the brauer group of del pezzo surfaces

We establish a criterion for theexistence of a non-trivial element of order ve in the Brauer group of S1in terms of certainGalois stable congurations of exceptional divisors on this surf

The Brauer Group of Del Pezzo Surfaces

A DISSERTATION

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ABSTRACTThe Brauer Group of Del Pezzo Surfaces

Andrea C Carter

Let S1 be a Del Pezzo surface of degree one over a number eld k, and let S1 denotethe base-change of S1 over the algebraic closure k of k We establish a criterion for theexistence of a non-trivial element of order ve in the Brauer group of S1in terms of certainGalois stable congurations of exceptional divisors on this surface

Soli Deo Gloria

Chapter 4 Five Torsion in the Brauer Group of Degree One Del Pezzo Surfaces 34

References 56

List of TablesA.1 Part I of the Intersection Table for a Ten-Tuple-Five 59A.2 Part II of the Intersection Table for a Ten-Tuple-Five 60

A.5 Part I of the Combined Intersection Table for a

Ten-Tuple-Five, Ten-Tuple-Three, and Ten-Tuple-One 62A.6 Part II of the Combined Intersection Table for a

Ten-Tuple-Five, Ten-Tuple-Three, and Ten-Tuple-One 63

CHAPTER 1IntroductionThe main result of this thesis is a criterion for detecting non-trivial order ve Brauer-group elements on a Del Pezzo surface of degree one, S1 Specically, the criterion is given

in terms of congurations of exceptional divisors on S1that are determined by intersectionproperties:

Theorem Let S1 be a degree one Del Pezzo surface dened over a number eld k.There exists a nontrivial element of (Br S1/Br k)[5] if and only if there exists a ten-tuple-ve on S1 that is stabilized by the absolute Galois group Gk such that

• Gk does not x any ve within the ten-tuple-ve;

• Gk does not x any quadruple-ve within the ten-tuple-ve;

• Gk does not x any ten-tuple-one within the ten-tuple-ve;

• Gk does not x any ten-tuple-three within the ten-tuple-ve

For an explanation of notation and terminology see chapter 2 below For a denition

of the congurations of exceptional divisors, see Denitions 16 and 17 For a proof, seeTheorem 20 below

We hope that such a criterion will be useful for, among other things, investigatingpossible Brauer-Manin obstructions to weak approximation in Del Pezzo surfaces of degreeone over number elds

In chapter 2 we recall background material and give notation, in chapter 3 we gatherand develop ideas particular to the problem we are considering, and in chapter 4 wedescribe the criterion and give the proof of the main theorem.

CHAPTER 2Background2.1 Preliminaries

An important question about a curve or a surface is, Does it contain a rational point,and if so, how are the rational points distributed? Or, more generally, given a variety

X dened over a eld k what is the distribution of the k-points of X? This can be adicult question to answer One approach to this problem is to determine if the so-called Local-to-Global principle applies Before discussing this principle, we recall somedenitions

Let X be a variety dened over a number eld k Denote the completion of k at aplace v by kv

Denition 1 A local point on X at a place v of k is a point on X dened over kv.The set of local points at v are denoted X(kv) We will be interested in consideringlocal points at v for all v simultaneously, i.e., an element of the product QvX(kv) If thisproduct is nonempty, then X is said to have local points everywhere  at all places, v.For example, a variety dened over Q has local points everywhere if it has points in Rand in the completions Qp for all primes p

Let Ov denote the ring of integers of kv Recall that the adèles of k are dened to be

Ak = {(xv) ∈Y

v∈Σ

kv×Yv6∈Σ

Ov|Σ is a nite set of places }

In the cases we are interested in, considering local points everywhere is equivalent toconsidering the set X(Ak)of adèlic points of X:

Lemma 1 For X a projective variety, X(Ak) =Q

Denition 2 A global point on X is a point on X dened over k

The set of global points are denoted X(k) The following principle, also called theHasse principle, holds for some families of varieties In such cases, the determination ofthe existence of global points may be considerably simplied:

