Genus one fibrations and vertical Brauer elements on del Pezzo surfaces of degree 4
aa r X i v : . [ m a t h . AG ] F e b GENUS ONE FIBRATIONS AND VERTICAL BRAUER ELEMENTS ONDEL PEZZO SURFACES OF DEGREE 4
VLADIMIR MITANKIN AND CECÍLIA SALGADO
Abstract.
We consider a family of smooth del Pezzo surfaces of degree four and studythe geometry and arithmetic of a genus one fibration with two reducible fibres for whicha Brauer element is vertical.
Contents
1. Introduction 12. Two conic bundles 33. Lines and Brauer elements 54. A genus 1 fibration and vertical Brauer elements 11References 161.
Introduction
A del Pezzo surface of degree four X over a number field k is a smooth projectivesurface in P given by the complete intersection of two quadrics defined over k . They arethe simplest class of del Pezzo surfaces that have a positive dimensional moduli space andfor which interesting arithmetic phenomena happen. Indeed, del Pezzo surfaces of degreeat least 5 with a k -point are birational to P k and, in particular, have a trivial Brauergroup. They satisfy the Hasse Principle and weak approximation. The Brauer group Br X = H ét ( X, G m ) of X is a birational invariant which encodes important arithmeticinformation such as failures of the Hasse principle and weak approximation via the Brauer–Manin obstruction. We refer the reader to [Poo17, §8.2] for an in-depth description of thisobstruction. The image Br X of the natural map Br k → Br X does not play a rôle indetecting a Brauer–Manin obstruction and thus one can consider the quotient Br X/ Br X instead of Br X . We say that X has a trivial Brauer group when this quotient vanishes. Date : February 18, 2021.2020
Mathematics Subject Classification
In contrast to del Pezzo surfaces of higher degree, the Hasse principle may fail for delPezzo surfaces of degree four [JS17]. Yet, they form a tractable class. Colliot-Thélène andSansuc conjectured in [CTS80] that all failures of the Hasse principle and weak approxi-mation are explained by the Brauer-Manin obstruction. This is established conditionallyfor certain families ([Wit07], [VAV14]).In [VAV14] Várily-Alvarado and Viray proved that del Pezzo surfaces of degree four thatare everywhere locally soluble have a vertical Brauer group. In particular, given a Brauerelement A , they show that there is a genus one fibration g , with at most two reduciblefibres, for which A ∈ g ∗ (Br( k ( P ))) . The aim of this paper is to study this fibration in detailfor a special family of quartic del Pezzo surfaces which we investigated from arithmeticand analytic point of view in [MS20].Let a = ( a , . . . , a ) be a quintuple with coordinates in the ring of integers O k of k .Define X a ⊂ P k by the complete intersection x x − x x = 0 ,a x + a x + a x + a x + a x = 0 (1.1)and we shall assume from now on that X a is smooth. The latter is equivalent to ( a a − a a ) Q i =0 a i = 0 . This altogether gives the following family of interest to us in this article: F = { X a as in (1.1) : a ∈ O k and ( a a − a a ) Y i =0 a i = 0 } . There are numerous reasons behind our choice of this family. Firstly, surfaces in F admittwo distinct conic bundle structures, making their geometry and hence their arithmeticconsiderably more tractable. Moreover, for such surfaces the conjecture of Colliot-Thélèneand Sansuc is known to hold unconditionally [CT90], [Sal86]. Secondly, our surfaces canbe thought of as an analogue of diagonal cubic surfaces as they also satisfy the interestingequivalence of k -rationality and trivial Brauer group. This is shown in Lemma 3.4 whichis parallel to [CTKS87, Lem. 1.1].Our aim is to take advantage of the two conic bundle structures present in the surfacesto give a thorough description of a genus one fibration with two reducible fibres for which aBrauer element is vertical. More precisely, after studying the action of the absolute Galoisgroup on the set of lines on the surfaces, we show that the two reducible fibres are of type I and that the field of definition of the Mordell–Weil group of the associated elliptic surfacedepends on the order of the Brauer group modulo constants which in our case is 1, 2 or4 [Man74], [SD93]. The presence of the two conic bundle structures plays an important rôle forcing a bound on the degree and shape of the Galois group of the field of definitionof the lines. We show in Theorem 1.1 that surfaces with Brauer group of size 2 are suchthat the genus one fibration only admits a section over a quadratic extension of k , whilethose with larger Brauer group, namely of order 4, have a section for the genus 1 fibrationalready defined over k . Theorem 1.1.
