DEL PEZZO SURFACES OF DEGREE 6 OVER AN ARBITRARY FIELD

MARK BLUNK

Abstract. We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field F. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen K-theory of the surface. As a consequence, we obtain an index reduction formula for the function field of the surface.

1. Introduction

If X is an algebraic variety defined over an arbitrary field F, a common method (cf. the introduction of [13]) for learning various properties of X is to first study \overline{X} := X ×_{Spec F} Spec(\overline{F}), the extension of scalars of X to a separable closure \overline{F} of F, and then to study the action of the Galois group Gal(\overline{F} /F) on algebraic groups and other algebraic objects associated to \overline{X}. This is particularly useful when dealing with a class of varieties that all become isomorphic over \overline{F}, e.g. Severi-Brauer varieties or involution varieties. A Severi-Brauer variety is determined by a central simple F -algebra A, and an involution variety is determined by a central simple F -algebra A with an orthogonal involution of the first kind (A, σ). In either case this algebraic data determines geometrical and topological information about the corresponding variety. In particular the Quillen K-groups of the variety are determined the algebra in the Severi-Brauer example, and the algebra with involution in the involution variety example. This was proved for Severi-Brauer varieties and involution varieties in [8] and [11], respectively. Panin proved in [7] a more general theorem computing the K-theory of projective homogeneous varieties, which contains both examples as special cases. In all of these examples, as in this paper, the action of algebraic groups plays a significant role. An immediate consequence of this computation of the K-theory is an index reduction formula, which determines how extending scalars of a division F -algebra to the function field of the variety reduces the index of the algebra. In this paper we will study del Pezzo surfaces of degree 6 over F, obtaining similar results.

A del Pezzo surface S is a smooth projective surface over a field F such that the anti-canonical bundle ω_{S}^{−1} is ample. The degree (the self-intersection number of ω_{S}) of any such surface can be any integer between 1 and 9. Such varieties were discussed in [1], [2], and [12]. As mentioned in some of these references, a del Pezzo surface of degree 6 is a toric variety for a particular two dimensional torus, which we will describe below. We explore this toric structure in Section 2. The result is Theorem 2.4, a classification of all such surfaces up to isomorphism preserving the action of the torus. Section 3 contains the main result of the paper, Theorem 3.5, where it is proved that a del Pezzo surface of degree 6 is determined by a pair B and Q of separable F -algebras, with centers K and L étale quadratic and cubic over F respectively, and both containing K ⊗_{F} L as a subalgebra. Moreover, cor_{K/F} (B) and cor_{L/F} (Q) must be split. As an immediate corollary of Theorems 2.4 and 3.5, we give a necessary and sufficient condition in terms of B and Q for determining when the corresponding surface will have a rational point. We then make some connections between this characterization of del Pezzo surfaces of degree 6 and the characterization developed in [1].

In Section 4, we relate the algebras B and Q to the endomorphism rings of locally free sheaves on the associated del Pezzo surface S. These sheaves are used in Theorem 4.2 to relate the Quillen K-theory of S to that of B and Q, by showing that the algebra A = F × B × Q is isomorphic to S in a certain K-motivic category C, constructed in [7]. This implies that for all n,

K_{n}(S) ≅ K_{n}(A) = K_{n}(F) ⊕ K_{n}(B) ⊕ K_{n}(Q).

As a corollary we obtain an index reduction formula for the function field of S.

I would like to thank my advisor Alexander Merkurjev, who posed this question to me, and answered several of my questions which developed along the way.

We use the following notations and conventions:

An F -variety is a separated scheme of finite type over Spec(F).

\overline{F} will denote a separable closure of F.

An F -algebra A is separable if A⊗_{F} L is semisimple for every field extension L of F. Such an algebra is Azumaya over its center, which is an étale extension of F.

Γ will denote the group Gal(\overline{F} /F).

For any F -variety X and any field extension E of F, we will denote X ×_{Spec F} Spec(E) (resp. X ×_{Spec F} Spec(\overline{F})) by X_{E} (resp. \overline{X}).

For any separable F -algebra A and any étale extension E of F, we will denote A ⊗_{F} E (resp. A ⊗_{F} \overline{F}) by A_{E} (resp. \overline{A}).

If D is a Cartier divisor on a variety X, L(D) will denote the corresponding invertible sheaf on X.

For any variety X and any separable algebra A, P(X; A) will denote the exact category of left A ⊗_{F} O_{X}-modules which are locally free O_{X}-modules. We will denote P(X; F) (resp. P(Spec F; A)) by P(X) (resp. P(A)).

For any integer n, K_{n}(X; A) will denote the Quillen group K_{n}(P(X; A)). As above, we will denote K_{n}(X; F) by K_{n}(X) and K_{n}(Spec F; A) by K_{n}(A).

For any algebraic torus T, T̂ will denote the Γ-module of characters Hom\overline{F} (\overline{T} , 𝔾_{m,\overline{F}}).

2. Toric Varieties

We first recall from [4], [5], and [12] some basic properties of the variety S̃, the blow up of ℙ^{2} at the 3 non-collinear points [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1]. The variety S̃ can be realized as a closed subvariety of ℙ^{2}×ℙ^{2}, defined by the equations x_{0}y_{0} = x_{1}y_{1} = x_{2}y_{2}. The projection onto the first factor of ℙ^{2} is the blow down of the three lines m_{0} = {x_{1} = x_{2} = 0}, m_{1} = {x_{0} = x_{2} = 0}, and m_{2} = {x_{0} = x_{1} = 0}. Similarly, the projection onto the second factor of ℙ^{2} is the blow down of the three lines l_{0} = {y_{1} = y_{2} = 0}, l_{1} = {y_{0} = y_{2} = 0}, and l_{2} = {y_{0} = y_{1} = 0}.

Proposition 2.1. Let S̃ be the blow up of ℙ^{2} at the three points [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1].

i. The variety S̃ is a del Pezzo surface of degree 6 over F, and if F is separably closed, any del Pezzo surface S of degree 6 over F is isomorphic to S̃.

ii. The group CH^{1}(S̃) is generated by the lines l_{0}, l_{1}, l_{2}, m_{0}, m_{1}, and m_{2}. iii. The intersection pairing on CH^{1}(S̃) is determined by the following relations: l_{i}^{2} = −1, m_{i}^{2} = −1, l_{i}m_{j} = 1, and l_{i}m_{i} = l_{i}l_{j} = m_{i}m_{j} = 0, for distinct i, j ∈ {0, 1, 2}.

iv. The group CH^{2}(S̃) is cyclic, generated by the class of any rational point.

As mentioned in [1], there is an action of the torus T̃ = 𝔾_{m}^{3}/𝔾_{m} on ℙ^{2}, described by:

(t_{0}, t_{1}, t_{2}) · [x_{0}, x_{1}, x_{2}; y_{0}, y_{1}, y_{2}] = [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}].

Here 𝔾_{m} embeds into 𝔾_{m}^{3} diagonally. This action sends S̃ to itself, and is faithful and transitive on the open subset Ũ of S̃, the complement of the subvariety defined by the equation x_{0}x_{1}x_{2}y_{0}y_{1}y_{2} = 0. This closed subvariety has 6 irreducible components, the lines l_{0}, l_{1}, l_{2}, m_{0}, m_{1}, and m_{2}, which by the proposition are arranged in a hexagon. Thus S̃ is a T̃-toric variety, with fan dual to the hexagon of lines. There is also an action of the symmetric groups S_{2} and S_{3} on S̃. The nontrivial element of S_{2} acts on ℙ^{2} ×ℙ^{2} by interchanging the x_{i} and y_{i}, and the S_{3} action on ℙ^{2} ×ℙ^{2} arises from the diagonal action of S_{3} on the coordinates x_{0}, x_{1}, x_{2} and y_{0}, y_{1}, y_{2}. The S_{2} and S_{3} actions commute with each other, and both groups send S̃ ⊂ ℙ^{2} × ℙ^{2} to itself. Therefore they induce an action of S_{2} × S_{3} on S̃, preserving the set of lines l_{0}, l_{1}, l_{2}, m_{0}, m_{1}, and m_{2}, and thus inducing an isomorphism from S_{2} × S_{3} onto the automorphism group of the hexagon of lines. The torus T̃ is the connected component of the identity of the algebraic group Aut_{F} (S̃) of automorphisms of S̃, and S_{2} × S_{3} is the group of connected components. The action of S_{2} × S_{3} on S̃ define a section S_{2} × S_{3} → Aut_{F} (S̃), so we have the following split exact sequence of algebraic groups:

1 → T̃ → Aut_{F} (S̃) → S_{2} × S_{3} → 1.

[figure: ℙ^{2}; x_{1} = 0; x_{2} = 0; x_{0} = 0; ℙ^{2}; y_{0} = 0; y_{1} = 0; y_{2} = 0; S̃; l_{0}; l_{1}; l_{2}; m_{0}; m_{1}; m_{2}; p_{1}; p_{2}]

Figure 1. The Hexagon of Lines.

