RATIONAL SURFACES AND CANONICAL DIMENSION OF

PGL_{6}

JEAN-LOUIS COLLIOT-THÉLÈNE, NIKITA A. KARPENKO, AND ALEXANDER S. MERKURJEV

Summary : The “canonical dimension” of an algebraic group over a field by definition is the maximum of the canonical dimensions of principal homogenous spaces under that group. Over a field of characteristic zero, we prove that the canonical dimension of the projective linear group PGL_{6} is 3. We give two distinct proofs, both of which rely on the birational classification of rational surfaces over a nonclosed field. One of the proofs involves taking a novel look at del Pezzo surfaces of degree 6.

1. Introduction

Let F be a field and let C be a class of field extensions of F. A field E ∈ C is called generic if for any L ∈ C there is an F -place of E with values in L.

Example 1.1. Let X be a variety over F and let C_{X} be the class of field extensions L of F such that X(L) ≠ ∅. If X is a smooth irreducible variety, the field F (X) is generic in C by [7, Lemma 4.1].

The canonical dimension cdim(C) of the class C is the minimum of tr. deg_{F} E over all generic fields E ∈ C. If X is a variety over F, we write cdim(X) for cdim(C_{X}) and call it the canonical dimension of X. If X is smooth irreducible then by Example 1.1,

(1)  cdim(X) ≤ dim X.

If X is smooth, proper and irreducible, the canonical dimension of X is the least dimension of a closed irreducible subvariety Y ⊂ X such that there exists a rational dominant map X −−→ Y [7, Cor .4.6].

Example 1.2. Let A be a central simple F -algebra of degree n. Consider the class C_{A} of all splitting fields of A. Let X be the Severi-Brauer variety SB(A) of right ideals in A of dimension n. We have dim X = n − 1. Since A is split over a field extension E/F if and only if X(E) ≠ ∅, we have C_{A} = C_{X} and therefore cdim(C_{A}) = cdim(X).

Let A be a central simple F -algebra of degree n = q_{1}q_{2} … q_{r} where the q_{i} are powers of distinct primes. Write A as a tensor product A_{1} ⊗ A_{2} ⊗ … ⊗ A_{r}, where A_{i} is a central simple F -algebra of degree q_{i}. A field extension E/F splits A if and only if E splits A_{i} for all i. By Example 1.2, the varieties SB(A) and Y := SB(A_{1}) × SB(A_{2}) × ⋯ × SB(A_{r}) have the same classes of splitting fields and hence

Date: November 24, 2006.

This paper is the outcome of a discussion during a hike at Oberwolfach.

(2)  cdim SB(A) = cdim(Y) ≤ dim(Y) = ∑_{i=1}^{r}(q_{i} − 1)

by inequality (1).

It looks plausible that the inequality in (2) is actually an equality. This is proven in [2, Th. 11.4] in the case when r = 1, i.e., when deg(A) is power of a prime.

In the present paper we prove the equality in the case n = 6.

Theorem 1.3. Let A be a division central algebra of degree 6 over a field of characteristic zero. Then cdim SB(A) = 3.

The proof builds upon the classification of geometrically rational surfaces. Starting from this classification, we give two independent proofs of the theorem, each of which seems to have its own interest. The first proof uses a novel approach to del Pezzo surfaces of degree 6 (Section 4). The second proof involves a systematic study of the kernel of the map from the Brauer group of a field F to the Brauer group of the function field of a geometrically rational surface over F (Section 5).

Let G be an algebraic group over F. The canonical dimension of G is the maximum of cdim(X) over all G-torsors X over all field extensions of F.

Corollary 1.4. The canonical dimension of PGL_{6} over a field of characteristic zero is equal to 3.

Proof. Isomorphism classes of PGL_{6}-torsors over a field extension E/F are in 1-1 correspondence with isomorphism classes of central simple E-algebras of degree 6. Moreover, if a torsor X corresponds to an algebra A then the classes of splitting fields of X and A coincide. Therefore cdim(X) = cdim SB(A) ≤ 3. There is a field extension E/F possessing a division E-algebra A of degree 6. By Theorem 1.3, cdim SB(A) = 3 and therefore, cdim(PGL_{6}) = 3. □

Remark 1.5. In view of results of Berhuy and Reichstein [2, Rem. 13.2] and Zainoulline [18], Corollary 1.4 completes classification of simple groups of canonical dimension 2 in characteristic zero. Those are SL_{3m} /μ_{3} with m prime to 3.

Let F be a field, \overline{F} an algebraic closure of F. An F -variety, or a variety over F, is a separated F -scheme of finite type. Let X be an F -variety. We let \overline{X} = X ×_{F} \overline{F}.

We shall use the following notation. For a variety X over a field F we write n_{X} for the index of X defined as the greatest common divisor of the degrees [F (x) : F] over all closed points x ∈ X. If there exists an F -morphism X → Y of F -varieties then n_{Y} divides n_{X}. If X is a nonempty open set of a smooth integral quasi-projective F -variety Y then n_{X} = n_{Y} (this may be proved by reduction to the case of a curve). Thus if X and Y are two smooth, projective, integral F -varieties which are F -birational, then n_{X} = n_{Y} (see also [13, Rem. 6.6]).