Let X a ∈ F and let E be the genus one fibration on X a described in §4.2.Then the following hold. (i) If Br X a / Br X a ≃ Z / Z , then the genus 1 fibration E is an elliptic fibration i.e.,admits a section, over a quadratic extension. Moreover, it admits a section ofinfinite order over a further quadratic extension. The Mordell–Weil group of E isfully defined over at most a third quadratic extension. (ii) If Br X a / Br X a ≃ ( Z / Z ) , then E is an elliptic fibration with a 2-torsion sectionand a section of infinite order over k . Moreover, the full Mordell–Weil group of E is defined over a quadratic extension. Not surprisingly, this is in consonance with the bounds obtained in our earlier paper[MS20, §1] when k = Q , as surfaces with Brauer group of size 2 are generic in the familywhile those with larger Brauer group are special.This paper is organized as follows. Section 2 contains some generalities on quartic delPezzo surfaces that admit two conic bundles. There we also describe the two conic bundleson the surfaces of interest to us. Section 3 is devoted to the study of the action of theabsolute Galois group on the set of lines on X a . We have also included there a descriptionof the Brauer elements using lines, by means of results of Swinnerton-Dyer, giving the toolsto, in Section 4, describe a genus one fibration with exactly two reducible fibres for whicha Brauer element is vertical. Acknowledgements.
We would like to thank Martin Bright, Yuri Manin and Bianca Virayfor useful discussions. We are grateful to the Max Planck Institute for Mathematics inBonn and the Federal University of Rio de Janeiro for their hospitality while working onthis article. Cecília Salgado was partially supported by FAPERJ grant E-26/202.786/2019,Cnpq grant PQ2 310070/2017-1 and the Capes-Humboldt program.2.
Two conic bundles
Let X be a quartic del Pezzo surface over a number field k . From this point on weassume that X is k -minimal and moreover that it admits a conic bundle structure over k . VLADIMIR MITANKIN AND CECÍLIA SALGADO
It follows from [Isk71] that there is a second conic bundle structure on X . In this context,given a line L ⊂ X then L plays simultaneously the rôle of a fibre component and of asection depending on the conic bundle considered.Fix a separable closure ¯ k of k . In what follows we analyse the possible orbits of linesunder the action of the absolute Galois group Gal(¯ k/k ) when Br X = Br X in the light ofthe presence of two conic bundle structures over k . Firstly, we recall [BBFL07, Prop. 13]that tells us the possible sizes of the orbits of lines. In the statement of this propositionthe authors consider a quartic del Pezzo surface over Q but its proof establishes the resultfor a del Pezzo surface of degree four over any number field. Lemma 2.1. [ [BBFL07, Prop. 13] ] Let X be a del Pezzo surface of degree four over k .Assume that Br X/ Br X is not trivial. Then the Gal(¯ k/k ) -orbits of lines in X are one ofthe following: (2 , , , , , , , , (2 , , , , , , (4 , , , , (4 , , , (8 , . Remark . Recall that we have assumed that X is minimal. In particular, every orbitcontains at least two lines that intersect. Since each conic bundle is defined over k and theabsolute Galois group acts on the Picard group preserving intersection multiplicities, wecan conclude further that each orbit is formed by conic bundle fibre(s). In other words,if a component of a singular fibre of a conic bundle lies in a given orbit, then the othercomponent of the same fibre also lies in it.2.1. A special family with two conic bundles.
We now describe the two conic bundlestructures over k in the del Pezzo surfaces given by 1.1. It suffices to consider F (1 , ,
0) = P ( O P (1) ⊕ O P (1) ⊕ O P ) which one can think of as (( A \ × ( A \ / G m , where G m acts on ( A \ × ( A \ as follows: ( λ, µ ) · ( s, t ; x, y, z ) = ( λs, λt ; µλ x, µλ y, µz ) . The map F (1 , , → P given by ( s, t ; x, y, z ) ( sx : ty : tx : sy : z ) defines anisomorphism between X a and ( a s + a t ) x + ( a s + a t ) y + a z = 0 ⊂ F (1 , , . (2.1)A conic bundle structure π : X a → P on X a is then given by the projection to ( s, t ) .Similarly, one obtains π : X a → P via ( s, t ; x, y, z ) ( tx : sy : ty : sx : z ) . It gives asecond conic bundle structure on X a as shown by the equation ( a t + a s ) x + ( a s + a t ) y + a z = 0 ⊂ F (1 , , . (2.2) This puts us in position to refine Lemma 2.1 upon restricting our attention to surfacesin the family F . Lemma 2.3.
