Del Pezzo Surfaces and the Brauer-Manin Obstruction. Patrick Kenneth Corn. A.B. (Harvard University) 1998

Transcription

1 Del Pezzo Surfaces and the Brauer-Manin Obstruction by Patrick Kenneth Corn A.B. (Harvard University) 1998 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY Committee in charge: Professor Bjorn Poonen, Chair Professor Paul Vojta Professor Eyvind Wichmann Spring 2005

2 The dissertation of Patrick Kenneth Corn is approved: Chair Date Date Date University of California, Berkeley Spring 2005

3 Del Pezzo Surfaces and the Brauer-Manin Obstruction Copyright 2005 by Patrick Kenneth Corn

4 Abstract Del Pezzo Surfaces and the Brauer-Manin Obstruction by Patrick Kenneth Corn Doctor of Philosophy in Mathematics University of California, Berkeley Professor Bjorn Poonen, Chair The Brauer-Manin obstruction is a way to explain the failure of the Hasse principle. In this thesis, we explore the computation of this obstruction for Del Pezzo surfaces of degree 2 and 3 over number fields. In Chapter 1, we introduce the main concepts needed to define and work with the obstruction, and compute the fundamental cohomological invariant H 1 (k, Pic X) which arises in the study of this obstruction. Previously, this invariant had only been computed for Del Pezzo surfaces of degree d 3; in Theorem we extend the computation to Del Pezzo surfaces of any degree. In Chapter 2, we discuss the computation of the obstruction for Del Pezzo surfaces of degree 3. Many of the main ideas come from the papers [SD93] and [SD99]. Our goal will be to implement this computation completely explicitly, and we give an example of this computation for an example of Swinnerton-Dyer which does not fall into the class of examples we will study in Chapter 3. In Chapter 3, we implement this computation for diagonal cubic surfaces in 3, and prove Theorem 3.1.1, an extension of a result in [CTKS87], to establish stronger computational evidence for the conjecture that any diagonal cubic surface which fails to satisfy the Hasse principle has a Brauer-Manin obstruction. This result has been conjectured to hold for much wider classes of varieties (see Conjecture 1.2.9). In Chapter 4, we study Del Pezzo surfaces of degree 2. The first case of an 1

5 algorithm for computing the Brauer-Manin obstruction on these surfaces appears in the paper [KT04], for diagonal Del Pezzo surfaces of degree 2; we give a generalization of this algorithm to Del Pezzo surfaces with an equation w 2 = f(x 2, y 2, z 2 ) where f is a homogeneous quadratic polynomial in three variables. Using computations from Chapter 1, we identify an element of the Brauer group of such surfaces and use it to partially compute the Brauer-Manin obstruction for various classes of such surfaces. Using this, we give some computational evidence that the Brauer-Manin obstruction completely explains the failure of the Hasse principle for these Del Pezzo surfaces as well. Professor Bjorn Poonen Dissertation Committee Chair 2

6 To my family, and Hermine i

7 Contents 1 Del Pezzo surfaces, the Brauer-Manin obstruction, and Galois cohomology Del Pezzo surfaces The Brauer-Manin obstruction Brauer-Manin obstructions on Del Pezzo surfaces; the cohomological invariant The main theorem Cubic surfaces in Computing the obstruction: preliminaries Cyclic algebras Computing the obstruction for cubic surfaces An example; local invariant computations Diagonal cubic surfaces in The main theorem: statement and outline of proof Computing the obstruction for diagonal cubic surfaces over Local computations Computing small points Proof of Theorem Examples Del Pezzo surfaces of degree The geometry of Del Pezzo surfaces of degree Quaternion algebras in Br X Semi-diagonal Del Pezzo surfaces of degree Computing invariants Results and examples A Computations with MAGMA 98 A.1 Computations from Chapter A.2 Computations from Chapter ii

8 A.3 Computations from Chapter A.4 Computations from Chapter Bibliography 195 iii

9 Acknowledgements Thanks go first and foremost to my advisor, Bjorn Poonen, whose patience and insight inform every line of this thesis. I thank Ronald van Luijk and John Voight for helpful conversations, and Martin Bright for providing the slides of his Paris talk. I also thank the MAGMA staff and programmers, who were extremely quick and informative with answers to my questions about their wonderful computer algebra package. Finally, I would like to acknowledge the hospitality of the Institut Henri Poincaré in Paris, at which I spent a very rewarding semester. iv

10 Chapter 1 Del Pezzo surfaces, the Brauer-Manin obstruction, and Galois cohomology In this chapter, we define Del Pezzo surfaces and give their basic properties. We also define and explain the Brauer-Manin obstruction for arbitrary varieties, and introduce the cohomological invariant which we will need in order to compute it. Then we proceed to compute it for Del Pezzo surfaces of arbitrary degree, with help from the computer algebra system MAGMA (cf. [BCP97]). 1.1 Del Pezzo surfaces Definition A Fano variety is a smooth, projective, geometrically integral variety whose anticanonical divisor is ample. A Del Pezzo surface is a Fano variety of dimension 2. We will be concerned with Del Pezzo surfaces X over fields k of arithmetic interest, namely number fields and local fields. In this thesis, we will assume that the characteristic of every field k we consider is zero, unless specified. To sum up the properties of Del Pezzo surfaces that we will use, we combine several facts taken from [Man74] into one proposition, about which we make several 1

11 remarks. Proposition Let X be a Del Pezzo surface over a field k. Let d be the selfintersection (ω 1 X, ω 1 X ) of the anticanonical divisor. Let X = X k k. (a) 1 d 9. (b) Pic X is a free abelian group of rank 10 d. (c) If X X is a birational morphism and X is a Del Pezzo surface, then X is a Del Pezzo surface. (d) Either X is isomorphic to the blowup of 2 at r = 9 d points {x 1,..., x r } in general position, or d = 8 and X = 1 1. Conversely, any surface X satisfying the above condition is a Del Pezzo surface. (For a definition of general position, see Definition below.) (e) For X isomorphic to the blowup of 2 at r = 9 d points {x 1,..., x r } in general position, let C be an exceptional curve on X; that is, C is a curve on X such that (C, C) = 1 and C = 1 k. Then the image of C under the blowing-down map to 2 is either: 1. one of the x i 2. a line passing through two of the x i 3. a conic passing through five of the x i 4. a cubic passing through seven of the x i such that one x i is a double point 5. a quartic passing through eight of the x i such that three x i are double points 6. a quintic passing through eight of the x i such that six x i are double points 7. a sextic passing through eight of the x i such that seven x i are double points and one is a triple point Conversely, each object in the preceding list gives rise to exactly one exceptional curve C. (f) The anticanonical map X d is a closed immersion for d 3, so X can be realized as a degree-d surface in d for d 3. For d = 2 the anticanonical map has degree 2. 2