2. Rational curves and surfaces

We shall need the following

Theorem 2.1. Let X be an integral projective variety of dimension at most 2 over a perfect field F. Then there is a smooth integral projective variety X′ over F together with a birational morphism X′ → X.

This special case of Hironaka’s theorem has been known for a long time. In dimension 1, it is enough to normalize. Modern proofs in the two-dimensional case ([10] [11] [1]) handle arbitrary excellent, noetherian two-dimensional integral schemes: given such a scheme X they produce a birational morphism X′ → X with X′ regular. A regular scheme of finite type over a perfect field F is smooth over F.

In this paper an integral variety X over F is called rational, resp. unirational if there exists a birational, resp. dominant F -rational map from projective space ℙ_{F}^{n} to X, for some integer n. A geometrically integral F -variety X is called geometrically rational, resp. geometrically unirational, if there exists a birational, resp. dominant \overline{F} -rational map from projective space ℙn ; \overline{F} to \overline{X}, for some integer n. Rational integral varieties are unirational. For varieties of small dimension the converse holds under mild assumptions as the following two well known statements show.

Theorem 2.2 (Lüroth). A unirational integral curve X over F is rational, i.e., X is birationally isomorphic to ℙ_{F}^{1} .

Theorem 2.3 (Castelnuovo). A unirational integral surface X over an algebraically closed field field F of characteristic zero is rational, i.e., X is birationally isomorphic to ℙ_{F}^{2} .

Proof. See [9, III.2, Theorem 2.4 p. 170] , or [3]. The assumption on char F is necessary (cf. [9, p. 171] ). Surfaces given by an equation z^{p} = f(x, y) in characteristic p are unirational but in general not rational. □

The following theorem has its origin in a paper of F. Enriques ([5], 1897). The theorem as it stands was proved by V. A. Iskovskikh (1980) after work by Yu. I. Manin (1966, 1967). A proof of the theorem along the lines of modern classification theory (the cone theorem) was given by S. Mori (1982).

For a smooth F -variety X one lets K = K_{X} ∈ Pic X denote the class of the canonical bundle.

A smooth proper F -variety X if called F -minimal if any birational F - morphism from X to a smooth proper F -variety is an isomorphism. By Castelnuovo’s criterion, a smooth projective surface over a perfect field F is not F -minimal if and only if X contains an exceptional curve of the first kind.

Theorem 2.4 (Iskovskikh, Mori). Let X be a smooth, projective, geometrically integral surface over a field F. Assume that X is geometrically rational. The group Pic X is free of finite type. Let ρ denote its rank. One of the following statements holds:

(i) The surface X is not F -minimal.

(ii) We have ρ = 2 and X is a conic bundle over a smooth conic.

(iii) We have ρ = 1 and the anticanonical bundle −K_{X} is ample.

Proof. See [6], [15, Thm. 2.7] and [9, Chapter III, Section 2]. See also the notes [3] (where characteristic zero is assumed). □

Smooth projective surfaces whose anticanonical bundle is ample are known as del Pezzo surfaces. They automatically are geometrically rational. Let X/F be a del Pezzo surface and d = deg(K_{X}^{2}). In particular, n_{X} divides d. We have 1 ≤ d ≤ 9. For all this, see [12], [9, Chap III.3], [3].

Over a separably closed field F, a del Pezzo surface is either isomorphic to ℙ^{1} ×_{F} ℙ^{1} or it is obtained from ℙ^{2} by blowing up a finite set of points (at most 8, in general position). The Picard group of ℙ^{2} is ℤh, where h is the class of a line, and K = −3h. The Picard group of ℙ^{1} ×_{F} ℙ^{1} is ℤe_{1} ⊕ ℤe_{2}, where e_{1} and e_{2} are the classes of the two rulings, and K = −2e_{1} −2e_{2}. Given the behaviour of the canonical class under blow-up [12, Chap. III, Prop. 20.10] we therefore have:

Lemma 2.5. Let X/F be a del Pezzo surface over a separably closed field F. Then one of the following mutually exclusive possibilities holds:

(i) X is isomorphic to ℙ^{2}.

(ii) X is isomorphic to ℙ^{1} ×_{F} ℙ^{1}.

(iii) The canonical class K_{X} is not a proper mutliple of another element in Pic X.

3. Reduction to a problem on rational surfaces

Lemma 3.1. Let W be a regular, proper, geometrically unirational variety over a field F of characteristic 0. Assume that the canonical dimension cdim(W) = d ≤ 2. Then there exists a geometrically rational closed F -subvariety X ⊂ W of dimension d and a dominant rational map W −−→ X.

Proof. By a property of canonical dimension recalled at the very beginning of this paper, there exist a closed irreducible F -subvariety X ⊂ W of dimension d and a dominant rational map W −−→ X. By assumption W and therefore X are geometrically unirational. By Theorems 2.2 and 2.3, X is a geometrically rational variety. □

Proposition 3.2. Let A be a division central algebra of degree 6 over a field F of characteristic zero. Write A = C ⊗ D, where C and D are central simple F -algebras of degree 2 and 3 respectively. Consider the Severi-Brauer varieties Y = SB(C) and Z = SB(D) of dimension 1 and 2 respectively. Assume cdim(SB(A)) ≤ 2. Then there exists a geometrically irreducible smooth projective F -surface X such that

(i) X is F -minimal.