Denition 3 (Global Principle) A family of varieties satises the Global principle if for every variety, X, in the family,

Local-to-YvX(kv) 6= ∅, implies X(k) 6= ∅

In other words, the Local-to-Global principle states that if X has a point in everycompletion of k, i.e., has local points everywhere, then it has a k-point, i.e., a globalpoint

If it is known that a variety does have at least one rational point, it is interesting toknow the density or distribution of these points A principle similar to the Hasse principleaddresses the issue of the distribution of k-points for X

Denition 4 (Weak Approximation) X is said to satisfy weak approximation if X(k)

is dense in QvX(kv)

Remark If X(kv) = ∅ for some v, then X(k) is dense in QvX(kv) since both setsare empty Otherwise, weak approximation holds if X(k) is dense in Qv∈ΣX(kv)for any

nite set of places Σ in the case of X a smooth and geometrically integral variety [Sko01,

pp 99-100] These last hypotheses will be satised for the varieties we will consider

Clearly, if weak approximation holds for X, then so does the Hasse principle

Remark In the context of the Hasse principle, one should note that the problem ofdetermining whether or not X(Ak)is non-emtpy requires only a nite number of compu-tations Indeed, if X is an integral model of X (i.e a projective scheme over the ring

of integers Ok of k with generic ber isomorphic to X), then there is an integer N, plicitly computable in terms of X, such that X(Ak)is non-empty if and only if X (Ok/N )

ex-is non-empty (as follows from Hensel's lemma, and the fact that X necessarily containspoints over Ok/p for any prime p of suciently large residue characteristic)

The Hasse principle holds for curves of degree two and, more generally, for degree twohypersurfaces in Pn[Poo01] However, there are varieties that violate the Hasse principle,such as some genus one curves For example, the curve with ane model

2w2 = 1 − 17z4

has points dened over Qp for every prime p and over R, but has no Q-rational point[Sil92, p 304] The failure of the Hasse principle for genus one curves is measured bycertain invariants called Shafarevich-Tate groups More precisely, a genus one curve, C,with local points everywhere but no global points is a nontrivial element of X(Jac(C)),the Shafarevich-Tate group of the Jacobian of C

There are also surfaces that violate the Hasse principle One example is the cubicsurface given by the homogeneous equation

5w3+ 9x3+ 10y3+ 12z3 = 0,

which has points in R and Qp for every prime p, but not points in Q [Cor05, p 63]

In the case of surfaces, obstructions to the Hasse principle and weak approximation cansometimes be detected by calculations involving what is called the Brauer group

Let S be a smooth projective surface dened over k; recall that S(k) denotes the

k-points of S and S(Ak) the adèlic points of S Manin's strategy is to construct a certainset, B, such that

S(k) ⊆ B ⊆ S(Ak),where B is closed in S(Ak) (see Denition 9 below) Then if S(Ak) 6= ∅ but B = ∅, theHasse principle fails for S In such a case we say there is a Brauer-Manin obstruction

to the Hasse principle on S If the Hasse principle fails for S, then weak approximationcertainly fails However, if S satises the Hasse principle, it is interesting to know if Ssatises weak approximation The existence of rational points on S does not imply thatS(k) is dense in QvS(kv) If B is strictly smaller than S(Ak) then weak approximationcannot hold, since B is closed In this case there is said to be a Brauer-Manin obstruction

to weak approximation Whether testing the Hasse principle or weak approximation, theset B is an approximation of S(k) that is hopefully easier to determine than S(k) itself

To dene B, we need to recall the precise denition of the Brauer group

2.2 The Brauer GroupThe Brauer group of S is a generalization of the Brauer group of the eld k Recallthat Br k = H2(Gk, k∗), where Gk= Gal(k/k) and k is a separable closure of k [Ser67].Alternately, Br k can be dened as the set of equivalence classes of central simple algebrasover k [Mil97].