We have another way to realize S̃ as a closed subvariety of a product of projective spaces. Define f_{i} : S̃:→ ℙ^{1} for i = 0, 1, 2 by

f_{0}([x_{0} : x_{1} : x_{2}; y_{0} : y_{1} : y_{2}]) = [x_{1} : x_{2}] or [y_{2} : y_{1}]
f_{1}([x_{0} : x_{1} : x_{2}; y_{0} : y_{1} : y_{2}]) = [x_{2} : x_{0}] or [y_{0} : y_{2}]
f_{2}([x_{0} : x_{1} : x_{2}; y_{0} : y_{1} : y_{2}]) = [x_{0} : x_{1}] or [y_{1} : y_{0}].

Each f_{i} is well defined, as the two definitions agree on the overlap, and thus is a morphism of varieties. These morphisms define a morphism f : S̃ → ℙ^{1} × ℙ^{1} × ℙ^{1}. If we denote the bi-homogeneous coordinates of ℙ^{1} × ℙ^{1} × ℙ^{1} by X_{0}, X_{1}, Y_{0}, Y_{1}, Z_{0}, and Z_{1}, it can be shown that f maps S̃ isomorphically onto the hypersurface of ℙ^{1} × ℙ^{1} × ℙ^{1} defined by the equation X_{0}Y_{0}Z_{0} = X_{1}Y_{1}Z_{1}.

The morphism f sends T̃ to the torus

Ker(𝔾_{m}^{2}/𝔾_{m} × 𝔾_{m}^{2}/𝔾_{m} × 𝔾_{m}^{2}/𝔾_{m} ⟶^{m} 𝔾_{m}^{2}/𝔾_{m}),

where m((t_{0}, t_{1}), (t′_{0}, t′_{1}), (t′′_{0}, t′′_{1})) = (t_{0}t′_{0}t′′_{0}, t_{1}t′_{1}t′′_{1}).

Now let S be a del Pezzo surface of degree 6 over an arbitrary field F. Then \overline{S} is a del Pezzo surface of degree 6 over \overline{F}, and thus by Proposition 2.1 is isomorphic over \overline{F} to \overline{S̃}. So S is an F -form of S̃. As the six lines of the hexagon form a full set of exceptional curves in \overline{S}, the action of Γ on \overline{S} is globally stable on the set of lines of the hexagon. Therefore, there is an open subvariety U whose complement Z is isomorphic over \overline{F} to the hexagon of lines. The action of Γ on \overline{Z} permutes its irreducible components, inducing an action of Γ on the hexagon.

Let T denote the connected component of the identity of Aut_{F} (S). The group of connected components G of Aut_{F} (S) is an étale group scheme: it is the group scheme determined (as in Proposition 20.16 of [3]) by the automorphism group of the hexagon of lines, with continuous Γ-action on this finite group as in the previous paragraph. So T is a torus, S is a T -toric variety, with an open set U which is a T -torsor, and Γ-action on the fan determined by the étale group scheme G.

This Γ-action on the hexagon determines a homomorphism γ : Γ → S_{2} × S_{3}. Projecting onto either factor yields cocycles with values in S_{2} and S_{3}, and thus γ determines a pair (K, L), where K and L are étale quadratic and cubic extensions of F, respectively. Note that while the fan, dual to the hexagon of lines, is the same for all del Pezzo surfaces of degree 6 over F, the possible Γ-actions on the fan are in a one-to-one correspondence with pairs (K, L).

For a fixed cocycle γ (i.e. a fixed pair (K, L)), we will classify all del Pezzo surfaces S of degree 6 where the Γ-action on \overline{Z} ⊂ \overline{S} is determined by γ.

We have from [12] the following short exact sequence of Γ-modules:

0 → T̂ → ℤ[KL/F] → Pic(\overline{S}) → 0.

Here KL denotes the algebra K ⊗_{F} L, and ℤ[KL/F] is the lattice of the six lines of \overline{Z}. The homomorphism ℤ[KL/F] → Pic(\overline{S}) takes a line to the corresponding Cartier divisor on \overline{S}. As described in [1], this short exact sequence can be extended into the exact sequence

(1)  0 → T̂ → ℤ[KL/F] → ℤ[K/F] ⊕ ℤ[L/F] → ℤ → 0.

Here ℤ[L/F] is the lattice of pairs of opposite lines, and ℤ[K/F] is the lattice of triangles, where each triangle is a triple of skew lines. The homomorphism ℤ[KL/F] → ℤ[L/F] sends each line to the pair containing it, and the homomorphism ℤ[KL/F] → ℤ[K/F] sends each line to the triangle containing it. The homomorphism ℤ[K/F] ⊕ ℤ[L/F] → ℤ is the difference of the augmentation maps. This sequence induces the following short exact sequence of Γ-modules:

(2)  0 → T̂ → ℤ[KL/F]/ℤ → ℤ[K/F]/ℤ ⊕ ℤ[L/F]/ℤ → 0.

where ℤ embeds into ℤ[K/F], ℤ[L/F], and ℤ[KL/F] diagonally.

In analogy with R_{K/F}^{(1)} (𝔾_{m}) := Ker(N_{K/F} : R_{K/F} (𝔾_{m}) → 𝔾_{m}), we define the following algebraic F -groups:

G_{L} := Ker(N_{KL/L} : R_{KL/F} (𝔾_{m}) → R_{L/F} (𝔾_{m}))
G_{K} := Ker(N_{KL/K} : R_{KL/F} (𝔾_{m}) → R_{K/F} (𝔾_{m}))

These groups are F -tori, dual to the Γ-modules ℤ[KL/F]/ℤ[L/F] and ℤ[KL/F]/ℤ[K/F], where ℤ[K/F] and ℤ[L/F] are diagonally embedded in ℤ[KL/F]. The embeddings of R_{K/F} (𝔾_{m}) and R_{L/F} (𝔾_{m}) into R_{KL/F} (𝔾_{m}) induce embeddings R_{K/F}^{(1)} (𝔾_{m}) → G_{L} and R_{L/F}^{(1)} (𝔾_{m}) → G_{K}. The description of T̃ ⊂ S̃ ⊂ ℙ^{2} × ℙ^{2} above descends to the following exact sequence:

(3)  1 → R_{K/F}^{(1)} (𝔾_{m}) → G_{L} → T → 1.

Similarly, the description of f(T̃) ⊂ ℙ^{1} × ℙ^{1} × ℙ^{1} above descends to

1 → R_{L/F}^{(1)} (𝔾_{m}) → G_{K} → T → 1.

We will use these sequences in Section 4.

Finally, from (1) and (2), we have corresponding sequences of F -tori:

(4)  1 → 𝔾_{m} → R_{K/F} (𝔾_{m}) × R_{L/F} (𝔾_{m}) → R_{KL/F} (𝔾_{m}) → T → 1,

and

1 → R_{K/F}^{(1)} (𝔾_{m}) × R_{L/F}^{(1)} (𝔾_{m}) ⟶^{ϕ} R_{KL/F}^{(1)} (𝔾_{m}) → T → 1.

Recall that for E = K, L, and KL,

H^{1}(F, R_{E/F}^{(1)} (𝔾_{m})) =F^{×}/N_{E/F} (E^{×})
H^{2}(F, R_{E/F}^{(1)} (𝔾_{m})) = Ker(cor_{E/F} : Br(E) → Br(F)).

Moreover, as N_{KL/F} ((KL)^{×}) is a subgroup of N_{K/F}(K^{×}) and N_{L/F} (L^{×}), it follows that the restriction of the homomorphism of H^{1} groups induced by ϕ to either factor is just factoring out the corresponding subgroup of the quotient, and thus ϕ will be surjective. Therefore, by the induced long exact sequence in cohomology, we obtain the following exact sequence:

1 → H^{1}(F, T) → Ker(cor_{K/F}) × Ker(cor_{L/F}) → Ker(cor_{KL/F}).

where the last homomorphism sends a pair (x, y) to res_{KL/K}(x)−res_{KL/L}(y) ∈ Br(KL).

Let C_{1} be the set of K-algebra isomorphism classes of Azumaya K-algebras B of rank 9 such that B_{L} = B ⊗_{K} KL and cor_{K/F} (B) are split, C_{2} the set of L-algebra isomorphism classes of Azumaya L-algebras Q of rank 4 such that Q_{K} = Q ⊗_{L} KL and cor_{L/F} (Q) are split, and set C = C_{1} × C_{2}. Then C is a pointed set with distinguished element (M_{3}(K), M_{2}(L)), and the map ψ : C → Ker(cor_{K/F}) × Ker(cor_{L/F}) sending a pair (B, Q) to ([B], [Q]) is a morphism of pointed sets. Moreover, res_{KL/K}([B]) = [B ⊗_{K} KL] and res_{KL/L}([Q]) = [Q ⊗_{L} KL] are trivial, so it follows that ψ maps into H^{1}(F, T).