(ii) n_{X} is divisible by 6.

(iii) X has a point over F (Y ×_{F} Z).

(iv) Y ×_{F} Z has a point over F (X).

Proof. Since cdim(Y ×_{F} Z) = cdim(SB(A)) ≤ 2, then by Lemma 3.1, there exist a geometrically rational closed F -subvariety X_{1} ⊂ Y × Z of dimension at most 2 and a dominant rational map Y ×_{F} Z −−→ X_{1}. By Theorem 2.1, there is a birational F -morphism X_{2} → X_{1} with X_{2} smooth and projective. Note that since A is a division algebra, we have n_{Y ×Z} = 6. Since we have F -morphisms X_{2} → X_{1} → Y ×_{F} Z, the numbers n_{X_{1}} and n_{X_{2}} are divisible by 6. There is a dominant rational map Y ×_{F} Z −−→ X_{2}.

Suppose that dim X_{2} = 1, i.e., X_{2} is a geometrically rational curve. Then X_{2} is a conic curve (twisted form of the projective line) and n_{X_{2}} divides 2, a contradiction. It follows that X_{2} is a surface. Let X_{2} → X be a birational F -morphism with X an F -minimal smooth projective surface. Since both X_{2} and X are smooth projective we have n_{X} = n_{X_{2}}. □

4. Del Pezzo surfaces of degree 6

In this Section, F is an arbitrary field.

Let us first recall a few facts about del Pezzo surfaces of degree 6. We refer to [12] for background and proofs.

Let us first assume that F is algebraically closed. A del Pezzo surface of degree 6 is the blow-up of ℙ^{2} in 3 points not on a line. Because PGL_{3} acts transitively on the set of 3 noncolinear points in ℙ^{2}, all del Pezzo surfaces of degree 6 are isomorphic. A concrete model is provided by the surface S in ℙ^{2} ×_{F} ℙ^{2} with bihomogeneous coordinates [x_{0} : x_{1} : x_{2}; y_{0} : y_{1} : y_{2}] defined by the system of bihomogeneous equations x_{0}y_{0} = x_{1}y_{1} = x_{2}y_{2}. Projection of S onto either ℙ^{2} identifies S with the blow-up of ℙ^{2} in the 3 points [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1]. There are 6 “lines” (exceptional curves of the first kind) on S, the inverse images E_{1}, E_{2}, E_{3} of the 3 points on the first ℙ^{2} and the inverse images F_{1}, F_{2}, F_{3} of the 3 points on the second ℙ^{2}. The configuration of these lines is that of a (regular) hexagon : two curves E_{i} do not meet, two curves F_{i} do not meet, and (E_{i}.F_{j}) = 1 if i ≠ j, while (E_{i}.F_{i}) = 0.

The torus T = (𝔾_{m})^{3}/𝔾_{m} over F where 𝔾_{m} is diagonally embedded in (𝔾_{m})^{3} acts on ℙ^{2} ×_{F} ℙ^{2} in the following manner: (t_{0}, t_{1}, t_{2}) sends [x_{0} : x_{1} : x_{2}; y_{0} : y_{1} : y_{2}] to [t_{0}x_{0} : t_{1}x_{1} : t_{2}x_{2}; t_{0}^{−1}y_{0} : t_{1}^{−1}y_{1} : t_{2}^{−1}y_{2}]. This action induces an action on S ⊂ ℙ^{2} × ℙ^{2}. The torus T sends each line into itself. The action of T on the complement U of the 6 lines in S is faithful and transitive. If one identifies U with T by the choice of a rational point in U, the variety S with its open set U = T has the structure of a toric variety. The symmetric group S_{2} = ℤ/2 acts on S ⊂ ℙ^{2} ×_{F} ℙ^{2} by permuting the factors. This globally preserves the lines, the generator of S_{2} induces on the hexagon of lines the permutation of each E_{i} with each F_{i}, i.e. opposite sides of the hexagon are exchanged. The group S_{3} acts on S ⊂ ℙ^{2} ×_{F} ℙ^{2} by simultaneous permutation on each factor. This globally preserves the lines. The actions of S_{2} and S_{3} commute. The induced action of the group H := S_{2} × S_{3} on the hexagon of lines realizes the automorphism group of the hexagon.

Let the group H act on T = (𝔾_{m})^{3}/𝔾_{m} in such a way that the generator of S_{2} sends t ∈ T to t^{−1} and S_{3} acts by permutation of the factors. Let T′ be the semidirect product of T and H with respect to this action. The above construction yields an isomorphism from T′ to the algebraic group Aut(S) of automorphisms of the surface S. Indeed any σ in Aut(S) may be multiplied by an element of H so that the action on the hexagon becomes trivial. By general properties of blow-ups, this implies that any of the projections S → ℙ^{2} factorizes through the contraction S → ℙ^{2}, i.e. comes from an automorphism of ℙ^{2} which respects each of the points [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1]. Any such automorphism is given by an element of T.