Denition 5 A central simple algebra, A, over k is a unital algebra A over k which

is nite dimensional as a vector space over k such that

•A is central, i.e., the center of A is k

•A is simple: for any two-sided ideal I of A, I = A or I = {0}

-More informally, an Azumaya algebra on S is a sheaf of central simple algebras over

Note that since E and F are locally free, around each point x ∈ S there is a neighborhood

U over which the above isomorphism becomes

This denition is equivalent to Grothendieck's cohomological denition of the Brauergroup on the surfaces we will consider (Such an equivalence holds for regular noetherianschemes of dimension 2 [Gro68] as cited in [Cor05, p.5])

The following map is essential to the denition of the set B

Denition 8 (Evaluation Map) Let K be k or a completion of k Given a point

x ∈ S(K) there is a map

Br(S) → Br(K) given by

A 7→ A (x),where A (x) = Ax⊗O x k(x), as dened above This map is called the evaluation map

In order to dene B, recall the short exact sequence from class eld theory:

This allows us to dene the set B

The image of this map is the set of constant algebras in Br S; at each point x ∈ S the

ber (A ⊗k OS)(x) is simply A Since A ∈ Br k, then Pvinvv((A ⊗kOS)(x)) = 0,

as just mentioned Therefore, to determine B we need only consider S(Ak)Ai for a set{Ai} of coset representatives of Br S/Br k In the situations we will consider, this is

a nite group By construction, the set B approximates the global points of S but isdetermined by concrete calculations The simplications to calculating B just mentionedwill hopefully make B an even more accessible approximation to S(k)

2.3 Constructing Brauer Group ElementsBrauer group elements can be constructed from cyclic algebras:

Denition 10 Let L/K be a nite cyclic eld extension of degree n with Galois groupgenerated by σ and c an invertible element of K Then there is a central simple K-algebradened as follows:

• Over L, there is a basis {1, x, , xn−1};

• xn = c;

• xl = (σl)x, for all l ∈ L

This is called a cyclic algebra over K and is denoted (L/K, c) [Poo03]

Although σ is not specied in the notation, the algebra (L/K, c) does depend on thechoice of σ

Note 1 Two cyclic algebras (L/K, c) and (L/K, c0)are equivalent if c

some l ∈ L If the basis for (L/K, c) over L is given by {1, x, , xn−1} and the basis for

(L/K, c0) over L is given by {1, y, , yn−1}, then the change of basis is given by

x 7→ ly

Then the image of xn is (ly)n= cc0yn = c, as required

Lemma 2 Let l/k be a cyclic extension and f ∈ k(S) Then (l(S)/k(S), f) can beextended to an Azumaya algebra on all of S if (f) = Nl/k(D) for some D ∈ Div Sl

sketch of proof The cyclic algebra (l(S)/k(S), f) over the generic point of S tends to an Azumaya algebra over U, the open set U of S where f is invertible At eachpoint x ∈ U the ber is (l/k, f(x)) So to obtain an Azumaya algebra on S, we simplyneed to be able to extend the algebra across the zeroes and poles of f, i.e., across (f)

ex-If (f) = Nl/k(D) for a divisor dened over l, then each point of S has a neighborhoodwhere (f) = Nl/k(g) for some g ∈ l(S) By Note 1, (l(S)/k(S), f) = (l(S)/k(S), 1)locally The algebra (l(S)/k(S), 1) is dened everywhere and since it is locally equiva-lent to (l(S)/k(S), f), the latter algebra can be extended over any open set where it isequivalent to (l(S)/k(S), 1) Since these open sets cover S, the cyclic algebra dened by(l(S)/k(S), f ) can be extended to an Azumaya algebra on all of S 