Let X be a k -minimal del Pezzo surface of degree four described by equation (1.1) . Then the Gal(¯ k/k ) -orbits of lines in X are one of the following: (2 , , , , , , , , (2 , , , , , , (4 , , , . Proof.
We only have to eliminate the possibility of orbits of size 8. One can see readilyfrom 2.1 and 2.2 that each line on X is defined over at most a biquadratic extension of k . (cid:3) Lines and Brauer elements
Following Swinnerton-Dyer [SD99] we detect the double fours that give rise to Brauerclasses. Firstly, we show that a del Pezzo surface of degree 4 given by (1.1) has a trivialBrauer group if and only if it is rational over the ground field (see Lemma 3.4). In partic-ular, no k -minimal del Pezzo surface of degree 4 given by (1.1) has a trivial Brauer group.We take a step further after Lemma 2.3 and note that for a del Pezzo surface of degree 4with a conic bundle structure the sizes of the orbits of lines are determined by the order ofthe Brauer group (but, of course, not vice-versa as a surface with eight pairs of conjugatelines can have both trivial or non-trivial Brauer group for example). On the other hand, ifone assumes that the Brauer group is non-trivial then the size of the orbits does determinethat of the Brauer group (see Lemma 3.10). Moreover, given a non-trivial Brauer element,we describe in detail a genus one fibration with exactly two reducible fibres as in [VAV14]for which this element is vertical. We obtain a rational elliptic surface by blowing up fourpoints, namely two singular points of fibres of the conic bundle (2.1) together with twosingular points of fibres of the conic bundle (2.2). The field of definition of the Mordell–Weil group of the elliptic fibration is determined by the size of the Brauer group of X a . Ingeneral, it is fully defined over a biquadratic extension. We also show that the reduciblefibres are both of type I .3.1. Conic bundles and lines.
Let X a be given by (1.1). Then it admits two conicbundle structures given by (2.1) and (2.2). Each conic bundle has two pairs of conjugatesingular fibres with Galois group ( Z / Z ) acting on the 4 lines that form each of the twopairs. The intersection behavior of the lines on X a is described in Figure 3.1. Together,these 8 pairs of lines give the 16 lines on X a . VLADIMIR MITANKIN AND CECÍLIA SALGADO
We now assign a notation to work with the lines. Given i ∈ { , · · · , } , the union of twolines L + i and L − i will denote the components of a singular fibre of the conic bundle (2.1).Similarly, the union of two lines M + i and M i will denote the singular fibres of the conicbundle (2.2). More precisely, using the variables ( x : x : x : x : x ) to describe theconic bundles, we have the following x x = x x = − r − a a , x = ± r d − a a x , ( L ± ) x x = x x = r − a a , x = ± r d − a a x , ( L ± ) x x = x x = − r − a a , x = ± r da a x , ( L ± ) x x = x x = r − a a , x = ± r da a x , ( L ± ) x x = x x = − r − a a , x = ± r da a x , ( M ± ) x x = x x = r − a a , x = ± r da a x , ( M ± ) x x = x x = − r − a a , x = ± r d − a a x , ( M ± ) x x = x x = r − a a , x = ± r d − a a x . ( M ± )One can readily determine the intersection behavior of these lines, which we describe inLemma 3.1. We also take the opportunity to identify fours and double fours defined oversmall field extensions. Recall that a four in a del Pezzo surface of degree 4 is a set of fourskew lines that do not all intersect a fifth one. A double four is four together with the fourlines that meet three lines from the original four ([SD93, Lemma10]). Lemma 3.1.