12 (g) Let X be a Del Pezzo surface of degree d. A G k -stable set of n pairwise skew exceptional curves on X can be blown down over k; that is, there is a birational morphism f : X Y exhibiting X as the blowup of Y at a set of n G k -stable points in general position. (By (c), Y is a Del Pezzo surface of degree d + n). Proof of proposition: Parts (a)-(g) are Theorem 24.3(i), Lemma , Corollary (i), Theorem 24.4, Theorem 26.2, Theorem 24.5 (and Remark ), and a special case of Theorem 21.5 of [Man74], respectively. Definition A set of points x 1,..., x r in 2 are in general position if no three of the points lie on a line, no six lie on a conic, and no eight lie on a singular cubic with a singularity at an x i. Remark We will often abuse notation and speak of exceptional curves on X instead of X. Part (e) shows that there are only finitely many exceptional curves on X, and in fact, for each value of r, the number c r of exceptional curves is easily computable from the description in part (e): r c r Remark The finite set of exceptional curves is G k -stable, because G k preserves intersections. 1.2 The Brauer-Manin obstruction Determining the structure of the set X(k) for X a variety over a field k is a fundamental problem in algebraic geometry; it is also usually much too difficult to solve completely. In fact, the same can be said of the easier problem of finding an algorithm which decides, given X and k, whether X(k) is empty (for k =, this is known as Hilbert s tenth problem over and is unsolved cf. [Shl00]). For specific classes of varieties, however, such a problem may be tractable, and the Brauer-Manin obstruction is a key ingredient of the solution for some of those classes. When k is a number field, a natural necessary condition to consider for X(k) to be nonempty is that X(k v ) be nonempty for all places v of k. This condition is usually 3

13 very easy to check in practice. For certain classes of varieties, this necessary condition is also sufficient; one says that these varieties satisfy the Hasse principle. For example, plane quadrics satisfy the Hasse principle. (See [Sko01], Theorem for a fuller list of results and references.) An instructive way to write this necessary condition is as follows: if we let X( k) be the set of adelic points of X, where X is a proper k-variety, then the natural map X( k) v X(k v), where the product runs over all places v of k, is a bijection. (Cf. [Sko01], pp ) There is a natural inclusion X(k) X( k), and so a necessary condition for X(k) to be nonempty is that X( k) be nonempty. Similarly, the Hasse principle, when it holds, says that X( k) X(k). The idea of the Brauer-Manin obstruction is to refine this condition by inserting a set between the two sets in the above inclusion. To construct this set, we will need to define the Brauer group of X. There are essentially two reasonable ways to make this definition: the first is by generalizing the central simple algebras used in the definition of the Brauer group of a field. Definition An Azumaya algebra on a scheme X is a locally free sheaf A of O X - algebras on X, such that for any point x X, the fiber A(x) = A x Ox k(x) is a central simple k(x)-algebra. (Here k(x) denotes the residue field at x.) Two Azumaya algebras A, B are said to be similar if there are locally free nonzero coherent O X -modules E and F such that A OX End E = B OX End F. Remark Standard arguments (as in [Gro68a], p. 47) show that the set of Azumaya algebras modulo equivalence form a group under tensor product (the inverse is induced by taking the opposite algebra, and the identity is the class of O X ). We will call this group Br Az X. The second way to define the Brauer group is somewhat more useful for theoretical purposes. Generalizing the cohomological identity Br k = H 2 (k, k ), we define Br X = H 2 et(x, m). If X has finitely many connected components, there are natural 4

14 injections Br Az X (Br X) tors Br X. It is conjectured that the composition is an isomorphism for any smooth k-variety X, and it is known ([Gro68b], p. 76) that it is an isomorphism for a regular noetherian scheme X of dimension 2, which will be enough for our purposes. Remark Regardless of which definition of Brauer group we use, the Brauer group of Spec k is always canonically isomorphic to Br k, for k any field. Since there is always a map Spec k(x) X for any integral scheme X (the generic point), and the association X Br X is a contravariant functor, we have an induced map Br X Br k(x), by Remark The following important fact about this map will be useful later: Proposition For X a regular integral quasi-compact scheme, the map Br X Br k(x) is injective. Proof of proposition: See [Mil80], III Now we are ready to define the intermediate set which will be used in the Brauer-Manin obstruction. Let X be a smooth, geometrically integral variety over a global field k. Let v be a place of k. Given a point x v X(k v ), which is a map Spec k v X, we get an evaluation map Br X Br k v by functoriality (and Remark 1.2.3). We denote this map by A A(x v ). Proposition For any point (x v ) X( k), the evaluation map Br X v Br k v has image inside v Br k v ; i.e. for any A Br X, A(x v ) = 0 in Br k v for all but finitely many places v of k. Proof of proposition: Spread out X to a scheme X over O k,s, and spread out A to an element B Br X, for some finite set S of places of k. Enlarging S if necessary, we may assume that x v X (O v ) for all v / S. Then A(x v ) comes from an element B(x v ) Br O v, but Br O v = 0 ([Mil80], IV.1). Definition For X a k-variety, k a global field, and A Br X, define X( k) A = {(x v ) X( k): v inv v A(x v ) = 0} 5