Remark It is relatively easy to compute the invariants of Azumaya algebras thatcan be extended from cyclic algebras of the form (l(S)/k(S), f) This suggests that thissource of Azumaya algebras may be useful for approximating S(k) by the set B

To summarize our situation thus far, we need to be able to do explicit computationswith Azumaya algebras in order to detect a Brauer-Manin obstruction We have seen

that cyclic algebras are a good source of Azumaya algebras if they satisfy the conditions

of Lemma 2 Because this construction of an Azumaya algebra uses information fromdivisors, there is a family of surfaces which suggests itself for our purposes, since there is

a highly structured conguration of lines on each of the surfaces in the family This is thefamily of Del Pezzo surfaces

2.4 Del Pezzo SurfacesDel Pezzo surfaces admit a range of equivalent descriptions The following denitionfrom [Har77] will be the most convenient for our purposes

Denition 11 Let n be a natural number with 1 ≤ n ≤ 9 A Del Pezzo surface ofdegree n is isomorphic over k to P2 blown up at 9 − n points, p1, , p9−n, that are ingeneral position, or else, in the case that n = 8, to the product P1

× P1.The requirement that p1, , p9−n be in general position is that no three of the pointslie on a line, no six on a conic, and no eight lie on a singular cubic with a singularity atone of the pi [Cor05, p 3] Denote such a surface of degree n by Sn

Let Sndenote the base-change of Snover k, the algebraic closure of k As mentioned,the divisors on Del Pezzo surfaces will play an important role in our discussion We willconsider the divisor class group over k, P ic (Sn), and be interested in the Galois-theoreticproperties of elements of the Picard group For ease of notation, we will denote P ic (Sn)

by P if it is clear which surface is being considered

Denote by Ei the image of pi under the blowing up map and by ` the strict transform

of a line in P2 Let ei and µ denote the divisor classes of Ei and `, respectively Then,except for the case n = 8 with Sn ∼= P1× P1, P ic (Sn) ∼= Ze1⊕ · · · ⊕ Zen⊕ Zµ Let D.E

denote the intersection pairing of two divisors, D and E (see [Har77]) The intersectionproperties of the divisor classes are determined by the rules

e2i = −1 ei.ej = 0 (i 6= j) µ2 = 1 ei.µ = 0

We will refer to any smooth rational curve on Snwith self-intersection −1 as an exceptionaldivisor on Sn, or simply as a line on Sn (The justication for this latter terminologybeing that when n ≥ 3, exceptional divisors are mapped to lines under the anti-canonicalembedding.) Any n skew lines on Sn can be realized as the ei under blowing up, so theintersection properties of a set of lines is their distinguishing characteristic

Remark In addition to having information about lines  and thus divisors which canpotentially be used to construct Azumaya algebras  Del Pezzo surfaces are good subjectsfor studying the Hasse principle because every element of Br(Sn) becomes trivial over

k, so nontriviality of elements in the Brauer group is due to arithmetic properties of Sn[Cor05, p.10] or [MH74, Theorem 42.8]

Del Pezzo surfaces of degree nine, S9, are isomorphic to P2 over k The case we will

be most interested in is degree one However, we begin by recalling the case of degreethree Del Pezzo surfaces, which has attracted much classical interest

2.4.1 Degree Three Del Pezzo Surfaces

Degree three Del Pezzo surfaces over k can be parametrized as P2 blown up at six points,

p1, , p6, and can be embedded as nonsingular cubic surfaces in P3 via the anticanonical

embedding (see [Har77] or [MH74]) Such a surface, S3, has exactly 27 lines which can

be described explicitly as the following divisors, relative to the blowing up map:

• the full transform of the points pi Denote these six lines by e1, , e6

• the strict transform of the lines through two points pi and pj (i 6= j) Denotethese fteen lines by fij

• the strict transform of the conics through each of the points except pi These sixlines are denoted by gi

2.4.2 Degree One Del Pezzo Surfaces

Similarly, there are 240 exceptional divisors on a degree one Del Pezzo surface They havethe following description:

Divisor Image in P2 Description in Number

gijk conic through pl such 2µ −P

that l 6= i, j, k

hij cubic through pk for k 6= j 3µ − 2ei−P

with pi a double point

lijk quartic through all pl with 4µ − 2ei− 2ej − 2ek 56

double points at pi, pj, and pk −P

all pj and a triple point at pi

There is a convenient description of the action of Gkon P ic (Sn)for Del Pezzo surfaces

of degree n ≤ 3, given in terms of Weyl groups

2.4.3 Weyl Groups

Since we are interested in Galois-theoretic properties of divisors on Del Pezzo surfaces,

it is important to know the action of the Galois group on P ic (Sn) Recall P ic (Sn) ∼=

Ze1⊕· · ·⊕Zen⊕Zµ In addition to the chosen basis elements, there is another distinguished

element of P ic (Sn), π = 3µ − e1− · · · − en, the anti-canonical class The class π has theproperty that gπ = π for all g ∈ Gk.

Lemma 3 Let L be a nite extension of k which is a eld of denition for each ofthe lines on Sn If GL/k is dened to be Gal(L/k), then

H1(Gk, P ic(Sn)) = H1(GL/k, P ic(Sn))

So the following description of GL/k will be useful to us

Lemma 4 Let L be as just described for a Del Pezzo surface, Sn, with 1 ≤ n ≤ 3.Then there is an embedding

GL/k ,→ W (E9−n)

This embedding is useful for our purpose because (in the case n = 1) the action of

W (E8)on P can be thought of as generated by

{(12), (23), (34), (45), (56), (67), (78)} & Q,

where Q is the quadratic transformation centered at p1, p2,and p3[SD99] The rst sevenelements act on the subscripts of the generators and the last element, Q, has the following

CHAPTER 3Nontrivial Brauer Group Elements

in terms of stabilizers of elements of the Picard group of Sn Since the Picard groups ofDel Pezzo surfaces are completely explicit, this is an advantage for computation

Lemma 6 Let P denote P ic (Sn) Let x ∈ P and let x denote the image of x inP/nP Furthermore, let G be a nite group that acts on P and let x ∈ P such that

x ∈ (P/nP )G Then there is an isomorphism

Proof Apply Hi(G, ·) to the short exact sequence 0 → P ×n

−→ P → P/nP → 0.The isomorphism then follows from the denition of the connecting homomorphism 

In the case of degree three Del Pezzo surfaces, Swinnerton-Dyer gave a description

of the three-torsion elements of the Brauer group In particular, he described theoretic conditions on congurations of the lines on degree three Del Pezzo surfaces thatcorrespond to nontrivial three-torsion elements in H1(Gk, P ic (S3)), which is isomorphic

Galois-to Br S3/Br k, by Lemma 5 The relevant congurations of lines on degree three DelPezzo surfaces are called nines and triple nines

Denition 12 A nine on a Del Pezzo surface of degree three is a set of three skewexceptional curves together with the six exceptional curves each intersecting exactly two ofthose three

Denition 13 A triple-nine on a Del Pezzo surface of degree three, S3, is a partition

of the 27 exceptional curves on S3 into three nines

Theorem 7 There exists a nontrivial element of H1(Gk, P ic (S3))[3] if and only ifthere exists a Gk-stable triple-nine on S3 satisfying the condition that each nine is Gk-stable, but no set of three skew lines in any nine is itself Gk-stable

Proof For Swinnerton-Dyer's proof that a nontrivial element in H1(Gk, P ic (S3))[3]gives rise to a triple-nine on S3, see [SD99] For a proof of the if-and-only-if statement,

A similar statement was made and rened for the two-torsion elements of the Brauergroup of degree three Del Pezzo surfaces in [SD99] and [Cor05], respectively One mightwonder if such a correspondence could be proven for Del Pezzo surfaces of other degrees.The following theorem describes possible torsion elements in the Brauer group ofdegree one Del Pezzo surfaces

Theorem 8 Let S1/k be a Del Pezzo surface of degree 1 Then H1(Gk, P ic (S1)) isisomorphic to one of the following groups

elements of the listed groups One might also hope that the calculations of the set B aresimplied if Br S/Br k has small rank For these reasons, we focus on nding 5-torsionelements of the Brauer group of Del Pezzo surfaces of degree one.