Let now F be an arbitrary field and S a del Pezzo surface of degree 6 over F. Over a separable closure \overline{F} of F the del Pezzo surface \overline{S} = S ×_{F} \overline{F} is split, i.e. isomorphic to the model given above by [4]. Since the 6 lines are globally stable under the action of the Galois group, there exists a Zariski open set U ⊂ X whose complement over \overline{F} consists of the 6 lines. The Galois action on the 6 lines induces an automorphism of the hexagon of lines, hence a homomorphism Gal(\overline{F} /F) → H = S_{2} × S_{3}. There is thus an associated étale quadratic extension K/F and an étale cubic extension L/F. Let T be the connected component of identity in the F -group Aut(S). Then T is an algebraic torus and U is a principal homogeneous space under T as they are so over \overline{F}. The group of connected components of Aut(S) is a twisted form of S_{2} × S_{3} : it is the F -group of automorphisms of the finite F -scheme associated to the configuration of the 6 lines on \overline{F}. The F -torus T will be identified below (Remark 4.4).

Let K be an étale quadratic F -algebra and B an Azumaya K-algebra of rank 9 over K with unitary involution τ trivial on F [8, §2.B]. Thus B is a central simple K-algebra of dimension 9 if K is a field, or B is isomorphic to the product A×A^{op}, where A is a central simple F -algebra of dimension 9 and A^{op} is the opposite algebra if K ≃ F × F.

We consider the F -subspace of τ-symmetric elements

Sym(B, τ) = {b ∈ B such that τ(b) = b}

of dimension 9.

Lemma 4.1. For a right ideal I ⊂ B of rank 3 over K, the F -subspace (I · τI) ∩ Sym(B, τ) of Sym(B, τ) is 1-dimensional. The correspondence I ↦ (I · τI) ∩ Sym(B, τ) gives rise to a closed embedding of varieties

s_{B,τ} : R_{K/F} SB(B) → ℙ(Sym(B, τ)).

Proof. It is sufficient to prove the lemma in the split case, i.e., when

B = End(V) × End(V^{∗}),

where V is a 3-dimensional vector space over F, and τ is the exchange involution τ(f, g^{∗}) = (g, f^{∗}) for all f, g ∈ End(V) (cf. [8, Prop. 2.14]). We identify Sym(B, τ) with End(V) via the embedding f ↦ (f, f^{∗}) into B.

A right ideal I of B of rank 3 over K = F × F is of the form

I = Hom(V, U) × Hom(V^{∗}, W^{∗}),

where U and W are a subspace and a factor space of V of dimension 1 respectively. We have

τI = Hom(W, V) × Hom(U^{∗}, V^{∗}),

and

I · τI = Hom(W, U) × Hom(U^{∗}, W^{∗}).

The F -space

(I · τI) ∩ Sym(B, τ) = Hom(W, U)

therefore is 1-dimensional. Under the identification of R_{K/F} SB(B) with ℙ(V)× ℙ(V^{∗}), the morphism s_{B,τ} takes a pair of lines (U, W^{∗}) to Hom(W, U) = U ⊗ W^{∗}, i.e., it coincides with the Segre closed embedding

ℙ(V) × ℙ(V^{∗}) → ℙ(V ⊗ V^{∗}) = ℙ(End(V)).  □

Let Trd : B → K be the reduced trace linear form. For any x, y ∈ Sym(B, τ) we have

τ Trd(xy) = Trd(τ(xy)) = Trd(τ(y)τ(x)) = Trd(yx) = Trd(xy),

hence Trd(xy) ∈ F. We therefore have an F -bilinear form 𝔟(x, y) = Trd(xy) on Sym(B, τ). The form 𝔟 is non-degenerate as it is so over \overline{F}.

Let L be a cubic étale F -subalgebra of B that is contained in Sym(B, τ). Write L^{⊥} for the orthogonal complement of L in Sym(B, τ) with respect to the form 𝔟. As L is étale, we have L ∩ L^{⊥} = 0. Consider the 7-dimensional F -subspace F ⊕ L^{⊥} of Sym(B, τ) and set

S(B, τ, L) = s_{B,τ}^{−1}(ℙ(F ⊕ L^{⊥})).

Thus S(B, τ, L) is a closed subvariety of R_{K/F} SB(B) (cf. [16]).

An isomorphism between two triples (B, τ, L) and (B′, τ′, L′) is an F -algebra isomorphism f : B →^{∼} B′ such that f ∘ τ = τ′ ∘ f and f(L) = L′. The automorphism group of (B, τ, L) is a subgroup of the algebraic group Aut(B, τ).

The construction of the scheme S(B, τ, L) being natural, an automorphism of a triple (B, τ, L) induces an automorphism of S(B, τ, L), i.e., we have an algebraic group homomorphism

(3)  μ : Aut(B, τ, L) → Aut(S(B, τ, L)).

Theorem 4.2. Let F be an arbitrary field. Let B be a rank 9 Azumaya algebra with unitary involution τ over a quadratic étale algebra K over F and a cubic étale F -subalgebra L of B contained in Sym(B, τ).