15 and define X( k) Br = A Br X X( k) A. Remark The evaluation map described above gives a pairing X(k) Br X Br k So for any A Br X, there is a natural map ev A : X(k) Br k. Similarly we obtain a natural map ev A : X(k v ) Br k v. By definition, the following diagram commutes: X(k) ev A X( k) ev A 0 Br k v Br k v i / 0 where i = v inv v. The exactness of the bottom row comes from class field theory (cf. [Mil97], p. 100). So this shows that, as promised, X(k) X( k) Br X( k). Definition If X( k) Br = and X( k), we say that there is a Brauer- Manin obstruction to rational points on X. If X( k) Br, we say that there is no Brauer-Manin obstruction to rational points on X, or (somewhat confusingly) that the Brauer-Manin obstruction is empty for X. If the statement X( k) Br X(k) holds for a class of varieties X, we say that the Brauer-Manin obstruction to rational points is the only one for the class. Conjecture ([CT03a]) The Brauer-Manin obstruction to rational points is the only one for rationally connected smooth projective geometrically integral varieties over number fields. 1.3 Brauer-Manin obstructions on Del Pezzo surfaces; the cohomological invariant For X a Del Pezzo surface over a number field k, we break up the question of rational points by degree. In general, as the degree decreases, the surfaces become more difficult 6

16 to deal with. Degree-9 surfaces are Severi-Brauer varieties, or twists of 2, of dimension 2; and it is well-known (cf. [Sko01], Theorem 5.1.1(b)) that such varieties satisfy the Hasse principle. So the question of rational points reduces to the question of local points, which is easily solvable in any specific case. Del Pezzo surfaces of degree d, 5 d 9, satisfy the Hasse principle, and in fact a Del Pezzo surface of degree 1, 5, or 7 must automatically have a k-point. (See [CT03b] for a summary, and [Sko01], Corollary for d = 5.) This leaves d = 2, 3, 4 as interesting cases. There are well-known counterexamples to the Hasse principle for each of these remaining degrees, and we give one for each degree here. Each of these can be explained by the existence of a Brauer-Manin obstruction, which is predicted by Conjecture Example The equation w 2 = 6x 4 3y 4 + 2z 4 represents a surface inside the weighted projective space (2, 1, 1, 1). This surface is a Del Pezzo surface of degree 2, with points everywhere locally, but no points over, as we will see in Chapter 4. (For an elementary proof of this, see [KT04].) Example The equation 5x 3 + 9y z t 3 = 0 defines a cubic surface in 3, which is a Del Pezzo surface of degree 3. It has points everywhere locally, but no points over, as we will see. This is originally due to Cassels and Guy: cf. [CG66]. Example The equations uv = x 2 5y 2, (u + v)(u + 2v) = x 2 5z 2 in 4 define a Del Pezzo surface of degree 4, which has points everywhere locally, but no points over. (Cf. [BSD75].) In what follows, we will concentrate on Del Pezzo surfaces of degree 2 and 3, for which the problem of computing the Brauer-Manin obstruction is more difficult, and give algorithms for computing it in many cases. For degree 4, the problem has essentially been solved by Adam Logan, who has created a MAGMA program for computing the obstruction in complete generality. Cf. [Log04]. The first step in computing the Brauer-Manin obstruction is understanding the cokernel of the natural map Br k Br X. Azumaya algebras A in the image of this map are called constant algebras, and it is clear from the commutative diagram used in Remark that if A is constant, v inv v A(x v ) = 0 for all points (x v ) X( k). So, to 7

17 determine X( k) Br, we need only find points (x v ) X( k) such that v inv v B(x v ) = 0, where B runs over a set of representatives in Br X for the cokernel of the natural map Br k Br X. It will be convenient here to prove a result about an assumption we will often make when necessary, namely that the surfaces X we are considering have local points everywhere. Of course we will usually only be interested in surfaces X with this property, but it will be useful to indicate how this property helps us. G k/k. Throughout this paper, we will use the notation G L/k := Gal(L/k) and G k := Proposition If X is a smooth projective geometrically integral k-variety, k is a number field, and v X(k v), then: 1. the natural map Br k Br X is injective 2. (Pic X L ) G L/k = Pic X, for any Galois extension L/k Proof of proposition: For (1), the injection Br k v Br k v factors as Br k Br X v Br k v, where the second map is the evaluation map coming from a point in v X(k v). Since the composition is injective, the first map must be injective. To see (2), apply the Hochschild-Serre spectral sequence H p (G L/k, H q et (X L, m)) H p+q et (X, m) for L any Galois extension of k, to obtain the following exact sequence (let G = G L/K ): 0 Pic X H 0 (G, Pic X L ) H 2 (G, L ) ker(br X Br X L ) H 1 (G, Pic X L ) H 3 (G, L ) (1.1) (This is because Pic X = Het(X, 1 m); for a reference, see [Mil80], III, Proposition 4.9). First, note that H 2 (G, L ) = Br(L/k), and so by part (1), the map H 2 (G, L ) ker(br X Br X L ) is injective. Therefore Pic X H 0 (G, Pic X L ) is an isomorphism, which proves (2). This result can also be proved without spectral sequences, as in [Bri02], Proposition We have one more related cohomological result that will be useful later. 8

18 Proposition ([Bri02], Lemma 4.8) Let L/K be a Galois extension of fields, and X a K-variety. Then the natural map i : Pic X (Pic X L ) G L/K fits into an exact sequence 0 Pic X i (Pic X L ) G L/K H 1 (G L/K, k(x L ) /L ) 0 and, in particular, i is an isomorphism if and only if H 1 (G L/K, k(x L ) /L ) = 0. Proof of proposition: The exact sequence of G L/K -modules relating Div X L and Pic X L gives the long exact sequence 0 H 0 (G L/K, k(x L ) /L ) H 0 (G L/K, Div X L ) H 0 (G L/K, Pic X L ) H 1 (G L/K, k(x L ) /L ) H 1 (G L/K, Div X L ) By Hilbert Theorem 90, the first term is k(x) /K. The second term is Div X. So the cokernel of the map Pic X (Pic X L ) G L/K, which is the natural map, is isomorphic to the kernel of H 1 (G L/K, k(x L ) /L ) H 1 (G L/K, Div X L ), and the result follows from the following lemma. Lemma H 1 (G L/K, Div X L ) = 0. Proof of lemma: By definition, we can write Div X L as a free abelian group on the irreducible divisors D, which can be partitioned into G L/K -orbits: Div X L = D D = R σ GL/K (σr) for representatives R of each orbit. This exhibits Div X L as an induced G L/K -module, which has trivial cohomology by Shapiro s Lemma ([Wei94], p. 171). The first main theorem of this thesis will be an enumeration of all the possible finite (abelian) groups which can occur as (Br X)/(Br k) for a Del Pezzo surface X. This has already been done in special cases: Manin, in [Man74], showed that (Br X)/(Br k) is trivial for Del Pezzo surfaces X of degree 5 and gave some elementary results about the exponent of the group for lower degree; Urabe, in [Ura96], extended tables of Manin for (Br X)/(Br k) in the special case when the exceptional curves on X were all defined over some cyclic extension; and Swinnerton-Dyer handled the cases d = 3, 4 in general in his paper [SD93] with the powerful cohomological method that we will use in our proof. 9