3.2 Outline of the Method

In this subsection we sketch the outline of our result and proof Our overall goal is tobetter understand the rational points of a degree one Del Pezzo surface, S1 Since the set

B is an approximation to S1(k), we may hope to accomplish our goal by determining theset B for a given Del Pezzo surface of degree one As mentioned above, we will focus on thesituation where (Br S1/Br k)[5] is of rank one or two Then B is determined by invariantcalculations involving only the generators of Br S1/Br k So our more manageable goal

is to nd the analogue of Theorem 7 in the degree one, ve-torsion case

Goal Determine a criterion for the existence of nontrivial order ve elements of

PG/5PG for groups Gthat are isomorphic to GL/k for some S1 Consequently, we need to be able to move freely

between discussing elements of (P/5P )G and the corresponding lifts in P To that end,

we introduce a new set

Denition 14 Let G be given Consider the following diagram, where q is the quotientmap, φ is an isomorphism by the second isomorphism theorem, and the remaining mapsare the obvious ones Let YG be dened to make the upper square Cartesian,

YGr

by forgetting x For simplicity, we will hereafter speak of elements x ∈ YG, considering

YG to be given by the second description As a result, we will consider the map r to besimply reduction modulo ve

As mentioned in Lemma 6, for any x ∈ P , we denote by x the class of x in P/5P

We dene YG because it will be important to us to consider elements of P itself, not justmodulo ve, but we are interested in the reduction of the elements modulo ve being inthe quotient (P/5P ) G

P G /5P G So, given a group G, we wish to consider the corresponding set YG.However, we also need to consider what groups G produce a nontrivial quotient (P/5P ) G

The embedding from Lemma 4 narrows our search:

PG/5PG A straightforward but important idea is found

in lemma 10 below, regardingHfx, the stabilizer of x in P/5P , and Hy, the stabilizer of y,any lift of x to P The key is that x ∈ YG yields a nontrivial element x of (P/5P )G

PG/5PG if andonly if Hy ( G ⊆ fHx for all lifts y of x Our method will be to nd what Galois groups

G can occur by using this condition and then to describe them in terms of stabilizers ofexceptional divisors on S1 This will give us the desired statement

3.3 Some Useful LemmasSince we are interested in the quotient (P/5P )G

PG/5PG but will be manipulating elements of

YG, it is important to know what elements of YG are in the kernel of φ ◦ q ◦ r

Lemma 9 The kernel of φ ◦ q ◦ r is PG+ 5P

Proof Recall that φ is an isomorphism by the Second Isomorphism Theorem:

PG+ 5P5P

∼= PG/5PG

Then the kernel of φ ◦ q ◦ r is the preimage of P G +5P

5P under r Since r is reduction modulo

As previously mentioned, it will be useful to have a description of when a group G gives

a nontrivial quotient, (P/5P )G

PG/5PG The following lemma will provide the needed criterion.Lemma 10 Let G be xed and x ∈ (P/5P )G Let Hfx be the stabilizer of x in P/5Pand let Hy be the stabilizer of y ∈ YG, any lift of x Then x is in a nontrivial class of thequotient (P/5P )G

PG/5PG if and only if the inclusions

Hy ( G ⊆ fHx

hold for all lifts y of x

Proof An element x ∈ YG corresponds to nontrivial elements in each of the groups

in (3.1) if and only if

(1) x is xed by G;