(i) The variety S(B, τ, L) is a del Pezzo surface of degree 6.

(ii) Any del Pezzo surface of degree 6 over F is isomorphic to S(B, τ, L) for some B, τ and L.

(iii) Two surfaces S(B, τ, L) and S(B′, τ′, L′) are isomorphic if and only if the triples (B, τ, L) and (B′, τ′, L′) are isomorphic.

(iv) The homomorphism μ is an isomorphism.

(v) The étale quadratic algebra K/F and the étale cubic algebra L/F are naturally isomorphic to the ones associated to the Galois action on the lines of the del Pezzo surface S(B, τ, L) over \overline{F}.

Proof. (i): We may assume that F is separably closed. We claim that any triple (B, τ, L) is isomorphic to the split triple (M_{3}(F)×M_{3}(F), ε, F^{3}), where:

(1) ε(a, b) = (b^{t}, a^{t}), (t denotes the transpose matrix), in particular Sym(M_{3}(F) × M_{3}(F), ε) consists of matrices of the shape (a, a^{t});

(2) F^{3} is identified with the subalgebra of diagonal matrices in Sym(M_{3}(F) × M_{3}(F), ε), i.e. those of the shape (a, a) with a diagonal;

(3) K = F × F ⊂ M_{3}(F) × M_{3}(F) is the obvious map from F × F to the center of M_{3}(F) × M_{3}(F).

Indeed, as K and B are split, (B, τ) is isomorphic to (M_{3}(F) × M_{3}(F), ε) by [8, Prop. 2.14]. Let L′ ⊂ M_{3}(F) × M_{3}(F) be the image of the (split) étale cubic subalgebra L under this isomorphism. In particular, (B, τ, L) ≃ (M_{3}(F)×M_{3}(F), ε, L′). Any of the two projections to M_{3}(F) identifies L′ with a split étale cubic subalgebra of M_{3}(F). Any two split étale cubic subalgebras of M_{3}(F) are conjugate, i.e., there is an a ∈ M_{3}(F)^{×} such that aL′a^{−1} = F^{3}. Then the conjugation by (a, (a^{−1})^{t}) yields an isomorphism between (M_{3}(F) × M_{3}(F), ε, L′) and (M_{3}(F) × M_{3}(F), ε, F^{3}). The claim is proved.

So we may assume that (B, τ, L) is the split triple. Then F ⊕ L^{⊥} is the space of all pairs (b, b^{t}) with a matrix b all diagonal elements of which are equal. Let [x_{0} : x_{1} : x_{2}; y_{0} : y_{1} : y_{2}] be the projective coordinates in ℙ^{2} ×_{F} ℙ^{2}. The Segre embedding s_{B,τ} takes [x_{0} : x_{1} : x_{2}; y_{0} : y_{1} : y_{2}] to the point of ℙ(M_{3}(F)) given by the matrix (x_{i}y_{j})_{i,j=1,2,3} (we here identify an element (a, a^{t}) ∈ Sym(M_{3}(F) × M_{3}(F), ε) with a ∈ M_{3}(F)).

Therefore S(B, τ, L) is a closed subvariety of ℙ^{2}×_{F} ℙ^{2} given by the equations x_{0}y_{0} = x_{1}y_{1} = x_{2}y_{2}, that is a split del Pezzo surfaces of degree 6.

(iv): We may assume that F is separably closed and hence we are in the split situation of the proof of (i). Let the torus T, the semidirect product T′ and the split del Pezzo surface S be as in the initial discussion of del Pezzo surfaces of degree 6. We let T′ act on B by F -algebra automorphisms as follows. The groups T, respectively S_{3}, act on B by the formula x(a, b) = (xax^{−1}, (x^{−1})^{t}bx^{t}), where x is in T, respectively is the monomial matrix corresponding to an element of S_{3}. The generator of S_{2} takes a pair (a, b) to (b, a). The action of T′ defined this way commutes with τ and preserves L elementwise and therefore induces an algebraic group homomorphism φ : T′ → Aut(B, τ, L). The composite map of φ with the homomorphism

Aut(B, τ, L) → Aut(K) × Aut(L) = S_{2} × S_{3}.

is a surjective homomorphism which coincides with the one described at the beginning of this section. We claim that φ is an isomorphism. Let G be the kernel of the above homomorphism. It suffices to show that the restriction ψ : T → G of φ is an isomorphism. We view G as a subgroup of the connected component Aut(B, τ)^{+} of identity in Aut(B, τ). We have an isomorphism between PGL_{3} and Aut(B, τ)^{+} taking an a to the conjugation by (a, (a^{−1})^{t}) (cf. [8, §23]). The composite map

T ⟶^{ψ} G ↪ Aut(B, τ)^{+} →^{∼} PGL_{3}

identifies T with the maximal torus T̃ of the classes of diagonal matrices in PGL_{3}. The image of G in PGL_{3} coincides with the centralizer of T̃ in PGL_{3}, hence it is equal to T̃. Thus ψ is an isomorphism. The claim is proved.