Let i, j, k, l ∈ { , · · · , } with j = i . Consider L + i , L − i , M + i and M − i as above.Then (a) L + i intersects L − i , M − i and M + j , while L − i intersects L + i , M + i , and M − j . (b) M + i intersects M − i , L − i and L + j , while M − i intersects M + i , L + i and L − j . (c) The lines L + i , L + j , M − k , M − l and the lines L − i , L − j , M + k , M + l , with i + j ≡ k + l ≡ , form two fours defined over the same field extension L/k of degree atmost 2. Together they form a double four defined over k .Proof. Statements (a) and (b) are obtained by direct calculations. For the line L , forinstance, one sees readily that it intersects L − , M − , M +2 , M +3 and M +4 respectively at thepoints ( − q − a a : 0 : 1 : 0 : 0) , ( − q − a a : − q − a a : 1 : − q a a : − q da a ) , ( − q − a a : q − a a :1 : q a a : q da a ) , ( − q − a a : − q − a a : 1 : a √ a a : − q da a a a ) and ( − q − a a : q − a a : 1 : − a √ a a : q da a a a ) . Part (c) follows from (a) and (b). To see that one of such fours is definedover an extension of degree at most 2, note that each subset { L + i , L + j } and { M − k , M − l } isdefined over the same extension of degree 2. For instance, taking i = 1 , j = 2 , k = 3 and l = 4 , we see that the four is defined over k ( √− a a d ) . The double four is defined over k since both { L + i , L + j , L − i , L − j } and { M + k , M + l , M − k , M − l } are Galois invariant sets. (cid:3) L +1 L − L +2 L − L +3 L − L +4 L − M − M +1 M − M +2 M − M +3 M − M +4 Figure 1.
The lines on X a and their intersection behaviour. The intersec-tion points of pairs of lines are marked with • .Among the 40 distinct fours on a del Pezzo surface of degree 4, the ones that appear inthe previous lemma are special. More precisely, given a four as in Lemma 3.1 such thatits field of definition has degree d ∈ { , } , the smallest degree possible among such fours,then any other four is defined over an extension of degree at least d . Definition 3.2.
Given a four as in Lemma 3.1 part (c), we call it a minimal four if thefield of definition of its lines has the smallest degree among such fours.
VLADIMIR MITANKIN AND CECÍLIA SALGADO
For the sake of simplicity and completion we state a result proved in [MS20, Prop. 2.2]that determines the Brauer group of X a in terms of the coefficients a = ( a , · · · , a ) . Weremark that the statement of the proposition below does not require that the set of adelicpoints X a ( A k ) of X a is non-empty and that the proof presented in [MS20] works over anarbitrary number field k . Proposition 3.3.
Let ( ∗ ) denote the condition that − a a d / ∈ k ( √− a a ) ∗ , − a a d / ∈ k ( √− a a ) ∗ and that one of − a a , − a a or a a is not in k ∗ . Then we have Br X a / Br X a = ( Z / Z ) if a a , a a , − a a ∈ k ∗ and − a a d k ∗ , Z / Z if ( ∗ ) , { id } otherwise. Recall the definition of a rank of a fibration [Sko96], which (as in [FLS18]) for thesake of clarity to be distinguished from the Mordell–Weil rank or the Picard rank we callcomplexity here. That is the sum of the degrees of the fields of definition of the non-splitfibres. It is clear that the conic bundles in X a have complexity at most four. This allowsus to obtain in our setting the following lemma. Lemma 3.4.
Let k be a number field and X a given by (1.1) . Assume that X a ( A k ) = ∅ .Then X a is k -rational if and only if Br X a = Br k .Proof. The if implication holds for any k -rational variety since Br X a is a birational in-variant. To prove the non-trivial direction, we make use of [KM17] which shows that conicbundles of complexity at most 3 with a rational point are rational. Firstly, note that theassumption X a ( A k ) = ∅ implies that Br k injects into Br X a . If Br X a / Br k is trivial,then either − a a d ∈ k ( √− a a ) ∗ or − a a d ∈ k ( √− a a ) ∗ by Proposition 3.3. Thusthe complexity of the conic bundle π is at most 2. It remains to show that X a admits arational point. This follows from the independent work in [CT90] and [Sal86] which showthat the Brauer–Manin obstruction is the only obstruction to the Hasse principle for conicbundles with 4 degenerate geometric fibres. There is no such obstruction when Br X a / Br k is trivial. Under the assumption X ( A k ) = ∅ we conclude that X a admits a rational pointand hence is rational. (cid:3) Remark . Lemma 3.4 is parallel to [CTKS87, Lem. 1] which deals with diagonal cubicsurfaces whose Brauer group is trivial. Moreover, a simple exercise shows that in our case,if the Brauer group is trivial, then the surface is a blow up of a Galois invariant set of fourpoints in the ruled surface P × P , while the diagonal cubic satisfying the hypothesis of [CTKS87, Lem. 1] is a blow up of an invariant set of six points in the projective plane.The Picard group over the ground field of the former is of rank four while that of the latterhas rank three.3.2. Brauer elements and double fours.