Theorem 2.2. ψ : C → H^{1}(F, T) is an isomorphism of pointed sets.

Proof. If ψ(B, Q) = ψ(B′, Q′), then [B] = [B′] ∈ Ker(cor_{K/F}) ⊂ Br(K). Then B and B′ are similar Azumaya K-algebras of the same rank, and thus must be isomorphic as K-algebras. Similarly, Q and Q′ are isomorphic, so that ψ is injective.

Now let (x, y) ∈ H^{1}(F, T), so that (x, y) ∈ Ker(cor_{K/F}) × Ker(cor_{L/F}), and res_{KL/K}(x) = res_{KL/L}(y). This implies that

3x = cor_{KL/K}(res_{KL/K}(x))
= cor_{KL/K}(res_{KL/L}(y)) = res_{K/F} (cor_{L/F} (y)) = 0.

Similarly, 2y = 0. Thus res_{KL/K}(x) = res_{KL/L}(y) has order divisible by 2 and 3, and therefore is trivial. If L is not a field, then L = F × E, where E is an étale quadratic extension of F. Then res_{KL/K}(x) = (x, res_{E⊗_{F} K/K}(x)), and so res_{KL/K}(x) = 0 implies x = 0. If L is a field, then x is split by a field extension of degree 3 (If K = F × F, and x = (x_{1}, x_{2}) ∈ Br(K) = Br(F) × Br(F) is split by KL if and only if x_{1} and x_{2} are split by L). Thus for all possible K and L, there is an Azumaya K-algebra B of rank 9 that represents x in Br(K). Since res_{KL/K}(x) and cor_{K/F} (x) are trivial, B ⊗_{K} KL and cor_{K/F} (B) are split. Similarly, there is an Azumaya L-algebra Q of rank 4 which represents y such that Q⊗_{L}KL and cor_{L/F} (Q) are split. Then (B, Q) ∈ A, and ϕ(B, Q) = (x, y), so ψ is surjective. □

Remark 2.3. As KL is an étale algebra of degree 3 over K, if KL splits B, then KL can be embedded as a subalgebra of B. Similarly, KL can be embedded as a subalgebra of Q. If (B, Q) = (B′, Q′) in C = H^{1}(F, T), then any K-isomorphism from B to B′ sends KL ⊂ B to a subalgebra of B′ isomorphic to KL. Moreover, if we choose a fixed embedding of KL into both B and B′, by applying Skolem-Noether to B′ we can find an isomorphism from B to B′ which restricts to the identity on KL. Similarly, we may assume that Q to Q′ are isomorphic via an isomorphism which is the identity on KL.

It follows that if K and L are étale quadratic and cubic extensions of F respectively, and T is the two dimensional torus induced from K and L as in the exact sequence (4), then elements of H^{1}(F, T) are determined by triples (B, Q, KL), where B is an Azumaya K-algebra of rank 9 such that cor_{K/F} (B) is split, Q is an Azumaya algebra over L of rank 4 such that cor_{L/F} (Q) is split, and we have a fixed embedding of KL as a subalgebra into both B and Q. Two triples (B, Q, KL) and (B′, Q′, KL) will determine the same element of H^{1}(F, T) if there are KL-algebra isomorphisms from B to B′ and Q to Q′.

If S is a del Pezzo surface of degree 6, and if T is the connected component of the identity of Aut_{F} (S), S is a T -toric variety, with Γ-action on the fan induced by the Γ-action γ on the connected components of \overline{Z}, the hexagon of lines. The T -torsors U ⊂ S is determined by an element of the pointed set H^{1}(F, T). Two surfaces S and S′ will be isomorphic as toric varieties if and only if T and T′ are isomorphic as algebraic groups, and there is an isomorphism from S to S′ which preserves the action of T ≅ T′ on S and S′, thus inducing isomorphisms Γ-actions on the fan and isomorphisms of the T -torsors determining S and S′. Thus we have proved the following:

Theorem 2.4. Let S be a del Pezzo surface of degree 6, and T be the connected component of the identity of the group Aut_{F} (S). Then S is a T -toric variety with Γ-invariant fan determined by a pair (K, L) and the T -torsor U determined by a triple (B, Q, KL). Two triples (B, Q, KL) and (B′, Q′, K′L′) will describe isomorphic toric varieties if and only if K and L are isomorphic to K′ and L′ as F -algebras, (so that T ≅ T′), and there exist KL-algebra isomorphisms from B to B′ and Q to Q′.

3. The Main Theorem

We would now like to classify these surfaces up to isomorphism as abstract varieties. This is less restrictive than isomorphism as toric varieties. We will see that a del Pezzo surface is still determined by a triple (B, Q, KL), but now we will allow F -algebra isomorphisms on B and Q, i.e. algebra isomorphisms which may not fix KL.

Let S be a del Pezzo surface of degree 6 over F. Then S is a T -toric variety, where T is the connected component of Aut_{F} (S), with the Γ-action on the fan determining a pair (K, L), and the T -torsor U ⊂ S determining a pair (B, Q, KL). Let G be the group of connected components of Aut_{F} (S). As we have shown above, G is an étale group scheme, determined by the action of Γ on the hexagon of lines. In particular, G(F) = Aut_{F} (KL) ≅ Aut_{F} (K)×Aut_{F} (L).

Consider the following action of G(F) on H^{1}(F, T): if (g, h) ∈ Aut_{L}(KL) × Aut_{K}(KL) and (B, Q, KL) ∈ H^{1}(F, T) then g nontrivial sends B to B^{op}, and sends the embedding i : KL → B to i^{g} : KL → B^{op}, where i^{g}(z) = i(g(z))^{op} ∈ B^{op}. As KL is a cyclic extension of L, Q is a cyclic L-algebra, so there is an element l ∈ L^{×} such that Q is generated by KL and an element y, subject to the relations y^{2} = l and zy = yσ(z) for every z ∈ KL, where σ ∈ Aut_{L}(KL) is the nontrivial automorphism of KL over L. Such an algebra is denoted (KL/L, l). Let h act on Q = (KL/L, L) by h · (KL/L, l) = (KL/L, h(l)), and send the embedding KL → Q to KL ⟶^{h} KL → (KL, h(l)). As l ∈ L^{×} determines Q = (KL/L, l) up to multiplication by an element of the subgroup N_{KL/L}((KL)^{×}) ⊂ L^{×}, and as Aut_{F} (L) takes N_{KL/L}((KL)^{×}) to itself, we see that this action is well defined.

The orbits of this group action can be described in terms of F -algebra isomorphisms on B and Q, as we will show below. We will use the following proposition several times: We will need the following proposition:

Proposition 3.1 (Proposition 4.18 of [3]). Let (B, τ) be a central simple F - algebra of degree n with unitary involution, and let K be the center of B. For every F -subalgebra L of B which is étale of dimension n over F, there exists a unitary involution of B fixing L.

Proposition 3.2. (B′, Q′) = g · (B, Q) for some g ∈ G(F) if and only if there are F -automorphisms ϕ_{B} : B → B′ and ϕ_{Q} : Q → Q′ such that ϕ_{B}|_{KL} = ϕ_{Q}|_{KL} = g.

Proof. Assume that (B′, Q′, KL) = g · (B, Q, KL) for some g ∈ G(F). Any F -automorphism g of KL can be expressed as the composition of two automorphisms, one fixing K and one fixing L. So it suffices to consider the separate cases where K and L are fixed by the automorphism.

We first consider the case where g fixes K, so that B′ = B. By Skolem-Noether, there is a K-automorphism ϕ_{B} of B such that ϕ_{B}|_{KL} = g. Now, Q = (KL/L, l), Q′ = (KL/L, g(l)) and as σ and g commute, g extends to an F -automorphism ϕ_{Q} from Q to Q′, by sending KL to KL via g, and y to y′. Then ϕ_{B} and ϕ_{Q} agree on KL.

Now assume that g = σ is the non-trivial L-automorphism of KL, so that Q′ = Q and B′ = B^{op}. As in the previous paragraph, by Skolem-Noether there is an L-automorphism ϕ_{Q} of Q such that ϕ_{Q}|_{KL} = g. Moreover, as cor_{K/F} (B) is split and L is an étale cubic extension of F, we know by Proposition 3.1 that B has a unitary involution τ which is the identity on L. The involution τ defines an F -isomorphism ϕ_{B} from B to B^{op}, such that ϕ_{B}|_{KL} = σ = g = ϕ_{Q}|_{KL}.