The composite map

T′ ⟶^{φ} Aut(B, τ, L) ⟶^{μ} Aut(S(B, τ, L))

coincides with the isomorphism in the initial discussion of del Pezzo surfaces of degree 6. Therefore, μ is an isomorphism.

(ii) and (iii): By the proof of (i), any triple (B, τ, L) over \overline{F} is isomorphic to the split triple. Moreover, any del Pezzo surface of degree 6 splits over \overline{F}. The homomorphism μ in (3) is an isomorphism by (iv), therefore the statements follow by the standard technique in [8, §26].

(v): The étale algebras K and L are associated to the Galois action on the set of 6 minimal diagonal idempotents e_{i} and f_{i} of the algebra F^{3} ×F^{3} (i = 1, 2, 3) where the e_{i} (resp. the f_{i}) are diagonal idempotents in F^{3} × 0 (resp. 0 × F^{3}). The statement follows from the fact that the correspondence e_{i} ↦ E_{i}, f_{i} ↦ F_{i} establishes an isomorphism of the (S_{2} × S_{3})-sets of minimal idempotents {e_{1}, e_{2}, e_{3}, f_{1}, f_{2}, f_{3}} and exceptional lines {E_{1}, E_{2}, E_{3}, F_{1}, F_{2}, F_{3}}. □

Remark 4.3. With notation as in the beginning of this section, the natural exact sequence of Galois modules

0 → \overline{F} [U]^{×}/\overline{F}^{×} → Div_{\overline{S}\\overline{U}}(\overline{S}) → Pic \overline{S} → Pic \overline{U},

where Div_{\overline{S}\\overline{U}}(\overline{S}) denotes the group of divisors of \overline{S} with support on the complement of \overline{U} and the first map is the divisor map, yields the exact sequence of Galois lattices:

0 → T̂ → ℤ[KL/F] → Pic\overline{S} → 0,

which defines the 2-dimensional F -torus T with character group T̂ = \overline{F} [U]^{×}/\overline{F}^{×}. The F -variety U is a principal homogeneous space under T. The 6-dimensional Galois module ℤ[KL/F] is the permutation module on the 6 lines.

Direct computation over \overline{F} shows that there is an exact sequence of Galois lattices

0 → T̂ → ℤ[KL/F] → ℤ[L/F] ⊕ ℤ[K/F] → ℤ → 0.

Here ℤ[L/F] is the 3-dimensional permutation lattice on the set of opposite pairs of lines in the hexagon and ℤ[K/F] is the 2-dimensional permutation lattice on the set of triangles of triples of skew lines in the hexagon. The map ℤ[KL/F] → ℤ[L/F] sends a line to the pair it belongs to, and the map ℤ[KL/F] → ℤ[K/F] sends a line to the triangle it belongs to. The map ℤ[L/F] ⊕ ℤ[K/F] → ℤ is the difference of the augmentation maps. Note that this Galois homomorphism has an obvious Galois equivariant section.

¿From this we conclude that there exist an isomorphism of Galois lattices

Pic \overline{S} ⊕ ℤ ≃ ℤ[L/F] ⊕ ℤ[K/F]

and an exact sequence of F -tori

1 → 𝔾_{m,F} → R_{L/F} 𝔾_{m} × R_{K/F} 𝔾_{m} → R_{KL/F} 𝔾_{m} → T → 1.

Taking F -points and using Hilbert’s theorem 90, we conclude that T (F) is the quotient of (KL)^{×} by the subgroup spanned by K^{×} and L^{×}. We also see that the F -torus is stably rational. More precisely T ×_{F} R_{K/F} 𝔾_{m} ×_{F} R_{L/F} 𝔾_{m} is F -birational to 𝔾_{m,F} ×_{F} R_{KL/F} 𝔾_{m}. The F -torus T actually is rational (Voskresenskiĭ proved that all 2-dimensional tori are rational).

Remark 4.4. It follows from the proof of Theorem 4.2 that T is a maximal F -torus of the connected component of the identity Aut(B, τ)^{+} of the automorphism group of the pair (B, τ). By [8, §23], the group of F -points of Aut(B, τ)^{+} coincides with

{b ∈ B^{×} | b · τ(b) ∈ F^{×}}/K^{×}.

It follows that

T (F) = {x ∈ (KL)^{×} | N_{KL/L}(x) ∈ F^{×}}/K^{×}.

We leave it to the reader to compare this description with the one produced in the previous remark.

Remark 4.5. If the quadratic algebra K is split, i.e., K = F × F and B = A × A^{op} with the switch involution τ, where A is a central simple F -algebra of dimension 9, the surface S(B, τ, L) is a closed subvariety of SB(A) ×_{F} SB(A^{op}) and the projection S(B, τ, L) → SB(A) is a blow-up with center a closed subvariety of SB(A) isomorphic to Spec L. In particular, the surface S(B, τ, L) is not minimal.

Lemma 4.6. Let S = S(B, τ, L) be a del Pezzo surfaces of degree 6. Then (i) If n_{S} = 6, then K and the K-algebra B are not split. (ii) If S(F) ≠ ∅, then the K-algebra B is split.