19 Proposition If X is a surface defined over a number field k, there is an isomorphism Br 1 X Br 0 X H1 (k, Pic X) (1.2) where Br 1 X = ker(br X Br X) and Br 0 X = im(br k Br X). Proof of proposition: We have that H 3 (k, k ) = 0 by [Tat86], p So, using the tail end of the exact sequence (1.1) with L = k, we get the isomorphism (1.2). Remark Del Pezzo surfaces are rational varieties; a rational k-variety X is a variety such that X is birational to 2 k. Theorem 42.8 in [Man74] implies that Br X = Br k = 0 for a rational surface X, so Br 1 X = Br X for all Del Pezzo surfaces X. Henceforth, we will often refer to the cokernel of Br k Br X as (Br X)/(Br k); usually, we will not need to assume that v X(k v) for our statements about this cokernel to be true, but at least the notation is justified by the fact that the map is injective if this assumption is true (by Proposition 1.3.4). Fortunately for us, when X is a Del Pezzo surface, (Br X)/(Br k) turns out to be finite and quite small. This is extremely useful for computation of the Brauer- Manin obstruction. For instance, if (Br X)/(Br k) is trivial, then the Brauer-Manin obstruction is automatically empty for X, since X( k) Br = X( k). Or if (Br X)/(Br k) is cyclic of prime order, which happens quite often for the surfaces we will be studying, then we will only need to find one nonconstant Azumaya algebra A, since in this case X( k) Br = X( k) A. Blowing up a surface X at a k-point adds a copy of to Pic X, corresponding to the class of the exceptional curve lying over the point (Corollary of [Man74]); and since Pic 2 =, generated by the class of O(1), we recover from this the result of Proposition 1.1.2(b) (except for the case d = 8, where it follows from a direct computation), together with two more facts. First, when X is isomorphic to the blow-up of 2 at r points, Pic X is generated by the class µ of a line in 2 together with the classes e 1,..., e r of the exceptional curves lying over the blown-up points x 1,..., x r (r = 9 d). Second, the canonical class, with respect to this basis, equals ( 3, 1,..., 1). (This follows from Proposition of [Man74].) 10

20 Remark Using Proposition 1.1.2(e), it is routine to determine the classes in Pic X of each exceptional curve with respect to the standard basis {µ, e 1,..., e r }. We will do this later for d = 3 and d = 2. For now, we will merely note that for d 7, Pic X is generated by the classes of the exceptional curves, because µ = (1, 0,..., 0) = (1, 1, 1, 0,..., 0) + (0, 1, 0,..., 0) + (0, 0, 1, 0,..., 0). Also, the intersection pairing is easy to determine with respect to this basis: r (a 0 µ + a 1 e a r e r, b 0 µ + b 1 e b r e r ) = a 0 b 0 a i b i. i=1 To compute H 1 (k, Pic X), we begin with a standard inflation-restriction argument. Proposition Let X be a Del Pezzo surface over a field k, and let M/k be the (finite, Galois) fixed field of the subgroup {σ G k : σ fixes every exceptional curve on X}. Then H 1 (k, Pic X) = H 1 (Gal(M/k), Pic X M ). is Proof of proposition: The inflation-restriction sequence for G k, G M, and Pic X 0 H 1 (Gal(M/k), (Pic X) G M ) H 1 (k, Pic X) H 1 (M, Pic X) Note that the injective map Pic X M (Pic X) G M is surjective in this case, because the classes of the exceptional curves, which generate Pic X, all come from Pic X M by definition. In fact, this gives an isomorphism Pic X M = Pic X. Now, G M acts trivially on Pic X, by Remark 1.3.9, and G M, a torsion group, can have no nontrivial homomorphisms into the torsion-free group Pic X. So H 1 (M, Pic X) = 0, and the proposition is proved. Now, since the set of exceptional curves on a Del Pezzo surface is G k -invariant, we have a natural map G k A, where A is the automorphism group of the exceptional curves preserving intersections (since the intersection pairing is preserved by the action 11

21 of G k as well). By the definition of the field M in Proposition , the kernel of this map is exactly G M, so we get an induced injection Gal(M/k) A. The group A can be obtained from the theory of lattices and root systems. For a thorough exposition, see [Man74], Ch. 4, especially Theorem When d 3, i.e. r 6, we will occasionally use the fact that A = W (E r ), the Weyl group of the lattice E r. At any rate, A is a finite group with a fixed action on r+1 = Pic X, and to determine the possible groups which appear as (Br X)/(Br k), we need only determine the possibilities for H 1 (H, r+1 ), where H is any subgroup of A. However, A is too large for us to solve the problem simply by listing its subgroups and computing the cohomology of each of them (at least when d = 1). This is where Swinnerton-Dyer s method enters the picture. Another advantage of this method, even in situations when brute force computations will solve the problem, is that it gives extra information about the nontrivial elements of H 1 (H, Pic X). Proposition ([SD93]) For a Del Pezzo surface of X of degree 4 over a field k of characteristic 0, H 1 (k, Pic X) = 1, /2, or ( /2) 2. For a Del Pezzo surface X of degree 3 over k, H 1 (k, Pic X) = 1, /2, ( /2) 2, /3, or ( /3) 2. Proof of proposition: It will be convenient to give a restatement of Swinnerton- Dyer s proof, in order to introduce some of the ideas we will use in the harder cases d = 2 and d = 1, and also so that we can clear up two errors in his original paper [SD93]. We will concentrate on the case d = 3; the case d = 4 follows from the analysis for d = 3, because we can blow up a Del Pezzo surface X of degree 4 at its generic point (over the function field k(x)) to get a Del Pezzo surface Y of degree 3, and there are isomorphisms H 1 (k, Pic X) H 1 (k(x), PicX) H 1 (k(x), PicY ) coming, respectively, from the fact that Gal(k k(x)/k(x)) Gal(k/k) is an isomorphism (X is geometrically integral), and from the G k(x) -module decomposition Pic Y = Pic X. So we need only see which possibilities for H 1 (k(x), PicY ) can occur when we are given that there is a G k(x) -invariant exceptional curve, which will follow from the work we do for d = 3. For the details, see [SD93]. 12