Conversely, assume that ϕ_{B} : B → B′ and ϕ_{Q} : Q → Q′ are F -isomorphisms such that ϕ_{B}|_{KL} = ϕ_{Q}|_{KL} = g ∈ Aut_{F} (KL). As in the arguments above, we will first consider the separate cases where g fix K and L.

If g fixes L, then ϕ_{Q} is an isomorphism of L-algebras, so that (B′, Q, KL) = (B′, Q′, KL) in H^{1}(F, T). If we restrict ϕ_{B} to the center of B, we get an F - isomorphism of K. If ϕ|_{K} is the identity (i.e. g is trivial), then B and B′ are isomorphic as K-algebras. If ϕ|_{K} is not the identity, then by pre-composing ϕ with the isomorphism from B^{op} to B induced by any unitary involution τ fixing L on B, we see that B^{op} and B′ are isomorphic as K-algebras. In either case, (B′, Q′, KL) = (B′, Q, KL) = g · (B, Q, KL) in H^{1}(F, T).

Now assume that g fixes K, so that ϕ_{B} : B → B′ is an isomorphism of Kalgebras, and then (B′, Q′, KL) = (B, Q′, KL) in H^{1}(F, T). If Q = (KL/L, l), and if l′ = ϕ_{Q}(l) = g(l), then Q is isomorphic over L to (KL/L, l′), and so (B′, Q′, KL) = (B, Q′, KL) = g · (B, Q) in H^{1}(F, T).

Finally, assume that g does not fix K or L. If σ is the nontrivial Lautomorphism of KL, σg does fix K. Moreover, by post-composing ϕ_{B} with the F -isomorphism ϕ_{B′} : B′ → (B′)^{op} induced by any unitary involution τ of B′ fixing L (which exist by Proposition 3.1), we get isomorphisms ϕ_{B′} ∘ ϕ_{B} : B → (B′)^{op} and ϕ_{Q} : Q → Q′ such that (ϕ_{B′} ∘ ϕ_{B})|_{KL} = ϕ_{Q}|_{KL} = σg. Therefore by our argument in the previous paragraph, σ · (B′, Q′, KL) = ((B′)^{op}, Q′, KL) = σg · (B, Q, KL) in H^{1}(F, T). Acting on both sides of this equation by σ, we get (B′, Q′, KL) = g · (B, Q, KL). □

The next theorem relates isomorphism classes of del Pezzo Surfaces of degree 6 with G(F)-orbits of H^{1}(F, T). We will need the following standard result from Galois cohomology: (cf. Corollary (28.10) of [3] or Chapter I, Section 5.5, Corollary 2 of [9].)

Proposition 3.3. Let Γ be a profinite group, A and B be Γ-groups with A a normal subgroup of B, and set C = B/A. If β ∈ H^{1}(Γ, B), and b a cocycle representing β, then the elements of H^{1}(Γ, B) with the same image as β in H^{1}(Γ, C) corresponding bijectively with (C_{b})^{Γ}-orbits of the set H^{1}(Γ, A_{b}).

Theorem 3.4. The isomorphism class of S corresponds to a G(F)-orbit of H^{1}(F, T).

Proof. As mentioned in Section 2, we have the following split exact sequence of algebraic groups:

(5)  1 → T̃ → Aut_{F} (S̃) → S_{2} × S_{3} → 1,

where T̃ is the connected component of the identity of Aut_{F} (S̃), and S_{2} × S_{3} is the group of connected components. This sequence induces the split exact sequence of pointed sets:

1 → H^{1}(F, T̃) → H^{1}(F, Aut_{F} (S̃)) → H^{1}(F, S_{2} × S_{3}) → 1.

The elements of H^{1}(F, Aut_{F} (S̃)) are in a one-to-one correspondence with the set of isomorphisms classes of F -forms of S̃, which by Proposition 2.1 are del Pezzo surfaces of degree 6. Let β ∈ Z^{1}(F, Aut_{F} (S̃)) be a cocycle whose cohomology class is determined by the isomorphism class of S, and let γ be the image of β in Z^{1}(F, S_{2} × S_{3}). The cocycle γ is determined by the action of the Galois group Γ on \overline{Z} ⊂ \overline{S}, and induces a pair (K, L). The twist of T̃ by γ is the torus T, determined by (K, L) as in the sequence (4), and the twist of S_{2} ×S_{3} is the étale group scheme G. The result follows by Proposition 3.3. □

Note that as H^{1}(F, Aut_{F} (S̃)) → H^{1}(F, S_{2} × S_{3}) is surjective, we see that all possible pairs (K, L) are realized by the action of Γ on \overline{Z}, for Z contained in some del Pezzo surface S of degree 6. So let S_{1} and S_{2} be two del Pezzo surfaces of degree 6, and let (B_{i}, Q_{i}, K_{i}L_{i}) be an element of the the G_{i}(F)-orbit of H^{1}(F, T_{i}) determined by S_{i}, for i = 1, 2. Then S_{1} and S_{2} induce isomorphic Γ-actions on the hexagon of lines (so that (K_{1}, L_{1}) ≅ (K_{2}, L_{2}), T_{1} ≅ T_{2}, and G_{1} ≅ G_{2}; we denote these algebraic objects (K, L), T, and G, respectively). It follows from Proposition 3.2 that two pairs (B_{1}, Q_{1}, KL) and (B_{2}, Q_{2}, KL) are in the same G(F)-orbit of H^{1}(F, T) if and only if there are isomorphisms ϕ_{B_{1}} : B_{1} → B_{2} and ϕ_{Q_{1}} : Q_{1} → Q_{2} such that ϕ_{B_{1}}|_{KL} = ϕ_{Q_{1}}|_{KL}. We have proved the following theorem:

Theorem 3.5. There are bijections, inverse to each other, between the following two sets:

• The set of isomorphism classes of del Pezzo surfaces of degree 6.

• The set of triples (B, Q, KL), modulo the relation: (B, Q, KL) ∼ (B′, Q′, K′L′) if there are F -algebra isomorphisms ϕ_{B} : B → B′ and ϕ_{Q} : Q → Q′ such that ϕ_{B}|_{KL} = ϕ_{Q}|_{KL}.

For the rest of this paper, S(B, Q, KL) will denote the del Pezzo surface of degree 6 determined by the triple (B, Q, KL).

Corollary 3.6. The surface S(B, Q, KL) contains a rational point if and only if B and Q are split.

Proof. Since S(B, Q, KL) is a T -toric variety for a two dimensional torus T, S has a rational point if and only if the corresponding T -torsor U is a trivial torsor (cf. Proposition 4 of [14]). By Theorem 2.4, this occurs precisely when B and Q are split. □

Remark 3.7. If S_{0} is a T -toric model (i.e. the T -torsor U ⊂ S_{0} is trivial), then the map H^{1}(F, T) → H^{1}(F, Aut_{F} (S)) induced by T ↪ Aut_{F} (S) takes a T - torsor U to the surface determined by U and the Γ-action on the fan determined by the pair (K, L). Thus for any surface S, the elements of H^{1}(F, T) in the fiber of the isomorphism class of S determine the possible non-isomorphic T - toric structures on S, where T is the connected component of the identity of the algebraic group Aut_{F} (S). In terms of the algebras B and Q, the map H^{1}(F, T) → H^{1}(F, Aut_{F} (S)) forgets the KL-algebra structure of B and Q, preserving only the F -algebra structure and the embedding of KL into B and Q.

The group Aut_{F} (K) always has order 2, but the group Aut_{F} (L) can have order 1, 2, 3, or 6. If Aut_{F} (L) has order less than 6, then the orbit of (B, Q, KL) in H^{1}(F, T) contains at most 6 elements. It Aut_{F} (L) has order 6, then L = F^{3} is not a field, and thus B is necessarily split. If B is split, then the pair (B, Q, KL) ∈ H^{1}(F, T) is fixed by the subgroup Aut_{F} (K) of G(F), and so again the G(F)-orbit of (B, Q, KL) in H^{1}(F, T) has at most 6 elements. Thus for a del Pezzo surface S of degree 6, there at most 6 non-isomorphic T -toric structures on S.

Remark 3.8. We would like to relate this characterization of del Pezzo surfaces of degree 6 by triples (B, Q, KL) with the characterization by triples (B, τ, L) found in [1]. A triple (B, τ, L) is an Azumaya K-algebra B of rank 9, a unitary involution τ on B, and a cubic étale F -algebra L such that L ⊂ Sym(B, τ). Two triples (B, τ, L) and (B′, τ′, L′) are isomorphic if there is an F -algebra isomorphism ϕ : B → B′ such that τ′ϕ = ϕτ and ϕ(L) = L′.