Proof. If K is split then n_{S} ≤ 3 by Remark 4.5. By the same remark, if B is split then S has a rational point over K, hence n_{S} ≤ 2. Finally, if S has a rational point, then so does R_{K/F} SB(B) as S is a closed subvariety of R_{K/F} SB(B), and therefore B is split. □

We may now give our first proof of Theorem 1.3. Let notation be as in Proposition 3.2.

By Theorem 2.4, X is either a conic bundle over a smooth conic or a del Pezzo surface of degree d = 1, …, 9. In the first case, X has a rational point over a field extension of degree dividing 4, therefore, n_{X} divides 4, a contradiction. In the latter case, we have 6 | n_{X} | d, i.e., d = 6 and X is a del Pezzo surface of degree 6.

By Theorem 4.2, we have X = S(B, τ, L) for a rank 9 Azumaya algebra B with unitary involution τ over a quadratic étale algebra K over F and a cubic étale F -subalgebra L of B contained in Sym(B, τ). It follows from Lemma 4.6 (i) that K and B are not split. By (iii) of Proposition 3.2 and by Lemma 4.6 (ii), the K(Y ×_{F} Z)-algebra B ⊗_{K} K(Y ×_{F} Z) is split. The field extension K(Y ×_{F} Z)/K(Z) is the function field of a conic over K(Z) and B ⊗_{K} K(Z) is an algebra of degree 3 over K(Z), hence B ⊗_{K} K(Z) is also split. By a theorem of Châtelet (recalled below), the K-algebra B, which is not split, is similar to D_{K} or to D_{K}^{⊗2}. Since B carries an involution of the second kind we have cor_{K/F} ([B]) = 0 by [8, Th. 3.1]. From 2[D] = cor_{K/F} ([B]) we conclude that D and therefore B is split, a contradiction.

5. Splitting properties of geometrically rational varieties of canonical dimension at most 2

In this section we study the kernel of the natural homomorphism of Brauer groups Br F → Br F (X) for a geometrically unirational smooth variety X of canonical dimension at most 2.

Let us recall the well known:

Proposition 5.1. Let F be a field, \overline{F} a separable closure, 𝔤 = Gal(\overline{F} /F) the absolute Galois group. Let X/F be a proper, geometrically integral variety. We then have a natural exact sequence

0 → Pic X → (Pic \overline{X})^{𝔤} → Br F → Br X,

where Br X = H_{ét}^{2}(X, 𝔾_{m}). If moreover X/F is smooth, then the map Br X → Br F (X) is injective, and we have the exact sequence

0 → Pic X → (Pic \overline{X})^{𝔤} → Br F → Br F (X).

We write Br(F (X)/F) for the kernel of Br F → Br F (X).

The following well known result is due to F. Châtelet. In dimension 1, i.e. for A a quaternion algebra and X a conic, it goes back to Witt.

Proposition 5.2. Let X = SB(A) be the Severi-Brauer variety of A. Then Br(F (X)/F) is the subgroup of Br F generated by the class of A.

Proposition 5.3. Let F be a perfect field and X a geometrically rational surface over F. Then we have one of the following possibilities:

(i) X is F -birational to a Severi-Brauer surface, i.e. a twisted form of ℙ^{2}. Then Br(F (X)/F) is 0 or ℤ/3, and is spanned by the class of a central simple algebra of degree 3.

(ii) X is F -birational to a twisted form of ℙ^{1} ×_{F} ℙ^{1}. Then Br(F (X)/F) is 0 or ℤ/2 (spanned by the class of a quaternion or biquaternion algebra) or ℤ/2 ⊕ ℤ/2 (spanned by the classes of two quaternion algebras).

(iii) X is F -birational to a conic bundle over a smooth projective conic. Then Br(F (X)/F) is 0 or ℤ/2, or ℤ/2 ⊕ ℤ/2, and is spanned by the classes of two quaternion algebras.

(iv) Br(F (X)/F) = 0, i.e., the natural map Br F → Br F (X) is injective.

Proof. By resolution of singularities (Theorem 2.1) we may assume that X is smooth, projective and F -minimal.

Assume X is the Severi-Brauer surface SB(A) associated to a central simple F -algebra A of index 3. That Br(F (X)/F) is 0 or ℤ/3, and is spanned by the class of a central simple algebra of degree 3, follows from Proposition 5.2.

Assume X is a twisted form of ℙ^{1} ×_{F} ℙ^{1}. As the automorphism group of ℙ^{1} ×_{F} ℙ^{1} is the semidirect product of PGL_{2} × PGL_{2} with the cyclic group of order 2 permuting the components, we have X = R_{K/F} (C) where K/F is an étale quadratic F -algebra and C is a conic curve over K. If K is a field then by [14, Cor. 2.12], we have

Br(F (C)/F)) = cor_{K/F} (Br(K(C)/K))).

By Proposition 5.2, Br(K(C)/K)) is generated by the class of a quaternion algebra over K and therefore, Br(F (C)/F)) is generated by the corestriction of a quaternion algebra that is either 0 or a quaternion algebra, or a biquaternion algebra.

If K = F × F then X = C ×_{F} C′ where C and C′ are conics over F. In this case Br(F (C)/F)) is a quotient of ℤ/2 ⊕ ℤ/2, spanned by the classes of the quaternion F -algebras associated with C_{1} and C_{2}.