22 For d = 3 (as well as for d = 1 and d = 2), we will look at the p-primary parts of the group H 1 (k, Pic X), and use the following exact sequence from cohomology to control these groups. Lemma Let G be a finite (or profinite) group, and A a (continuous) G-module. Then there is an isomorphism (A/mA) G A G /ma G H1 (G, A)[m] defined by (class of x A) (σ 1 (σx x)) m Proof of lemma: Consider the exact sequence 0 A m A A/mA 0 of G-modules. This gives the long exact sequence of cohomology which begins 0 A G m A G (A/mA) G H 1 (G, A) m H 1 (G, A) and the lemma follows. (The description of the isomorphism is simply the definition of the boundary map.) Lemma Let H be any subgroup of G, for G and A as above. For integers d and n with d n, we have a commutative diagram 0 A H /na H (A/nA) H H 1 (H, A)[n] 0 n/d 0 A H /da H (A/dA) H H 1 (H, A)[d] 0 where the unlabeled vertical arrows are the natural maps given by reduction mod d. Proof of lemma: This follows directly from the proof of Lemma Remark Let H be a subgroup of G, for G and A as above. If H is a conjugate of H, there is a natural isomorphism H 1 (H, A) H 1 (H, A). We will use this fact quite often in what follows. 13

23 Remark Similarly, if an element x A/mA maps to c x H 1 (G, A)[m], then c gx = gc x for any g G, so c x will be a coboundary if and only if c gx is. If we are looking for elements x A/mA which map to nontrivial cocycles, we need only consider one element from each G-orbit of A/mA. Lemma If G is a finite group, A is a G-module, p is a prime number, and G p is a Sylow p-subgroup of G, then the restriction map H 1 (G, A) p H 1 (G p, A) is injective, where H 1 (G, A) p is the unique Sylow p-subgroup of the finite abelian group H 1 (G, A). Proof of lemma: The composition cores res: H 1 (G, A) H 1 (G p, A) H 1 (G, A) is multiplication by [G : G p ] ([Mil97], p. 54), which is prime to p, so passing to H 1 (G, A) p gives an isomorphism. The first map, restriction, must therefore be injective. Remark We will apply Lemmas , , and , and Remarks and with A = Pic X and G = G k or some quotient of G k (e.g. Gal(M/k) for M as in Proposition ). For future reference, let P = Pic X and P = P Gk. Since the group W (E 6 ) in which G k embeds is of order = (cf. [Man74], p. 139), the only possible prime orders of elements of this cohomology group are 2, 3, and 5, by Lemma We now analyze each of these possibilities in turn; Lemma will be useful in this analysis as well. For d = 3, the set E of exceptional curves on X has cardinality 27; with respect to the basis µ, e 1,..., e 6, we obtain the following list of their classes by using the description in Proposition 1.1.2(e): 6 classes e i, 1 i 6 15 classes f ij = µ e i e j, 1 i < j 6 14

24 6 classes g i = µ (e e 6 ) + e i, 1 i 6 (Denote the curves corresponding to these classes by E i, F ij, G i, respectively.) The first step in our analysis will be to show that H 1 (k, P ) can have no elements of order 5. To see this, note that a 5-Sylow subgroup G 5 of W (E 6 ) is cyclic of order 5. By Remark , we may take G 5 to be the subgroup generated by a 5-cycle cyclically permuting the last five entries of a vector in 7. But then it is evident that 7 is isomorphic to the G 5 -module 2 [G 5 ], with the trivial action on the first summand. Thus H 1 (G 5, 7 ) = H 1 (G 5, 2 ) H 1 (G 5, [G 5 ]) = 0 because there are no nontrivial homomorphisms G 5 2, and by Shapiro s Lemma ([Wei94], p. 171) the cohomology of an induced module is trivial. We could carry out the analysis for m = 2 and m = 3 by analyzing the corresponding Sylow subgroups, since their orders are not too large, but we can obtain more information about these cases by using a different method. This will be useful when we go on to look for elements of composite orders. In general, we look for elements x P /mp whose stabilizer H x in W (E 6 ) is strictly larger than the stabilizer H x of any lift of x to P. This is precisely what is needed for there to be a nontrivial element on the left side of the isomorphism in Lemma (where G will be contained in H x but not H x ). For m = 2, we note that we can add π to x if necessary, where π is the anticanonical class (3, 1,..., 1) P, to obtain an element x which is the same as x mod P/2P such that the first coordinate of x is 0. Also, since W (E 6 ) contains a subgroup isomorphic to S 6, consisting of the permutations of the last six coordinates, the number of coordinates of x which are 1 s is all that is important; in other words, x is in the same W (E 6 )-orbit as x l = l i=1 e i, for some l, 1 l 6. By Remark , we are reduced to looking at the x l. It is not hard to see that x 1, π + x 4, x 5 are all in the same orbit in P /2P, and x 2, π + x 3, x 6 are as well. (For example: x 1 is in the same orbit as f 56, which is congruent mod 2 to π + x 4.) It is immediate from the description of the classes of exceptional curves in E that these classes are distinct in P /2P, so x 1 (P /2P ) H x 1 P H. Therefore, the same holds for x 4 and x 5. 15