The following two results of Swinnerton-Dyerallow one to describe Brauer elements via the lines in a double four, and to determine theorder of the Brauer group.The first result is contained in [SD99, Lem. 1, Ex. 2].
Theorem 3.6.
Let X be a del Pezzo surface of degree 4 over a number field k and α a non-trivial element of Br X . Then α can be represented by an Azumaya algebra in thefollowing way: there is a double-four defined over k whose constituent fours are not rationalbut defined over k ( √ b ) , for some non-square b ∈ k . Further, let V be a divisor defined over k ( √ b ) whose class is the sum of the classes of one line in the double-four and the classes ofthe three lines in the double-four that meet it, and let V ′ be the Galois conjugate of V . Let h be a hyperplane section of S. Then the k -rational divisor D = V + V ′ − h is principal,and if f is a function whose divisor is D then α is represented by the quaternion algebra ( f, b ) . The following can be found at [SD93, Lem. 11].
Lemma 3.7.
The Brauer group Br X cannot contain more than three elements of order2. It contains as many as three if and only if the lines in X can be partitioned into fourdisjoint cohyperplanar sets T i , i = 1 , .., , with the following properties: (1) the union of any two of the sets T i is a double-four; (2) each of the T i is fixed under the absolute Galois group; (3) if γ is half the sum of a line λ in some T i , the two lines in the same T i that meet λ , and one other line that meets λ , then no such γ is in Pic X ⊗ Q + Pic ¯ X . We proceed to analyse how the conic bundle structures in X a and the two results abovecan be used to describe the Brauer group of X a .3.3. The general case.
We first describe the general case, i.e., on which there are fourGalois orbits of lines of size four.
Proposition 3.8.
Let X a ∈ F and assume that a satisfies hypothesis ( ∗ ) of Proposition3.3. Then there are exactly two distinct double fours on X a defined over k with constituentfours defined over a quadratic extension. In other words, there are exactly 4 minimal fourswhich pair up in a unique way to form two double fours defined over k . Proof.
Part (c) of Lemma 3.1 tells us that the minimal fours are given by the dou-ble four formed by the fours { L +1 , L +2 , M − , M − } , { L − , L − , M +3 , M +4 } and that formed by { L +3 , L +4 , M − , M − } and { L − , L − , M +1 , M +2 } . By the hypothesis, each four is defined overa quadratic extension and the two double fours are defined over k . The hypothesis on thecoefficients of the equations defining X a also imply that any other double four is definedover a non-trivial extension of k . For instance, consider a distinct four containing L +1 .For a double four containing this four to be defined over k , we need that the second fourcontains L − and that one of the fours contains L +2 and the other L − . The hypothesis thateach four is defined over a degree two extension gives moreover that L +2 is in the same fouras L +1 and hence, due to their intersecting one of the lines, M +1 and M +2 cannot be in thesame four. We are left with L +3 , L +4 , M +3 , M +4 and their conjugates. But if L +3 is in oneof the fours then L − would be in the other four. This is impossible as neither L +3 nor L − intersect L − or L − , and each line on a double four intersects three lines of the four thatdo not contain it. (cid:3) Corollary 3.9.
Let X a be as above. Then Br X a / Br X a is of order 2.Proof. This is a direct consequence of Proposition 3.8 together with Theorem 3.6. (cid:3)
We shall now allow further assumptions on the coefficients of X a to study how theyinfluence the field of definition of double fours and hence the Brauer group.3.4. Trivial Brauer group.
Suppose that one of − a a d, − a a d, a a d, a a d is in k ∗ .Assume, to exemplify, that − a a d is a square. Consider the conic bundle structure givenby (2.1). Then the lines L +1 and L +2 are conjugate and, clearly, do not intersect. Indeed,they are components of distinct fibres of (2.1). Contracting them we obtain a del Pezzosurface of degree 6. If X a has points everywhere locally, the same holds for the del Pezzosurface of degree 6 by Lang–Nishimura [Lan54], [Nis55]. As the latter satisfies the Hasseprinciple, it has a k -point. In particular, X a is rational, which gives us an alternative proofof Lemma 3.4.3.5. Brauer group of order four.