So let (B, Q, KL) be a triple as in Theorem 3.5. Then B is an Azumaya K-algebra of rank 9, and is classified up to isomorphism as an F -algebra. As B_{L} is split, so B contains KL, and hence L, as a subalgebra. Since cor_{K/F} (B) is split, we know that B has a unitary involution. Moreover, since L is an étale cubic extension of F contained in B, there is some unitary involution τ such that L ⊂ Sym(B, τ), by Proposition 3.1. So the B and L in our characterization match with the B and L described in [1]. The rest of the remark seeks to relate Q and the involution τ. That is, we want to classify all triples (B, τ, L) with B and L fixed. This should correspond to fixing K, L, and B, and trying to determine all possible Q.

As K/F and KL/L are cyclic,

Br(K/F) ≅ F^{×}/ N_{K/F}(K^{×})

and

Br(KL/L) ≅ L^{×}/ N_{KL/L}((KL)^{×}) .

The restriction homomorphism res_{L/F} : Br(F) → Br(L) sends the subgroup Br(K/F) to Br(KL/L). As K and L have coprime degrees, res_{L/F} |_{Br(K/F)} : Br(K/F) → Br(KL/L) is injective, and the cyclic algebra Q corresponds to an element of Br(KL/L)/ res_{L/F} (Br(K/F)), i.e. an element of

L^{×}/ N_{KL/L}((KL)^{×}) / res_{L/F} (F^{×}/ N_{K/F}(K^{×}) )
= L^{×}/ N_{KL/L}((KL)^{×}) /F^{×} N_{KL/L}((KL)^{×}) / N_{KL/L}((KL)^{×})
≅ L^{×}/F^{×} N_{KL/L}((KL)^{×}) .

If τ is a unitary involution on B which is the identity on L, and u ∈ L^{×}, then τ_{u} := Int(u) ∘ τ is also a unitary involution on B fixing L. Moreover, τ_{u} is conjugate to τ_{v} if uv^{−1} ∈ F^{×} N_{KL/L}((KL)^{×}). Thus, after a choice of a particular involution τ, we have a morphism of pointed sets from L^{×}/F^{×} N_{KL/L}((KL)^{×}) to the set of conjugacy classes of unitary involutions of B which are the identity on L, sending u to τ_{u}. By Corollary 19.3 of [3], this map is a surjection. So we have a surjective map from L^{×}/F^{×} N_{KL/L}((KL)^{×}) to the set of isomorphism classes of triples (B, τ, L).

Theorem 3.5 should say that the fibers of this surjection should be the orbit of the group Aut_{F} (L) in L^{×}/F^{×} N_{KL/L}((KL)^{×}), corresponding to the Aut_{F} (L)- orbit of (B, Q). However, to make this statement correct, we need to choose a particular involution τ on B. It is not clear in general what this involution should be. The involution τ should be chosen so that the surface S(B, τ, L) should correspond to the pair (B, M_{2}(L)). In particular, if B = M_{3}(K) is split, the surface described by the triple (M_{3}(K), τ, L) should have a rational point, by Corollary 3.6. The next remark constructs the involution in this case.

Remark 3.9. Given K and L, we will find a triple (B, τ, L), (i.e. a central simple algebra of degree 3 over K with an involution τ such that L ⊂ Sym(B, τ),) so that the corresponding del Pezzo surface S(B, τ, L) constructed in [1] has a rational point.

As KL is a three dimensional vector space over K, B := End_{K}(KL) is a Azumaya K-algebra of rank 9. Left multiplication by an element of KL determines an embedding of KL into B. If σ is the nontrivial L-automorphism of KL, h(x, y) = Tr_{KL/K}(σ(x)y) defines a hermitian form on KL. This hermitian form on KL induces an involution of the second kind τ on B, such that L ⊂ Sym(B, τ). So we have a triple (B, τ, L). Let S denote the corresponding del Pezzo surface of 6, constructed in [1].

I claim that S contains a rational point. According to [1], it suffices to show that there is a right ideal I of B of reduced dimension 1 such that (I · τ(I)) ∩ Sym(B, τ) ⊂ F ⊕ L^{⊥}, where L^{⊥} = {x ∈ Sym(B, τ)| Trd(lx) = 0, for all l ∈ L}.

Let W = span_{K}(1) ⊂ LK, so that W is a one dimensional K-subspace of KL. and then I = Hom_{K}(KL, W) is a right ideal of B of reduced dimension 1, generated by the linear map t = Tr_{KL/K} : KL → K ↪ KL. We want to show that t ∈ Sym(B, τ). First, note that for any x ∈ KL, Tr_{KL/K}(σ(x)) = σ(Tr_{LK/L}(x)). If x, y ∈ KL,

h(x, t(y)) = Tr_{KL/K}(σ(x) Tr_{KL/K}(y))
= Tr_{KL/K}(σ(x)) Tr_{KL/K}(y)
= Tr_{KL/K}(σ(Tr_{KL/K}(x))y)
= h(t(x), y).

So t ∈ Sym(B, τ), which implies that τ(I) is a left ideal of B, also generated by t. Thus, I · τ(I) = tBt.

In order to prove I · τ(I) ∩ Sym(B, τ) ⊂ F ⊕ L^{⊥}, it suffices to consider the case where L = F^{3} is split. So we can choose a basis e_{1}, e_{2}, e_{3} of idempotents for KL over K. In this basis, τ is the standard adjoint involution, Sym(B, τ) is the set of hermitian matrices, and F ⊕ L^{⊥} is the set of hermitian matrices where the diagonal entries agree. Moreover, t is the matrix with ones in every entry, so t ∈ F ⊕L^{⊥}. A direct calculation shows I ·τ(I) = span_{K}(t), and hence (I · τ(I)) ∩ Sym(B, τ) = span_{F} (t) ⊂ F ⊕ L^{⊥}.

4. K_{0} of del Pezzo Surfaces

Let S = S(B, Q, KL) be a del Pezzo surface of degree 6, a T -toric variety for a two dimensional torus T. Let Z ⊂ S be the closed variety such that \overline{Z} is the union of six lines l_{0}, l_{1}, l_{2}, m_{0}, m_{1}, and m_{2}. Recall the exact sequence (2) from Section 2, where ℤ[KL/F] is the lattice of connected components of \overline{Z} ⊂ \overline{S}, the lines l_{i} and m_{i}, and the homomorphism ℤ[KL/F] → Pic(\overline{S}) sends each line to the corresponding invertible sheaf on \overline{S}. From the exact sequence (1), we see that T̂ is the subgroup of ℤ[KL/F] generated by l_{0}−l_{1}−(m_{0}−m_{1}), l_{0} − l_{2} − (m_{0} − m_{2}), and l_{1} − l_{2} − (m_{1} − m_{2}). Note that any one of these 3 generators can be expressed as a linear combination of the other 2. So Pic(\overline{S}) is generated by the invertible sheaves L(−l_{i}), L(−m_{j}), and we have that the invertible sheaves L(−l_{i} −m_{j}) and L(−l_{j} −m_{i}) are isomorphic for i, j = 0, 1, 2.

There is another way to recover these generators and relations, which does not depend on the theory of toric varieties. Recall that there is a morphism p_{1} : \overline{S} → ℙ^{2}, obtained by blowing down the lines m_{0}, m_{1}, and m_{2}. If x_{0}, x_{1}, x_{2} are the homogeneous coordinates of ℙ^{2} and D_{i} = {x_{i} = 0} for i = 0, 1, 2, then D_{0}, D_{1}, and D_{2} are all linearly equivalent divisors on ℙ^{2}, and thus their strict transforms m_{1}+l_{0}+m_{2}, m_{0}+l_{1}+m_{2}, and m_{0}+l_{2}+m_{1} are all linearly equivalent divisors on \overline{S}. Therefore the corresponding invertible sheaves L(−m_{1}−l_{0}−m_{2}), L(−m_{0} − l_{1} − m_{2}), and L(−m_{0} − l_{2} − m_{1}) on \overline{S} are isomorphic. From this we can conclude that Pic(\overline{S}) is generated by the invertible sheaves L(−m_{0}), L(−m_{1}), L(−m_{2}), L(−m_{1}−l_{0}−m_{2}), L(−m_{0}−l_{1}−m_{2}), and L(−m_{0}−l_{2}−m_{1}), and we have that L(−l_{i} − m_{j} − l_{k}) and L(−l_{j} − m_{i} − l_{k}) are isomorphic for any i, j = 0, 1, 2. This presentation is equivalent to that in the previous paragraph. Similarly, this presentation can be obtained by considering the morphism p_{2} : \overline{S} → ℙ^{2} obtained by blowing down the lines l_{0} l_{1}, and l_{2}.

We define the following locally free sheaves on \overline{S}:

I_{1} = L(−m_{1} − l_{0} − m_{2}) ⊕ L(−m_{0} − l_{1} − m_{2}) ⊕ L(−m_{0} − l_{2} − m_{1})
I_{2} = L(−l_{1} − m_{0} − l_{2}) ⊕ L(−l_{0} − m_{1} − l_{2}) ⊕ L(−l_{0} − m_{2} − l_{1})
J_{1} = L(−l_{0} − m_{1}) ⊕ L(−l_{1} − m_{0})
J_{2} = L(−l_{0} − m_{2}) ⊕ L(−l_{2} − m_{0})
J_{3} = L(−l_{1} − m_{2}) ⊕ L(−l_{2} − m_{1}).