25 On the other hand, let H 2 be the mod-2 stabilizer of x 2, that is, the stabilizer of the image of x 2 in P /2P. H 2 must fix the set T 1 = {C E : (c, x 2 ) 1(mod 2)} = {E 1, E 2, G 1, G 2 } {F 1j, F 2j : 3 j 6}. Definition If X is a Del Pezzo surface of degree 3, a double-six on X is a set of twelve exceptional curves {L 1,..., L 6 } {M 1,..., M 6 } on X such that 1. the L i are pairwise skew 2. the M i are pairwise skew 0 if i = j 3. (L i, M j ) = 1 if i j The sets {L 1,..., L 6 } and {M 1,..., M 6 } are called (skew) sixes, and we will call the sets {L i, M i } opposite pairs. Note that the set T 1 = {E 1, G 1, F 23, F 24, F 25, F 26 } {E 2, G 2, F 13, F 14, F 15, F 16 } is a double-six. Lemma There is a one-to-one correspondence between nonzero elements of H 1 (k, Pic X)[2] and G k -stable double-sixes on X with the following three properties: 1. neither subset of six skew exceptional curves is itself G k -stable 2. no opposite pair is G k -stable 3. no set of three opposite pairs is G k -stable Proof of lemma: Consider a nonzero element c of order 2 in H 1 (k, P ). As we have seen, we can assume without loss of generality that c corresponds to x 2 (P /2P ) G k, or in other words that G k embeds via the natural map into the subgroup H 2 of W (E 6 ) which fixes x 2 mod 2P. This gives rise to a double-six T 1 which is G k -invariant, as above. Now we already know that the orbit of the ordered pair (e 1, e 2 ) under H 2 consists of pairs of skew exceptional curves in T 1 whose sum is congruent to e 1 + e 2 mod 2P. The only possible sums satisfying this requirement are e 1 + e 2, g 1 + g 2, f 1j + f 2j (3 j 6). 16

26 So the orbit of (e 1, e 2 ) under H 2 has at most 12 elements, consisting of (e 1, e 2 ), (g 1, g 2 ), (f 2j, f 1j ) (3 j 6), and the other six pairs resulting from switching coordinates. But since W (E 6 ) acts transitively on pairs of skew lines, all 12 of these elements lie in the orbit. For each of the six given ordered pairs (x, y) above, the difference x y is the same in P. It follows from this that the orbit of e 1 e 2 under the action of H 2 consists of only two elements, e 1 e 2 and its negative; moreover, the stabilizer of e 1 e 2 is an index-2 subgroup which equals the stabilizer of (either of) the sixes. Because we began with a nontrivial element of H 1 (k, P)[2], the image of G k in H 2 cannot be contained in this index-2 subgroup; otherwise there would be an element, namely e 1 e 2, in the class of x 2 mod 2P which was G k -stable. So G k does not stabilize either of the sixes. Moreover, the sum of the classes of an opposite pair is congruent to the difference of those classes mod 2P, which is equal to ±(e 1 e 2 ), so again, since we began with a nontrivial element of H 1 (k, P )[2], this sum cannot be fixed by G k. Finally, the sum of three opposite pairs is congruent to the difference of those three opposite pairs mod 2P, which is equal to ±(e 1 e 2 ) ± (e 1 e 2 ) ± (e 1 e 2 ), (the ± are independent of each other), which is congruent to e 1 e 2 mod 2P, and again it follows that this sum cannot be fixed by G k. For the converse, assume that there is a G k -stable double-six with the requisite properties. Then it is clear that the difference between the two classes of an opposite pair gives rise to an element of H 1 (k, Pic X)[2]; it is harder, however, to show that this element is nonzero. Here we will need to use MAGMA. The stabilizer of a double-six in W (E 6 ) has index 36, because each skew six determines a unique double-six and W (E 6 ) acts transitively on skew sixes (cf. [Har77], Proposition V.4.10), and there are skew sixes, hence 36 double-sixes ! = 72 We call this stabilizer H 1440, and use the MAGMA command SubgroupLattice to produce the full list of conjugacy classes of subgroups of H 1440, together with poset 17

27 relations determined by inclusion (i.e. C 1 C 2 if and only if there is a subgroup in class C 1 contained in a subgroup in class C 2 ). There are 194 classes of subgroups of H Then, we compute H 1 (H, P) for representative subgroups H of each class, and search for classes C such that H 1 (H, P ) = 0 for a representative H of C, and there is no class C > C such that H 1 (H, P) = 0 for a representative H of C. We find that there are three classes of subgroups with this property, so that any subgroup of H 1440 with trivial H 1 is contained in a subgroup in one of these three classes. Then we compute the vectors in P which are fixed by these classes, and discover that the classes are the classes of 1. the stabilizer of one of the sixes in the fixed double-six 2. the stabilizer of an opposite pair 3. the stabilizer of a set of three opposite pairs This completes the proof. See Algorithm A.1.1 for the MAGMA code. Remark In [SD93], Swinnerton-Dyer asserts that condition (1) alone is sufficient for a double-six to give rise to a nontrivial element of H 1 (k, Pic X)[2], but this is false: certainly, if the image of G k W (E 6 ) equals the stabilizer of a set {E 1, E 2 } of two skew exceptional curves, then the double-six T 1 constructed above is fixed by G k, and condition (1) is satisfied, but in fact it is easy to see that H 1 (k, Pic X) = 0, because we may blow down {E 1, E 2 } over k (by Proposition 1.1.2(g)) to get a Del Pezzo surface Y of degree 5, for which it is known that H 1 (k, Pic Y ) = 0; and now Theorem 23.3 of [Man74] implies that H 1 (k, Pic X) = 0 as well. A similar analysis works for m = 3. There is a different configuration of lines corresponding to these elements. Definition A nine on a Del Pezzo surface X of degree 3 is a set consisting of three skew curves together with the six curves intersecting exactly two of those three. A triple-nine on a Del Pezzo surface X of degree 3 is a partition of the 27 exceptional curves on X into three nines. 18