For the last case, assume that a a , a a , − a a ∈ k ∗ , − a a d, − a a d, a a d, a a d k ∗ . We produce two double fours that give distinctBrauer classes. Firstly note that all the singular fibres of the two conic bundles are definedover k . In particular, their singularities are k -rational points and thus there is no Brauer–Manin obstruction to the Hasse principle and Br X a = Br k . Moreover, every line is definedover a quadratic extension, but no pair of lines can be contracted since each line intersects its conjugate. Secondly, note that since − a a is a square, thus k ( √− a a d ) = k ( √ a a d ) .We have the double four as above, given by L +1 , L +2 , M − , M − and the correspondent inter-secting components, and a new double four given by { L +1 , L +3 , M − , M − } , { L +2 , L +4 , M − , M − } ,which under this hypothesis is formed by two minimal fours.The Picard group of X a is generated by L +1 , L +2 , L +3 , L +4 , a smooth conic and a section,say M +1 of the conic fibration (2.1). We can apply Lemma 3.7 with T i = { L + i , L − i , M + i , M − i } to check that in this case the Brauer group has indeed size four. Lemma 3.10.
Let X a be as in (1.1) . Assume that X a does not contain a pair of skewconjugate lines or, equivalently, X a is not k -rational. Then the following hold: (i) X a / Br X a = 4 if and only if the set of lines on X a has orbits of size (2 , , , , , , , . (ii) X a / Br X a = 2 if and only if the set of lines on X a has orbits of size (2 , , , , , or (4 , , , .Proof. This is an application of [SD93, Lem. 11] or a reinterpretation of Proposition 3.3together with the description of the lines given in this section and the construction ofBrauer elements via fours given by Swinnerton-Dyer (see for instance [SD93, Lem. 10] and[BBFL07, Thm 10.] for the construction of the Brauer elements via fours). (cid:3) A genus 1 fibration and vertical Brauer elements
In what follows we will give a description of the genus 1 fibration X a P from [VAV14]for which a given Brauer element is vertical. First we recall some basic facts about ellipticsurfaces. We then obtain the Brauer element and the genus 1 fibration as in [VAV14]to afterwards reinterpret it in our special setting of surfaces admitting two non-equivalentconic bundles over the ground field. We study how the order of the Brauer group influencesthe arithmetic of this genus 1 fibration. More precisely, after blowing up the base points ofthe genus one pencil, we show that the field of definition of its Mordell–Weil group dependson the size of the Brauer group.4.1. Background on elliptic surfaces.
Let k be a number field. Definition 4.1. An elliptic surface over k is a smooth projective surface X together witha morphism E : X → B to some curve B whose generic fibre is a smooth curve of genus ,i.e., a genus 1 fibration. If it admits a section we call the fibration jacobian . In that case,we fix a choice of section to act as the identity element for each smooth fibre. The set ofsections is in one-to-one correspondence with the k ( B ) -points of the generic fibre, hence it has a group structure and it is called the Mordell–Weil group of the fibration, or of thesurface if there is no doubt on the fibration considered.
Remark . If X is a rational surface and an elliptic surface, we call it a rational ellipticsurface . If the fibration is assumed to be minimal, i.e., no fibre contains ( − -curves ascomponents, then by the adjunction formula the components of reducible fibres are ( − -curves. In that case, the sections are precisely the ( − -curves and the fibration is jacobianover a field of definition of any of the ( − -curves.Given a smooth, projective, algebraic surface X its Picard group has a lattice structurewith bilinear form given by the intersection pairing. If X is an elliptic surface then, thanksto the work of Shioda, we know that its Mordell–Weil group also has a lattice structure,with a different bilinear pairing [Shi90]. Shioda also described the Néron–Tate heightpairing via intersections with the zero section and the fibre components. This allows us todetermine, for instance, if a given section is of infinite order and the rank of the subgroupgenerated by a subset of sections. We give a brief description of the height pairing below. Definition 4.3.