(2) y 6∈ PG for any y ≡ x (mod 5)

To see this, note that for x even to be in the quotient (P/5P )G

PG/5PG requires the rst condition,which is guarranteed by x ∈ YG From Lemma 9, an element x ∈ YG is nontrivial in thegroups of (3.1) if and only if x 6∈ PG+ 5P That is equivalent to the second statement

So x ∈ YGgives a nontrivial class [cx]in H1(G, P ic (S1))[5]if and only if every element

g ∈ G xes x modulo ve, but does not actually x any y such that x ≡ y This can berestated as the following set inclusions, which must hold for any lift, y, of x:

Hy ( G ⊆ fHx,

Remark Note that x ≡ y implies that Hfx = fHy, but it does not necessarily followthat Hx = Hy.

Since we wish to nd subgroups G ⊆ W (E8) that give a nontrivial quotient (P/5P ) G

we are on the hunt for groups G that satisfy the inclusions of Lemma 10 for some x ∈ YG

We will shift from thinking of G as a xed group and considering what elements x ∈ YGcorrespond to nontrivial elements in (P/5P ) G

P G /5P G, to considering elements x ∈ P (and x ∈P/5P) and determining for what groups, G, the element x is in the corresponding YG

We cannot consider Hx and Hfx for every possible x ∈ P , so we use the followingsimplications

Lemma 11 Let w ∈ W (E8) and x ∈ YG Then x is nontrivial in (P/5P )G

PG/5PG if andonly if wx is trivial in (P/5P )G

holds for all y ≡ x modulo ve By denition, Hy = {h ∈ W (E8)|h(y) = y}, so wHyw−1 =

Hwy Similarly, Hfx = {h ∈ W (E8)|h(x) ≡ x(mod 5)}, so wHfxw−1 = gHwx Therefore,

Hwy( wGw−1 ⊆ gHwx

holds for all y ≡ x modulo ve and therefore for all wy ≡ wx modulo ve Then byLemma 10, wx is nontrivial in (P/5P )G

So to nd a nontrivial element in H1(G, P )[5], we only need to nd a set of orbitrepresentatives for P/5P under W (E8)and then determine which of those representativesmight give a nontrivial cohomology class

To further simplify the computation, we only need to consider orbit representativesmodulo π:

Lemma 12 Let x ∈ (P/5P )G Then cx is a coboundary if and only if cx+π is

Proof Again we use Lemma 9 Recall that π ∈ PG Then x ∈ PG+ 5P if and only

CHAPTER 4Five Torsion in the Brauer Group of Degree One Del Pezzo

of P/5P under W (E8), modulo π

Lemma 13 There are 75 orbits of P/5P under W (E8) They have the followingrepresentatives and sizes, where i = 0, 1, 2, 3, or 4:

Orbit Representative Orbit Size

Proof See the MAGMA output in section 1 of Appendix B 

The orbit sizes, when considering the representatives with all ve values of i sum to

59 as expected Denote the representatives with i = 0 by r1, , r15 Recall by Lemma

12 that these are the only representatives we need to consider as we search for nontrivialquotients (P/5P )G

PG/5PG

To determine which orbits correpond to nontrivial cohomology classes, we considerthe mod 5 stabilizers and the condition of Lemma 10

Hy ( G ⊂ fHx,for any y a representative of x Recall that we can eliminate some trivial cohomologyclasses, [cx], by considering whether the single inclusion Hx ( fHx holds That is, we startwith ri ∈ P as a representative for ri ∈ P/5P So, for each i we compute Hr i and Hfr i.Lemma 14 Except for the cases i = 6, 8, 9, 13, 14, and 15, elements of P/5P in orbitsrepresented by ri+ jπ are trivial in the quotient (P/5P ) G

P G /5P G for every G ⊆ W (E8).Proof Adapting some of Patrick Corn's MAGMA code (see section 2 in AppendixB), we compute both Hr i and Hfri for i = 1, , 15 For all i except i = 6, 8, 9, 13, 14, and

