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

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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-Th´el`eneand 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 ad`elering 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 points± 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-Th´el`eneand Sansuc in 1980 conjectured that the Brauer-Manin obstruction is sufficient for geometrically rational surfaces. Conjecture 1.1. (Colliot-Th´el`eneand 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: ad`elering of k Xpkq: k-points of X XpAkq: ad`elicpoints 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 !X 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 ¥ 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 ¥ 1 and a division algebra D P Azk such that A » MnpDq and B » MmpDq as k-algebras. 3 (b) There exists m; n ¥ 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 ¡ 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.
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