Let E : X → B be an elliptic surface with Euler characteristic χ . Let O denote the zero section and P, Q two sections of E . The Néron–Tate height pairing is givenby h P, Q i = χ + P · O + Q · O − P · Q − X F ∈ reducible fibres contr F ( P, Q ) , where contr F ( P, Q ) denotes the contribution of the reducible fibre F to the pairing anddepends on the type of fibre (see [Shi90, §8] for a list of all possible contributions). Uponspecializing at P = Q we obtain a formula for the height of a section (point in the genericfibre): h ( P ) = h P, P i = 2 χ + 2 P · O − X F ∈ reducible fibres contr F ( P ) . Remark . The contribution of a reducible fibre depends on the components that P and Q intersect. In this article we deal only with fibres of type I , thus for the sake ofcompletion and brevity we give only its contribution. Denote by Θ the component thatis met by the zero section, Θ and Θ the two components that intersect Θ , and let Θ be the component opposite to Θ . If P and Q intersect Θ i and Θ j respectively, with i ≤ j then contr I ( P, Q ) = i (4 − j )4 .4.2. Vertical elements. Definition 4.5.
Let X be a smooth surface. Given a genus 1 fibration π : X → P , thevertical Picard group, denoted by Pic vert , is the subgroup of the Picard group generated bythe irreducible components of the fibres of π . The vertical Brauer group Br vert is given bythe algebras in Br k ( P ) that give Azumaya algebras when lifted to X (see [Bri06, Def. 3]).There is an isomorphism Br X/ Br X ≃ H ( k, Pic ¯ X ) and, as described by Bright[Bri06, Prop.4], a further isomorphism between B := {A ∈ Br k ( P ); π ∗ A ∈ Br X } and H ( k, Pic vert ) . Combining these with Theorem 3.6, allows us to describe vertical Brauerelements as those for which the lines in Theorem 3.6 are fibre components of π . Definition 4.6.
We call a Brauer element horizontal w.r.t. π if the lines used in Theorem3.6 to describe it are sections or multisections of π . Remark . As a line cannot be both a fibre component and a (multi)-section simultane-ously, a Brauer element that is horizontal cannot be vertical and vice-versa. For a generalfibration π some Brauer elements might be neither horizontal nor vertical.The following result shows that for a specific elliptic fibration, all Brauer elements areeither horizontal or vertical. Lemma 4.8.
Assume that the Brauer group of X a is non-trivial. Let F = L +1 + L +2 + M +3 + M +4 and F ′ = L − + L − + M − + M − . The pencil of hyperplanes spanned by F and F ′ gives a genus one fibration with exactly two reducible fibres on X a which are of type I ,for which a non-trivial element of its Brauer group is vertical. The other Brauer elementsare horizontal.Proof. The linear system spanned by F and F ′ is a subsystem of | − K X a | . Hence it givesa genus one pencil on X a . Its base points are precisely the four singular points of thefollowing fibres of the conic bundle fibrations: L +1 ∪ L − , L +2 ∪ L − , M +3 ∪ M − , M +4 ∪ M − .The blow up of these four base points produces a geometrically rational elliptic surface with two reducible fibres given by the strict transforms of F and F ′ . Since each of the latteris given by four lines in a square configuration and the singular points of this configurationare not blown up, these are of type I . There are no other reducible fibres as the only ( − -curves are the ones contained in the strict transforms of F and F ′ . Let E denote thefibration map.The Azumaya algebra ( f, b ) with f and b as in Theorem 3.6 taking as double fourthe components of F and F ′ , gives a Brauer element which is vertical for the genus one not necessarily with a section over the ground field Θ , = L +1 Θ , = M +3 Θ , = L +2 Θ , = M +4 E E E E Θ , = L − Θ , = M − Θ , = L − Θ , = M − Figure 2.