15, Hr i = fHr i, so it is impossible for there to be a G such that Hr i ( G ⊆ fHr i Then by

However, the representatives r6, r8, r9, r13, r14, r15each could still fail to give a ial cohomology class if there is another lift of ri to P whose stabilizer coincides with themod 5 stabilizer, as in the condition mentioned at the start of the proof of Lemma 10

nontriv-So, we check to see what each of the Hfri actually stabilizes to determine if there are any

si ≡ ri modulo ve with Hs i = fHr i

Lemma 15 Elements of P/5P in orbits represented by ri + jπ are trivial in thequotient (P/5P ) G

P G /5P G for every G if and only if i 6= 6

Proof This is true for i 6= 6, 8, 9, 13, 14, and 15 by Lemma 14 For the remainingsix values of i, it is unrealistic to consider Hs i for every si such that si ≡ ri (mod 5).However, we only need to know ifHfri is the stabilizer of such an si So, we look to see iff

Hri stabilizes any si ≡ ri Here is a table of the bases for the xed elements of P underthe action ofHfri, where an element a0µ+a1e1+· · ·+a8e8 ∈ P is written as [a0, a1, , a8]:

Representative Basis for the Fixed Space of P under Hfri

This information is veried by the MAGMA code in section 3 of Appendix B For

r6 = e1+ e2 + e3 + e4, the only vector in P that is xed by gHr 6 is not congruent to r6modulo ve, so r6 gives a nontrivial class in (P/5P ) G

P G /5P G for some G

However, for i = 8, 9, 13, 14, and 15, there are elements of P xed by Hfri that arecongruent to ri:

r8 ≡ b8 2, r9 ≡ b9 2, r13≡ 2b13 2, r14≡ 2b14 2, and r15≡ b15 2 + 2b15 3

Therefore, by Lemma 10, these ve ri are trivial in (P/5P ) G

P G /5P G for all possible G 

So we only need to consider one orbit representative: r6 = e1 + e2+ e3+ e4 For allother ri, every possible group G results in a trivial quotient (P/5P ) G

P G /5P G In the case of r6,there exists at least one group, G, such that r6 will be in a nontrivial class in (P/5P ) G

in the MAGMA code in section 3 of Appendix B So we are looking for G such that theinclusions

Hy ( G ⊆ H,

hold for any y ≡ x modulo ve Again it is unreasonable to compute Hy for all possible ysuch that y ≡ x However, we can utilize MAGMA to determine which groups can playthe roles of the Hy.

Lemma 16 There are exactly four conjugacy classes of subgroups of H, represented

by subgroups h1, h2, h3, and h4, with the property that

(1) H1(G, P )[5] 6= 0

if and only if(2) G ⊆ H and G 6⊆ ki, where ki is a conjugate of hi for some i = 1, 2, 3 or 4

In addition, hi has order 25325 = 1440 and ten conjugate subgroups under H for i = 1

or 2 and, in the case i = 3 or 4, hi has ve conjugate subgroups under H and order

26325 = 2880 Finally, each of the hi are maximal in H

Proof Consider the subgroup lattice of H Adapting a method from the three case found in [Cor05], we compute H1(h, P )for representatives, h, of each conjugacyclass Then we search for classes Ci such that H1(hi, P )[5] = 0 for a representative hi

degree-of Ci with the additional condition that there is no C0

i > Ci such that H1(h0i, P )[5] = 0for a representative h0

i In other words, we look for representatives hi of classes Cithat are maximal in the lattice of H with respect to having trivial ve-torsion in the rstcohomology group

As seen in the MAGMA code of section 4 of Appendix B, there are four such classes,

C1, C2, C3, and C4, represented by h1, h2, h3, and h4, respectively So every class thatgives trivial ve-torsion in the rst cohomology is included in C1, C2, C3, or C4 As is