The Γ-action on the hexagon of lines induces an action on the locally free sheaves I_{1} ⊕ I_{2} and J_{1} ⊕ J_{2} ⊕ J_{3}, compatible with the action on \overline{S}. Therefore I_{1} ⊕ I_{2} and J_{1} ⊕ J_{2} ⊕ J_{3} descend to sheaves I and J on S.

We will consider the following endomorphism rings: B′ = End_{O_{S}}(I)^{op}, and Q′ = End_{O_{S}}(I)^{op}. As S is projective, End_{O_{S}}(O_{S})^{op} = F, and since I and J are O_{S}-modules, it follows that B′ and Q′ are F -algebras. For i, j, and k not equal, End_{O\overline{S}}(L(−m_{i} − l_{j} − m_{k})) = End_{O\overline{S}}(L(−l_{i} − m_{j} − l_{k})) = \overline{F}, so we see that \overline{F}^{6} embeds diagonally into End_{O\overline{S}}(I_{1} ⊕ I_{2}). Moreover, since L(−l_{i} − m_{j}) and L(−l_{j} − m_{i}) are isomorphic for any i, j, I_{1} ⊕ I_{2} = (L(−m_{1} − l_{0} − m_{2}) ⊕ L(−l_{1} − m_{0} − l_{2})) ⊗\overline{F} V, where V is an \overline{F} -vector space of dimension 3. An element of Hom_{O\overline{S}}(L(−m_{1} − l_{0} − m_{2}), L(−l_{1} − m_{0} − l_{2})) is given by a global section of L(m_{2} − l_{2}). Any non-zero global section of L(m_{2} − l_{2}) would give a function defined on a neighborhood of l_{2} ⊂ \overline{S} with vanishing set l_{2}. Blowing down the lines l_{i}, this function would then correspond to a function defined on an open subset of ℙ^{2} with vanishing set a point, which is impossible, since a point is a codimension 2 subvariety of ℙ^{2}. Thus L(m_{2} − l_{2}) has no nonzero global sections. Similarly Hom_{O\overline{S}}(L(−l_{1}−m_{0}−l_{2}), L(−m_{1}−l_{0}−m_{2})) = 0, and so End_{O\overline{S}}(I_{1}⊕I_{2}) = End_{O\overline{S}}(L(−m_{i}−l_{j} −m_{k})×L(−l_{i}−m_{j} −l_{k}))⊗\overline{F} End\overline{F} (V) = End\overline{F}^{2}(V\overline{F}^{2}). So the center of End_{O\overline{S}}(I_{1} ⊕ I_{2}) is a copy of \overline{F}^{2}, contained in \overline{F}^{6}. This chain \overline{F}^{2} ⊂ \overline{F}^{6} ⊂ End_{O\overline{S}}(I_{1} ⊕ I_{2}) descends to K ⊂ KL ⊂ End_{O_{S}}(I), with K the center of End_{O_{S}}(I).

Now let E be any separable field extension of F over which the lines l_{i}, m_{j} are defined. This is equivalent to E splitting both K and L. the above arguments show that End_{O_{S}}(I) ⊗_{F} E ≈ M_{3}(E^{2}), where E^{2} ≈ K ⊗_{F} E. Therefore, we conclude that B′ is an Azumaya K-algebra of rank 9 which contains KL as a subalgebra. A similar argument shows that Q′ is an Azumaya L-algebra of rank 4 which also contains a copy of KL.

Theorem 4.1. B′ = End_{O_{S}}(I)^{op} and B are isomorphic as K-algebras. Similarly, Q′ = End_{O_{S}}(J)^{op} and Q are isomorphic as L-algebras.

Proof. Let S̃ be as in Proposition 2.1, and let Ĩ and J̃ be the sheaves associated to S̃ as above. Twisting Ĩ and J̃ by the Γ-action corresponding to the pair (K, L), we get sheaves I_{0} and J_{0} associated to the T -toric model S_{0}. Let B′_{0} = End_{O_{S}}(I_{0})^{op} and Q′_{0} = End_{O_{S}}(J_{0})^{op}. The embeddings of KL into B′_{0} and Q′_{0} described above induce the following commutative diagrams (cf. (3)):

\xymatrix{1 \ar[r] & R_{K/F}^{(1)} (𝔾_{m}) \ar[r] \ar[d] & G_{L} \ar[r] \ar[d] & T \ar[r] \ar[d] & 1 \\ 1 \ar[r] & R_{K/F} (𝔾_{m}) \ar[r] & R_{K/F} (GL(B′_{0})) \ar[r] & R_{K/F} (PGL(B′_{0})) \ar[r] & 1,}

and

\xymatrix{1 \ar[r] & R_{L/F}^{(1)} (𝔾_{m}) \ar[r] \ar[d] & G_{K} \ar[r] \ar[d] & T \ar[r] \ar[d] & 1 \\ 1 \ar[r] & R_{L/F} (𝔾_{m}) \ar[r] & R_{L/F} (GL(Q′_{0})) \ar[r] & R_{L/F} (PGL(Q′_{0})) \ar[r] & 1.}

The first diagram induces the following commutative diagram of cohomology sets:

\xymatrix{ & H^{1}(F, T) \ar[r] \ar[d] & Ker(cor_{K/F} : Br(K) → Br(F)) \ar[d] \\ 1 \ar[r] & H^{1}(K, PGL(B′_{0})) \ar[r] & Br(K).}

The left vertical arrow sends the triple (B, Q, KL) to the endomorphism ring B′ = End(I)^{op}, where I is the sheaf associated to the surface S(B, Q, KL) constructed above. The upper horizontal arrow sends the triple (B, Q, KL) to the class of [B] ∈ Br(K), which lands in the subgroup of elements of trivial norm. The right vertical map is the inclusion homomorphism. The lower horizontal arrow sends a K-algebra to its corresponding element in Br(K). By the commutativity of the diagram, we see that [B] = [B′] in Br(K). But these algebras have the same rank, so they must be isomorphic as K-algebras. A similar argument shows that Q′ = End_{O_{S}}(J)^{op} and Q are isomorphic as L-algebras. □

Let A be the separable F -algebra F ×B ×Q, so that P(A) = P(F)×P(B)× P(Q). Define the exact functors u_{F} from P(F) to P(S) by M_{1} ↦ O_{S} ⊗_{F} M_{1}, u_{B} from P(B) to P(S) by M_{2} ↦ I ⊗_{B} M_{2}, and u_{Q} from P(Q) to P(S) by M_{3} ↦ J ⊗_{Q} M_{3}. If we set P = O_{S} ⊕I ⊕J, then the respective right actions of F, B, and Q on O_{S}, I, and J combine to give a right action of A = F ×B ×Q on P. Therefore, we can define an exact functor from P(A) to P(S) by sending M to P ⊗_{A} M. This exact functor induces a homomorphism:

ϕ : K_{0}(A) → K_{0}(S).

More generally, if Y is any F -variety, then we have an exact functor from P(Y; A) to P(Y × S), sending M to p_{2}^{∗}(P) ⊗_{O_{Y ×S}⊗_{F} A} p_{1}^{∗}(M), where p_{1} : Y × S → Y and p_{2} : Y × S → S are the projection morphisms. This induces a homomorphism ϕ_{Y} : K_{0}(Y; A) → K_{0}(Y × S). Furthermore, if E is any field extension of F, then ϕ naturally extends to a homomorphism ϕ_{E} : K_{0}(A_{E}) → K_{0}(S_{E}).

Theorem 4.2. ϕ : K_{0}(A) → K_{0}(S) is an isomorphism.

We will prove this in several stages. Let us first consider the case where F is separably closed. By Proposition 2.1, S is isomorphic to the blow up of the projective plane at the 3 non-collinear points [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1]. Recall that we have the filtration 0 = K_{0}(S)^{(3)} ⊂ K_{0}(S)^{(2)} ⊂ K_{0}(S)^{(1)} ⊂ K_{0}(S)^{(0)} = K_{0}(S) by codimension of support, and homomorphisms CH^{i}(S) → K_{0}(S)^{(i/i+1)}, which send the class of a subvariety V to the equivalence class [O_{V}]. These homomorphisms are isomorphisms for i = 0, 1, 2. So by Proposition 2.1, if P is a rational point of S, K_{0}(S) is generated by [O_{S}], [O_{l_{0}}], [O_{l_{1}}], [O_{l_{2}}], [O_{m_{0}}], [O_{m_{1}}], [O_{m_{2}}], and [O_{P}]. Moreover, as CH^{0}(S), CH^{1}(S), and CH^{2}(S) are free abelian groups with ranks 1, 4, and 1, respectively, K_{0}(S) is free abelian with rank 6. Since F is separably closed, K, L, B, and Q are split, and thus K_{0}(A) is also free abelian of rank 6. Therefore ϕ will be an isomorphism provided it is surjective. So it suffices to show that [O_{S}], [O_{l_{0}}], [O_{l_{1}}], [O_{l_{2}}], [O_{m_{0}}], [O_{m_{1}}], [O_{m_{2}}], and [O_{P}] are in the image of ϕ.