28 Lemma There is a one-to-one correspondence between nontrivial elements of H 1 (k, Pic X)[3] and G k -stable triple-nines on X satisfying the condition that each nine is G k -stable, but no set of three skew lines in any nine is itself G k -stable. Proof of lemma: There are more elements in ( /3 ) 7 than in ( /2 ) 7, but the process is the same: we split ( /3 ) 7 into W (E 6 )-orbits, and consider only one representative from each orbit (we will see below that this process can be streamlined). There are 20 nontrivial orbits under the action of W (E 6 ); here we give a representative for each, and its size. 1. 0, 1 2. π, π, 1 4. µ, π + µ, π + µ, e 1, π + e 1, π + e 1, e 1, π + 2e 1, π + 2e 1, µ 2e 1, π + µ 2e 1, π + µ 2e 1,

29 16. 2µ e 1, π + 2µ e 1, π + 2µ e 1, f 12 e 2, e 1 + e 2 + e 3, 240 See Algorithm A.1.2 for the MAGMA code. If two elements are related by adding multiples of π or multiplying by 2, then the stabilizer of one element equals the stabilizer of the other, so we are reduced to considering stabilizers of representatives of orbits #1, 4, 7, 13, 19, and 20. Since 0 does not give rise to a nontrivial element of H 1 (k, P )[3], we are reduced to considering the five elements µ, e 1, µ 2e 1, f 12 e 2, and e 1 +e 2 +e 3. (Incidentally, the computations done in this argument are all essentially done by hand in [SD93].) We will now show that only the last of these five elements can give a nontrivial element of order 3, by showing that the other four elements satisfy the implication x (P /3P ) H x P H, for H W (E 6 ). Before we begin, it will be helpful to introduce the following notation: if we are considering an element β (P /3P ) H, then S r will denote the set of exceptional curves whose intersection with (a lift of) β is congruent to r mod 3. The set S r must be fixed by H. Suppose µ (P /3P ) H. Then S 0 = {E 1,..., E 6 }, so e e 6 P H. Thus 3µ = e e 6 π P H, so µ P H. Suppose e 1 (P /3P ) H. From the description of the classes of the 27 exceptional curves, we see that none of them are congruent to e 1 mod 3, so we must have e 1 P H. Suppose µ 2e 1 (P /3P ) H. Then S 2 = {E 1, G 1, F 12, F 13,..., F 16 }. Since E 1 and G 1 are the only curves which intersect five of the other curves in S 2, the set {E 1, G 1 } is fixed by H; so e 1 + g 1 P H, whence µ 2e 1 = π (e 1 + g 1 ) P H. 20

30 Suppose f 12 e 2 (P /3P ) H. Then S 1 = {F 12, G 2, E 1, F 34, F 35, F 36, F 45, F 46, F 56 }. Since F 12 is the only curve which intersects eight of the curves in S 1, we must have f 12 P H. Similarly looking at S 2, we find that e 2 P H. So the difference is in P H as well. The only element left to consider is e 1 + e 2 + e 3. If this lies in (P /3P ) G k, then the sets S 0, S 1, S 2 form a triple-nine with each nine G k -stable, and if e 1 + e 2 + e 3 P Gk, the corresponding element of H 1 (k, P)[3] is trivial; so if we assume that e 1 +e 2 +e 3 gives rise to a nontrivial element of H 1 (k, P)[3], the set {E 1, E 2, E 3 } cannot be G k -stable. Moreover, let x = e 1 + e 2 + e 3 ; then it is not hard to check that the sum of any three skew lines in any nine is congruent mod 3P to one of the following possibilities: x, π x, π x, π + x, x, π + x. Since none of these elements can be G k -stable by the nontriviality assumption on x, no set of three skew lines in a nine can be G k -stable. For the converse, we must again examine subgroups of W (E 6 ) in MAGMA. To determine the index of the simultaneous stabilizer of the three nines in a given triplenine, we reason as follows: a nine is determined by a skew triple, of which there are ! = 720. We could have chosen any of 6 skew triples in a given nine, so there are actually 120 nines. Any nine gives rise to exactly one triple-nine, so there are 40 triple-nines. So the stabilizer of a triple-nine has index 40, but it permutes the three nines inside the triple-nine in any of 3! ways, so the simultaneous stabilizer of the three nines has index 240, and hence order 216. This is a subgroup H 216 with the property that any subgroup H W (E 6 ) such that H 1 (H, P)[3] 0 must be contained in a conjugate of H 216, so again we employ the SubgroupLattice command in MAGMA to enumerate its subgroups (there are 162 conjugacy classes of subgroups). Searching again for the classes of subgroups which are maximal with respect to the property H 1 (H, P) = 0, we find three classes, and searching for fixed elements of P, we find that each class is the class of a stabilizer of three skew lines in a nine. Since any subgroup with trivial H 1 is contained in one of these, the 21

31 condition that the subgroup not fix three skew lines in a nine is sufficient to ensure that the element of H 1 (H, P)[3] corresponding to the sum of three skew lines in a nine is nontrivial. See Algorithm A.1.3 for the MAGMA code. Remark This lemma appears in [SD93] as stated, but to prove the if direction, Swinnerton-Dyer simply asserts that a triple-nine with the given properties must give rise to an element of order 3 in H 1 (k, P ). As in the order-2 case, the nontriviality of this element is certainly not clear without looking explicitly at the subgroups of the stabilizer of a triple-nine, as we did above using MAGMA. To finish the proof of the proposition, we need to rule out elements of order 4, 6, and 9, as well as obtain bounds on the number of elements of order 2 and 3 which can appear in the H 1. All of these results follow from the same techniques used in the proofs of Lemmas and , along with Remark ; cf. [SD93]. (If this seems an abrupt way to finish the proof, we will atone for this by going into great detail about the analogous computations for d = 2 and d = 1.) Before we go on to lower degrees, we note a corollary of Lemma from [SD93] which illustrates the benefits of this more explicit approach to computing cohomology: Corollary If X is a Del Pezzo surface of degree 3 and H 1 (k, Pic X) has even order, then X satisfies the Hasse principle. Proof of corollary: By Lemma , there is a quadratic extension L of k and a set of six skew lines which are G L -stable. Thus we may blow down these six skew lines over L, by Proposition 1.1.2(g), to obtain a Del Pezzo surface X defined over L of degree 9, which is known to satisfy the Hasse principle. Now if X has local points everywhere, then of course so does X L, and therefore so does X. So X (L). This implies that X(L), by the Lang-Nishimura lemma (see [Nis55]). The corollary then follows from the following lemma: Lemma Let X/k be a Del Pezzo surface of degree 3. extension of k. If X(L), then X(k). Let L be a quadratic 22

