZ-torsors over k are those representing homomorphisms of order n 1 in HompGalpk{kq, Z{nZq “ H pSpec k, Z{nZq. Proof of Theorem 4.8. ([Bri15, Lemma 5.12, Theorem 5.16, Corollary 5.17]) We will prove that if the evaluation map evA is constant on X pOkq then BkA “ 0. So BLA “ 0. Then we will prove that BLA “ 0 implies the evaluation map is constant on X pOLq. Assume that evA : X pOkq Ñ Br k is constant. Since A is normalized, evA is trivial. Suppose 1 ˝ by way of contradiction that BkA ‰ 0. Then BkA has order 2 in H pX0 , Z{2q. So BkA also has 1 ˝ order 2 in H pX0 , Q{Zq by Lemma 4.9. 1 Let α be a class in Br kr2s and let rαs denote its isomorphic image in H pF, Z{2q. We will prove ˝ that there exists P P X pOkq such that A pP q “ α. Consider Y Ñ X0 the torsor representing BA

17 1 ˝ α in H pX0 , Z{2q. By Lemma 4.10, Y is geometrically irreducible and so is the twist Y . By the α ˝ 1 ˝ Weil Conjectures applied to Y Ñ X0 in H pX0 , Z{2q there exists B such that whenever |F| ą B, α Y has an F-rational point. Under the assumption |F| ą B, we have Y pFq is nonempty. Hence α ˝ there is a point P0 is the image of Y pFq Ñ X0 pFq. This F-rational point extends to P P X pOq α α by Hensel’s Lemma. By our choice of P0, we have the ﬁber pY qP0 pFq “ pYP0 q pFq is nonempty. ˚ α This is equivalent to the existence of a section of the torsor pP0 BA q over F. By proposition 2.30 ˚ α ˚ 1 we deduce that the class of pP0 BA q is trivial. Untwisting we get that P0 BA “ rαs in H pF, Z{2q. Recall the commutative diagram in Lemma 4.7:

Br Xr2s H1pX ˝, {2q B 0 Z ˚ P ˚ P0 1 Br kr2s » H pF, Z{2q.

˚ The diagram above implies that A pP q “ P pA q “ α. This contradicts the fact the evA is constant and so we should have BkA “ 0. It follows by the following base extension diagram that pBqLA “ 0:

1 ˝ Br Xkr2s H ppX0 qk, Z{2q Bk

1 ˝ Br XLr2s H ppX0 qL, Z{2q. BL

Finally we want to prove that evA : X pOLq Ñ Br L is constant. Let P be an arbitrary point in ˚ X pOLq. Because BLA “ 0, P0 pBLA q “ 0 as well. Hence by the commutative diagram in Lemma ˚ 4.7, we duduce that P A “ 0. Hence evA : X pOLq Ñ Br L is trivial.

Corollary 4.11. Let X be a del Pezzo surface of degree 4 over k{Q and L{k be a ﬁnite degree extension. Consider a place v P Ωk and let O be the ring of integers of kv. Assume that X admits a proper regular model X over O. Fix A P Br X a normalized algebra of order 2. Furthermore we ˝ assume the special ﬁber X0 is geometrically irreducible, and |F| ą B, where B is a positive constant. 1 1 If invvA : Xpkvq Ñ 2 Q{Z is constant then so is invwA : XpLwq Ñ 2 Q{Z for every place w of L lying over v.

Proof. The map invvA is the composite of the inavriant isomorphism and the evaluation map 1 evA : Xpkvq Ñ Br kv. Similarly for invwA . If invvA : Xpkvq Ñ 2 Q{Z is constant then evA : Xpkvq Ñ Br kv is constant because invv : Br kv Ñ Q{Z is an isomorphism. Now we apply Theorem 4.8 to deduce that evA : XpLwq Ñ Br Lw is constant when F is big enough. Hence invwA “ invwpevA q is constant on XpLwq for every place w lying over v. The proof of Corollary 4.11 relies on the assumptions that a del Pezzo surface over a local ring kv admits a proper regular model over the valuation ring O, in the case when the special fiber is geometrically connected, and the residue field is sufficiently big. The problem of existence of a proper regular model is the same as resolution of singularities over Spec O. The status of resolution of singularities over a field is as follows. In 1964 Hironka proved resolution of singularities in all dimensions and characteristic 0. Abhyankar, in 1956, proved resolution of singularities of a surface in characteristic p ą 0, and of threfolds in characteristic p ą 5 in 1966 [Abh66]. Moreover, Corti in his paper rCor96s studies the existence of a nice integral model of a del Pezzo surface of degree 4 over a dedekind scheme. We will continue to work on proving the existence of this proper regular model. We will also find other tools when the special fiber is not geometrically irreducible.

18 References

[Abh66] Shreeram Shankar Abhyankar. Resolution of singularities of embedded algebraic surfaces. Pure and Applied Mathematics, Vol. 24. Academic Press, New York-London, 1966.

[BBFL07] M. J. Bright, N. Bruin, E. V. Flynn, and A. Logan. The Brauer-Manin obstruction and Sh[2]. LMS J. Comput. Math., 10:354–377 (electronic), 2007.

[Bri15] Martin Bright. Bad reduction of the Brauer-Manin obstruction. J. Lond. Math. Soc. (2), 91(3):643–666, 2015.

[Coo88] Kevin R. Coombes. Every rational surface is separably split. Comment. Math. Helv., 63(2):305–311, 1988.

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