Clearly, [O_{S}] = [O_{S} ⊗_{F} F] is in the image of ϕ. As F is separably closed, I = (L(−m_{1}−l_{0}−m_{2})⊕L(−l_{1}−m_{0}−l_{2}))⊗_{F} V, where V is an F -vector space of dimension 3, K = F ×F ≅ End_{O_{S}}(L(−m_{1}−l_{0}−m_{2}))×End_{O_{S}}(L(−l_{1}−m_{0}−l_{2})), and End_{O_{S}}(I)^{op} ≅ End_{F^{2}}(V_{F^{2}}).

Now Hom_{F^{2}}(V_{F^{2}}, F × 0) is a right End_{O_{S}}(I)-module, and thus a left Amodule, where the F and Q component of A = F × B × Q act trivially. Therefore,

ϕ( Hom_{F^{2}}(V_{F^{2}}, F × 0)) = [I ⊗_{B} Hom_{F^{2}}(V_{F^{2}}, F × 0)]
= [(L(−m_{1} − l_{0} − m_{2}) ⊕ L(−l_{1} − m_{0} − l_{2}))

⊗_{F^{2}} (F × 0)]
= [L(−m_{1} − l_{0} − m_{2})],

where we use Morita equivalence in the second line. A mirror argument shows that ϕ( Hom_{F^{2}}(V_{F^{2}}, 0 × F)) = [L(−l_{1} − m_{0} − l_{2})], and a similar argument applied to J and Q shows that [L(−l_{0}−m_{1})], [L(−l_{0}−m_{2})], and [L(−l_{1}−m_{2})] are in the image of ϕ.

Now let i, j ∈ {0, 1, 2} and not equal. By Proposition 2.1, the lines l_{i} and m_{j} have intersection a rational point P of S, with multiplicity 1, the lines m_{i} and m_{j} are skew, and the lines l_{i} and l_{j} are skew. Thus we have the following resolutions of O_{P} and O_{S}:

0 → L(−l_{i}−m_{j}) \underrightarrow{(\begin{matrix}⊗L(m_{j}) \\ ⊗L(l_{i}) \end{matrix})} L(−l_{i})⊕L(−m_{j}) ⟶^{(⊗L(l_{i}),−⊗L(m_{j}))} O_{S} → O_{P} → 0,

0 → L(−m_{i} − m_{j}) \underrightarrow{(\begin{matrix}⊗L(m_{i}) \\ ⊗L(m_{j}) \end{matrix})} L(−m_{i}) ⊕ L(−m_{j}) ⟶^{(⊗L(m_{i}),−⊗L(m_{j}))} O_{S} → 0,

and

0 → L(−l_{i} − l_{j}) \underrightarrow{(\begin{matrix}⊗L(l_{i}) \\ ⊗L(l_{j}) \end{matrix})} L(−l_{i}) ⊕ L(−l_{j}) ⟶^{(⊗L(l_{i}),−⊗L(l_{j}))} O_{S} → 0.

In addition, when D = l_{i} or m_{i}, we have the standard resolution

0 → L(−D) ⟶^{⊗L(D)} O_{S} → O_{D} → 0.

So 0 = [O_{S}] − [L(−m_{i})] − [L(−m_{j})] + [L(−m_{i} − m_{j})] in K_{0}(S). If we take k ∈ {0, 1, 2} not equal to i or j and multiply this equation by [L(−l_{k})], we see that

0 = [L(−l_{k})] − [L(−l_{k} − m_{i})] − [L(−l_{k} − m_{j})] + [L(−m_{i} − l_{k} − m_{j})].

Therefore,

[O_{l_{k}}] = [O_{S}] − [L(−l_{k})]
= [O_{S}] − [L(−l_{k} − m_{i})] − [L(−l_{k} − m_{j})] + [L(−m_{i} − l_{j} − m_{k})]

is in the image of ϕ. The same argument with l and m interchanged shows that [O_{m_{k}}] is in the image of ϕ for k ∈ {0, 1, 2}. Finally, [O_{P}] = [O_{S}] − [L(−l_{0})] − [L(−m_{1})] + [L(−l_{0} − m_{1})] is in the image of ϕ, and thus ϕ is surjective when F is separably closed.

Proposition 4.3. ϕ : K_{0}(A) → K_{0}(S) is an isomorphism if B and Q are split.

Proof. By the preceding argument, ϕ\overline{F} : K_{0}(\overline{A}) → K_{0}(\overline{S}) is an isomorphism. Moreover, ϕ\overline{F} commutes with the action of Γ on both K_{0}(\overline{A}) and K_{0}(\overline{S}), and thus it descends to an isomorphism on the Γ-invariant subgroups. Therefore, we have the following commutative diagram:

\xymatrix{K_{0}(A) \ar[d] \ar[r]^{ϕ} & K_{0}(S) \ar[d] \\ K_{0}(\overline{A})^{Γ} \ar[r]^{ϕ\overline{F}} & K_{0}(\overline{S})^{Γ}.}

As ϕ\overline{F} and the left vertical map K_{0}(A) → K_{0}(\overline{A})^{Γ} are isomorphisms, ϕ must be injective. Moreover, if the right vertical map is injective, then ϕ is surjective, and hence an isomorphism. So it suffices to show that K_{0}(S) → K_{0}(\overline{S})^{Γ} is injective.

To see this, note that the rank and wedge homomorphisms rank : K_{0}(\overline{S}) → ℤ and ∧ : K_{0}(\overline{S})^{(1)} → Pic(\overline{S}) commute with the action of Γ. Thus we have the following short exact sequences of Γ-modules:

0 → K_{0}(\overline{S})^{(2)} → K_{0}(\overline{S})^{(1)} ⟶^{∧} Pic(\overline{S}) → 0

and

0 → K_{0}(\overline{S})^{(1)} → K_{0}(\overline{S}) ⟶^{rank} ℤ → 0.

These sequences of Γ-modules induce the following long exact sequences:

0 → (K_{0}(\overline{S})^{(2)})^{Γ} → (K_{0}(\overline{S})^{(1)})^{Γ} ⟶^{∧} Pic(\overline{S})^{Γ} → H^{1}(F, K_{0}(\overline{S})^{(2)})

and

0 → (K_{0}(\overline{S})^{(1)})^{Γ} → (K_{0}(\overline{S}))^{Γ} ⟶^{rank} ℤ → H^{1}(F, K_{0}(\overline{S})^{(1)}).

The map K_{0}(S) → K_{0}(\overline{S})^{Γ} induces the following commutative diagrams:

\xymatrix{0 \ar[r] & K_{0}(S)^{(2)} \ar[r] \ar[d] & K_{0}(S)^{(1)} \ar[r]^{∧} \ar[d] & Pic(S) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & (K_{0}(\overline{S})^{(2)})^{Γ} \ar[r] & (K_{0}(\overline{S})^{(1)})^{Γ} \ar[r]^{∧} & Pic(\overline{S})^{Γ} \ar[r] & H^{1}(F, K_{0}(\overline{S})^{(2)}),}

and

\xymatrix{0 \ar[r] & K_{0}(S)^{(1)} \ar[r] \ar[d] & K_{0}(S) \ar[r]^{rank} \ar[d] & ℤ \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & (K_{0}(\overline{S})^{(1)})^{Γ} \ar[r] & K_{0}(\overline{S})^{Γ} \ar[r]^{rank} & ℤ \ar[r] & H^{1}(F, K_{0}(\overline{S})^{(1)}).}

As B and Q are split, S has a rational point by Corollary 3.6. Thus the homomorphism K_{0}(S)^{2} → (K_{0}(\overline{S})^{(2)})^{Γ} is a surjective homomorphism of free abelian groups of rank 1, and therefore an isomorphism. Moreover, the homomorphism Pic(S) → Pic(\overline{S})^{Γ} is injective. Thus, by applying the Snake Lemma to the first diagram and then to the second, we see that K_{0}(S) → (K_{0}(\overline{S}))^{Γ} is injective. □

Remark 4.4. If P and P′ are rational points of \overline{S}, they define equal classes in CH^{2}(\overline{S}) by Proposition 2.1, and hence [O_{P}] = [O_{P′}] in K_{0}(\overline{S})^{(2)}. So K_{0}(\overline{S})^{(2)} is generated by the Γ-invariant element [O_{P}], and thus is a trivial Γ-module. Similarly, our arguments above show that the set {[O_{l_{0}} ⊕ O_{m_{1}}], [O_{l_{0}} ⊕ O_{m_{2}}], [O_{l_{1}} ⊕ O_{m_{2}}], [O_{m_{1}} ⊕ O_{l_{0}} ⊕ O_{m_{2}}], [O_{l_{1}} ⊕ O_{m_{0}} ⊕ O_{l_{2}}]} is a Γ-invariant basis of K_{0}(\overline{S})^{(1)}, and thus K_{0}(\overline{S})^{(1)} is a permutation module. It follows that H^{1}(F, K_{0}(\overline{S})^{(2)}) = H^{1}(F, K_{0}(\overline{S})^{(1)}) = 0.