32 Proof of lemma: By Proposition 1.1.2(f), we can view X as a cubic surface in 3. Now take P X(L), and let σ be the generator of Gal(L/k). If σp = P, then we are done, so suppose the two points are not equal. But then the line through them in 3 is defined over k, and if this line does not lie on X (that is, if it is not one of the 27 exceptional curves), it intersects X in exactly three points (with multiplicities), by Bézout s Theorem, and its third intersection point with X must be G k -invariant. If the line lies on X, then X has many k-points; any plane defined over k not containing the line will intersect the line in exactly one point, which must be a k-point. In either case, X(k). 1.4 The main theorem For the cases d = 2 and d = 1, we will try to preserve the flavor of the clever manipulations with the sets S r employed in the proofs of the last section, but we will have to rely more heavily on MAGMA to carry out the various orbit and stabilizer computations that arise. Theorem Let X/k be a Del Pezzo surface of degree d. Then H 1 (k, Pic X) is isomorphic to one of the following groups: 5 d 9: {1} d = 4: any of the above groups, plus /2, ( /2 ) 2 d = 3: any of the above groups, plus /3, ( /3 ) 2 d = 2: any of the above groups, plus ( /2 ) s (3 s 6), /4 ( /2 ) t (0 t 2), ( /4 ) 2 d = 1: any of the above groups, plus ( /2 ) 7, ( /2 ) 8, ( /3 ) s (s = 3, 4), /4 ( /2 ) s (s = 3, 4), ( /4 ) 2 ( /2 ) t (t = 1, 2), /5, ( /5 ) 2, /6, /6 /2, /6 /3, ( /6 ) 2 Proof of theorem: All but d = 2 and d = 1 have been done already (the analysis for 5 d 9 is done by Theorem 29.3 of [Man74]). First we tackle d = 2. Before we begin, let us fix our notation for the 56 exceptional curves on our Del Pezzo surface X. 23

33 With respect to the standard basis referred to in Remark 1.3.9, we have the following list of their classes, again using Proposition 1.1.2(e): 7 classes e i, 1 i 7 21 classes f jk = µ e i e j, 1 j < k 7 21 classes g jk = π f jk, 1 j < k 7 7 classes h i = π e i, 1 i 7 As before, the curves corresponding to these classes are denoted E i, F jk, G jk, H i, respectively. Step 1: H 1 (H, P)[5] = H 1 (H, P )[7] = 0 for all H W (E 7 ). Note that W (E 7 ) contains W (E 6 ) as a subgroup of order 56, as the stabilizer of one of the exceptional curves. So any 3-Sylow or 5-Sylow subgroups of W (E 6 ) are also 3-Sylow or 5-Sylow subgroups of W (E 7 ). Furthermore, if H is such a subgroup, and we assume without loss of generality that the copy of W (E 6 ) containing H is the stabilizer of e 7, the decomposition P = 7 with respect to the standard basis is an H-module decomposition, so H 1 (H, P) = H 1 (H, 7 ) H 1 (H, ) = H 1 (H, 7 ) because H acts trivially on the last. Since we have already computed the possibilities for H 1 (H, 7 ), this implies that the 3-torsion and 5-torsion groups in our list for d = 2 must be the same as those in the list for d = 3, by Lemma In particular, there are no elements of order 5 anywhere in our list. And elements of order 7 can be ruled out by an argument similar to the one we did for elements of order 5 when d = 3; a Sylow 7-subgroup G 7 W (E 7 ) is cyclic of order 7, so we can take it to be the subgroup generated by an element which acts as a cyclic permutation of the last seven coordinates in Pic X = 8 = µ ( e 1 e 7 ). So immediately we have a decomposition of Pic X as a trivial G 7 -module plus an induced G 7 -module, so its cohomology is 0. This implies there are no elements of order 7 anywhere, by Lemma

34 of H 1 (H, P)[2]. Step 2: A description of the x P /2P that can give rise to nontrivial elements As with d = 3, we can reduce to the study of the elements x l = l i=1 x i, for 1 l 7. And an easy computation shows that x 1, x 5 are conjugate mod 2, as well as x 2, π + x 4, x 6 and x 3, x 7. (For example, e 1 is conjugate to g 67, which is congruent to x 5 mod 2P. And e 1 + e 2 + e 3 is conjugate to e 1 + e 2 + g 12, which is congruent to x 7. The other facts are proved similarly.) Since, as before, no exceptional curve is congruent mod 2P to x 1, it cannot give rise to a nontrivial element of H 1 (H, P )[2]. But x 2 and x 3 both can, and they are not in the same W (E 7 )-orbit mod 2P, or even mod π + 2P. We will explore this situation in more detail in Chapter 4; for now, we will simply note that this is different from the case d = 3, when there was only one orbit of elements mod π + 2P that gave rise to nontrivial elements of H 1 (H, P)[2]. of H 1 (H, P)[3]. Step 3: A description of the x P /3P that can give rise to nontrivial elements This is not necessary for computing H 1 (H, P )[3] in general, because we have already determined the possible groups that can arise in Step 1. However, this will still be useful for our examination of elements of order 6. Again, we make a list of orbits of ( /3 ) 8 under the action of W (E 7 ). This time, there are 18, which split up into six groups of 3: 1. Orbits of 0, π, 2π: 1 element each 2. Orbits of e 1, π + e 1, 2π + e 1 : 56 elements each 3. Orbits of e 1 e 2, π + e 1 e 2, 2π + e 1 e 2 : 126 elements each 4. Orbits of e 1 e 2 + e 3, π + e 1 e 2 + e 3, 2π + e 1 e 2 + e 3 : 576 elements each 5. Orbits of e 1 + e 2 + e 3, π + e 1 + e 2 + e 3, 2π + e 1 + e 2 + e 3 : 672 elements each 6. Orbits of e 1 + e 2, π + e 1 + e 2, 2π + e 1 + e 2 : 756 elements each See Algorithm A.1.4 for the MAGMA code. 25