Let X/F be a conic bundle over a conic Y. Then Br(F (Y)/F ) is 0 or ℤ/2, spanned by the class of the quaternion algebra Q associated to Y and the kernel of Br F (Y) → Br F (X) is 0 or ℤ/2. Thus the order of the kernel of Br F → Br F (X) divides 4. Let A/F be a nontrivial division algebra in Br(F (X)/F)) different from Q. It suffices to show that Br(F (X)/F)) contains a division quaternion algebra different from Q. The index of A over the function field F (Y) is at most 2. By the index reduction formula [17, Th. 1.3], the index of one of the F -algebras A and A ⊗_{F} Q is at most 2, i.e., one of these two algebras is similar to a division quaternion algebra different from Q.

If X is not F -isomorphic to a twisted form of ℙ^{2}, to ℙ^{1} ×_{F} ℙ^{1} or to a conic bundle over a conic then according to Theorem 2.4 and Lemma 2.5, X is a del Pezzo surface with Pic X of rank 1 such that the canonical class, which is in Pic X, is not divisible in Pic \overline{X}. As the cokernel of the natural map Pic X → (Pic \overline{X})^{𝔤} is torsion, the group (Pic \overline{X})^{𝔤} is free of rank 1. Therefore (Pic \overline{X})^{𝔤} is generated by the canonical class and hence the map Pic X → (Pic \overline{X})^{𝔤} is an isomorphism. By Proposition 5.1, this implies that Br(F (X)/F) = 0. □

Remark 5.4. The proof of this proposition uses Theorem 2.4 and Lemma 2.5 in a critical fashion but it requires no discussion of del Pezzo surfaces other than Severi-Brauer surfaces and twisted forms of ℙ^{1}×_{F} ℙ^{1}. The same comment applies to Theorem 5.5 and Corollary 5.7 hereafter, hence to the second proof of Theorem 1.3 given at the end of this section.

Theorem 5.5. Let W be a smooth, proper, geometrically unirational variety over a field F of characteristic zero.

(i) If cdim(W) = 1 then Br(F (W)/F ) is 0 or ℤ/2.

(ii) If cdim(W) = 2 then Br(F (W)/F ) is one of 0, ℤ/2, ℤ/2 ⊕ ℤ/2 or ℤ/3. The kernel is spanned either by a quaternion algebra, or two quaternion algebras, or one biquaternion algebra, or a cubic algebra.

Proof. By Lemma 3.1, there exists a closed geometrically rational F -subvariety X ⊂ W of dimension 1 in case (i) and of dimension 2 in case (ii). As Br W injects into Br F (W), we have

Br(F (W)/F ) ⊂ Br(F (X)/F ).

If the dimension of X is 1, then X is a smooth conic. The kernel of Br F → Br F (X) is 0 or ℤ/2.

If the dimension of X is 2, then the possibilities for Br(F (X)/F ) were listed in Proposition 5.3. □

Remark 5.6. The same theorem holds if the hypothesis that W is geometrically unirational is replaced by the hypothesis that W is a geometrically rationally connected variety. These hypotheses indeed imply that the variety X is geometrically rationally connected. Since X is of dimension at most 2 and char(F) = 0 this forces X to be geometrically rational.

Corollary 5.7. Let W/F be a smooth, proper, geometrically unirational variety over a field F of characteristic zero. Assume cdim(W) ≤ 2. Let A and A′ be central division F -algebras. If there is an F -rational map from W to the product SB(A) ×_{F} SB(A′) then one of the following occurs:

(1) A and A′ are cubic algebras.

(2) A and A′ are quaternion or biquaternion algebras.

Proof. If there is such a rational map, then the classes of A ∈ Ker[Br F → Br SB(A)] and A′ ∈ Ker[Br F → Br SB(A′)] belong to Br(F (W)/F ). The result then follows from Theorem 5.5. □

Remark 5.8. Corollary 5.7 holds if the hypothesis that W is geometrically unirational is replaced by the hypothesis that W is a geometrically rationally connected variety.

We now give our second proof of Theorem 1.3. This proof does not use Section 4. Let notation be as in Proposition 3.2, so that Y, resp. Z, is the Severi-Brauer variety attached to a quaternion algebra, resp. to an algebra of degree 3. Assume cdim(SB(A)) ≤ 2. Then cdim(Y ×_{F} Z) ≤ 2. If we apply the above corollary 5.7 to the identity map of Y ×_{F} Z then we get a contradiction. We could also combine Proposition 3.2 and Proposition 5.3. It is then clear that we here use statement (iv) of Proposition 3.2, as opposed to our use of statement (iii) of that same proposition in our first proof (end of Section 4) of Theorem 1.3.

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CNRS, Mathématiques, UMR 8628, Bâtiment 425, Université Paris-Sud, F-91405 Orsay, FRANCE

E-mail address: Jean-Louis.Colliot-Thelene à math.u-psud.fr

Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, 4, place Jussieu, F-75252 Paris CEDEX 05, FRANCE

E-mail address: karpenko à math.jussieu.fr

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

E-mail address: merkurev at math.ucla.edu