The reducible fibres of the genus one fibration E . The eight • denote fibre components and the four ◦ denote sections given by the blowup of the 4 base points.fibration E . Indeed, the lines that give such a double four are clearly in Pic vert and hencethe algebra ( f, b ) lies in the image of H ( k, Pic vert ) → H ( k, ¯ X a ) . By [Bri06, Prop. 4] itgives an element of the form E ∗ A , where A is in Br k ( P ) .The other Brauer elements on X a are described by double fours, i.e., pairs of sets of four ( − -curves on X a , subject to intersection conditions. Since each component intersectseach reducible fibre in exactly one point, after passing to its field of definition, these givesections of the genus one fibration. That is, such Brauer elements are horizontal withrespect to this genus one fibration. (cid:3) Remark . The genus one fibration for which a Brauer element is vertical describedin [VAV14] has in general two reducible fibres given as the union of two geometricallyirreducible conics, i.e., they are of type I . In our setting all the conics are reducible andhence give rise to fibres of type I . More precisely, let C ∪ C and C ′ ∪ C ′ be the tworeducible fibres with C i and C ′ i conics, then C ∪ C ′ is linearly equivalent to one of thefours, say L +1 ∪ L +2 ∪ L − ∪ L − and C ∪ C ′ is linearly equivalent to M +3 ∪ M +4 ∪ M − + M − .This seems to be very particular of the family considered in this note. More precisely,the presence of two conic bundle structures does not seem to be enough to guarantee thatthe reducible fibres of the genus one fibration are of type I . For that one needs that thelargest Galois orbit of lines has size at most four and moreover that the field of definitionof two of such orbits is the same. Mordell–Weil meets Brauer.
In what follows we will keep the letter E for thegenus one fibration on the blow up surface just described. We now give a proof of our mainresult, Theorem 1.1. Proof.
To prove (i) notice that the hypothesis of Proposition 3.3 imply that the four blownup points form two distinct orbits of Galois conjugate points. To exemplify, we work withthe genus one fibration given by F and F ′ as in Lemma 4.8. Let P i be the intersectionpoint of L + i and L − i for i = 1 , and that of M + i and M − i , for i = 3 , . Denote by E i theexceptional curve after the blow up of P i . Then { E , E } and { E , E } give two pairs ofconjugate sections of E . Moreover, the sections on a pair intersect opposite, i.e., disjoint,components of the fibres given by F and F ′ . Fixing one as the zero section of E , say E ,then a height computation gives that E is the 2-torsion section of E . Indeed, as we havefixed E as the zero section, the strict transform of L +1 and L − are the zero componentsof the fibres F and F ′ , respectively. We denote them by Θ ,j with j = 1 , respectively.Keeping the standard numbering of the fibre components, the strict transforms of L and L ′ are denoted by Θ ,j , with j = 1 , , respectively. Finally, in this notation, M +3 and M − correspond to Θ ,j while M +4 and M − correspond to Θ ,j , for j = 1 , respectively.To compute the height of the section E we need the contribution of each I to thepairing which in this case is (see [SS10, §11] for details on the height pairing on ellipticsurfaces and the contribution of each singular fibre to it). We have thus h E , E i = 2 − − − . In particular, E is a torsion section. Since E is distinct from the zero section E and suchfibrations admit torsion of order at most (see [Per90] for the list of fibre configurationsand torsion on rational elliptic surfaces), we conclude that E is a 2-torsion section. Thetwo other conjugate exceptional divisors E and E give sections of infinite order as onecan see, for example, after another height pairing computation.To show (ii) it is enough to notice that the hypothesis of Proposition 3.3 imply thatthe four base points of the linear system spanned by F and F ′ are defined over k . Fromthe discussion above we have that the zero section, the 2-torsion and also a section ofinfinite order, say E , are defined over k since each of them is an exceptional curve abovea k -point. The height matrix of the sections E and E has determinant zero, hence thesection E is linearly dependent on E . Moreover, it follows from the Shioda–Tate formulafor Pic( X ) Gal(¯ k/k ) that any section defined over k of infinite order is linearly dependent on E . Indeed, the rank of the Picard group of the rational elliptic surface is 6 since that of X a has rank 2 and we blow up 4 rational points. The non-trivial components of the two fibresof type I give a contribution of 3 to the rank. The other 3 come from the zero section, asmooth fibre and a section of infinite order, say E . For a second section of infinite orderwhich is independent in the Mordell–Weil group of E one can consider the pull-back of aline in X a . The hypothesis on the Brauer group implies that X a has no line defined over k but each is defined over a quadratic extension. (cid:3) References [BBFL07] M. J. Bright, N. Bruin, E. V. Flynn, and A. Logan,
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Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Math-ematik, Welfengarten 1, 30167 Hannover, Germany
Email address : [email protected] Mathematics Department, Bernoulli Institute, Rijksuniversiteit Groningen, The Nether-lands
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