We will need the following proposition (cf. Proposition 6.1 of [6]).

Proposition 4.5. If Y is a variety such that the homomorphism ϕ_{F (y)} : K_{0}(A_{F (y)}) → K_{0}(S_{F (y)}) is an isomorphism for every y ∈ Y, then ϕ_{Y} : K_{0}(Y; A) → K_{0}(S × Y) is surjective.

Proof. We do this by double induction on the dimension of Y and the number of irreducible components of Y.

If Y has a proper irreducible component Y′, with complement U, we have the following localization exact sequence (cf. [8]):

\xymatrix{K_{0}(Y′; A) \ar[r] \ar[d]^{ϕ_{Y′}} & K_{0}(Y; A) \ar[r] \ar[d]^{ϕ_{Y}} & K_{0}(U; A) \ar[r] \ar[d]^{ϕ_{U}} & 0 \\ K_{0}(Y′ × S) \ar[r] & K_{0}(Y × S) \ar[r] & K_{0}(U × S) \ar[r] & 0}

By our inductive assumption, the vertical maps on the right and on the left are surjective. This implies that the middle vertical map is surjective as well. So we may assume that Y is irreducible. If Y is not reduced, then we the natural Y_{red} → Y induces the commutative diagram

\xymatrix{K_{0}(Y_{red}; A) \ar[r] \ar[d]^{ϕ_{Y_{red}}} & K_{0}(Y; A) \ar[d]^{ϕ_{Y}} \\ K_{0}(Y_{red} × S) \ar[r] & K_{0}(Y × S),}

where the horizontal arrows are isomorphisms. Thus we may also assume that Y is reduced.

Now let x ∈ K_{0}(Y × S). By assumption, ϕ_{F (Y)} : K_{0}(A_{F (Y)}) → K_{0}(S_{F (Y)}) is an isomorphism, and hence there exists an open set U in Y such the image of x in K_{0}(U × S) is in the image of ϕ_{U}. We again consider the localization exact sequence:

\xymatrix{K_{0}(Z; A) \ar[r] \ar[d]^{ϕ_{Z}} & K_{0}(Y; A) \ar[r] \ar[d]^{ϕ_{Y}} & K_{0}(U; A) \ar[r] \ar[d]^{ϕ_{U}} & 0 \\ K_{0}(Z × S) \ar[r] & K_{0}(Y × S) \ar[r] & K_{0}(U × S) \ar[r] & 0,}

where Z is the complement of U in Y. By assumption, Z has a strictly smaller dimension than Y, and so by our inductive hypothesis, ϕ_{Z} is surjective. A standard diagram chase shows that x ∈ Im(ϕ_{Y}). □

Proof of Theorem 4.2. Let SB(B) be the Severi-Brauer K-variety associated to B, SB(Q) be the Severi-Brauer L-variety associated to Q, and Y = R_{K/F} (SB(B))× R_{L/F} (SB(Q)) be the product of the restriction of scalars of both varieties. Then for any field extension E of F, Y (E) is nonempty if and only if SB(B)(K⊗_{F} E) and SB(Q)(L ⊗_{F} E) are nonempty if and only if B_{E} = B ⊗_{K} (K ⊗_{F} E) and Q_{E} = Q ⊗_{L} (L ⊗_{F} E) are split.

The projection p : Y → Spec(F) induce the following diagram:

\xymatrix{K_{0}(A) \ar[r]^{ϕ} \ar[d]^{p^{∗}} & K_{0}(S) \ar[d]^{(p_{Y} )^{∗}} \\ K_{0}(Y; A) \ar[r]^{ϕ_{Y}} \ar[u]^{p_{∗}} & K_{0}(Y × S), \ar[u]^{(p_{Y} )_{∗}}}

where p_{Y} : Y × S → S is the projection induced by p. Both squares commute, and p_{∗}p^{∗} is the identity homomorphism, as Y is a geometrically rational variety. For every y ∈ Y, Y (F (y)) ≠ ∅, so B_{F (y)} and Q_{F (y)} are split, and thus ϕ_{F (y)} : K_{0}(A_{F (y)}) → K_{0}(S_{F (y)}) is an isomorphism by Proposition 4.3. So by Proposition 4.5, ϕ_{Y} : K_{0}(Y; A) → K_{0}(Y × S) is surjective. A diagram chase shows the top horizontal map ϕ is also surjective.

Now let E be any field extension of F such that S(E) ≠ ∅. Then, B_{E} and Q_{E} are split, and so ϕ_{E} : K_{0}(A_{E}) → K_{0}(S_{E}) is an isomorphism, again by Proposition 4.3. The homomorphisms ϕ and ϕ_{E} fit into the following commutative diagram:

\xymatrix{K_{0}(A) \ar[r]^{ϕ} \ar[d] & K_{0}(S) \ar[d] \\ K_{0}(A_{E}) \ar[r]^{ϕ_{E}} & K_{0}(S_{E}),}

where the vertical homomorphisms are induced by the inclusion F ⊂ E. The bottom horizontal map is an isomorphism, and the left vertical map is injective. It follows that ϕ is injective, and hence an isomorphism. □

As ϕ is an isomorphism, the hypothesis on the variety V in Proposition 4.5 is always true, we obtain the following corollary.

Corollary 4.6. ϕ_{V} : K_{0}(V; A) → K_{0}(V × S) is surjective for any F -variety V. □

5. Higher K-theory

We have shown that the K_{0} groups of S and A coincide. We will show that this is also true for the higher Quillen K-groups.

For any F -varieties X, Y, and Z, and separable F -algebras A, B, and C, consider the functor:

P(Y × Z; B^{op} ⊗_{F} C) × P(X × Y; A^{op} ⊗_{F} B) → P(X × Z; A^{op} ⊗_{F} C),

sending a pair (M, N) to (p_{13})_{∗}(p_{23}^{∗}(M) ⊗_{B} p_{12}^{∗}(N)), where p_{12}, p_{23}, and p_{13} are the projections of X × Y × Z onto its factors. This functor is bi-exact, and thus induces a product map

K_{n}(Y × Z; B^{op} ⊗_{F} C) ⊗_{ℤ} K_{m}(X × Y; A^{op} ⊗_{F} B) → K_{n+m}(X × Z; A^{op} ⊗_{F} C).

We will denote the image of u ⊗ x under this map by u •_{B} x.

We recall the K-Motivic Category C and some of its properties. The details can be found in [6] and [7]. Objects of C are pairs (X, A), where X is an F -variety and A is a separable F -algebra. For two pairs (X, A) and (Y, B) in C, we set Mor_{C}((X, A), (Y, B)) := K_{0}(X × Y; A^{op} ⊗_{F} B). The composition law is g ∘ f = g •_{B} f, for f : (X, A) → (Y, B) and g : (Y, B) → (Z, C) in C. For any pair (X, A) with X smooth, the identity element 1_{(X,A)} ∈ K_{0}(X ×X; A^{op}⊗_{F} A) is the element [O_{Δ} ⊗_{F} A], where Δ ⊂ X ×X is the diagonal. For any F -variety X and any separable F -algebra A, we will write X for the pair (X, F) and A for the pair (Spec F, A). Finally, for any F -variety V and any nonnegative integer n, we have a realization functor K_{n}^{V}, which sends an object (X, A) to K_{n}(V × X; A), and K_{n}^{V} (f)(x) = f •_{A} x ∈ K_{n}(V × Y; B) for any morphism f ∈ Mor_{C}((X, A), (Y, B)) = K_{0}(X × Y; A^{op} ⊗_{F} B) and x ∈ K_{n}(V × X; A). We will denote K_{n}^{Spec F} by K_{n}.

As we mentioned in the beginning of Section 4, there is a left action of A^{op} = End_{O_{S}}(O_{S})×End_{O_{S}}(I)×End_{O_{S}}(J) on the locally free sheaf P = O_{S} ⊕I ⊕J. So P ∈ P(X; A^{op}). The corresponding element [P] ∈ K_{0}(S; A^{op}) defines a morphism u : A → S in C. It follows from the construction of the realization functor that ϕ_{V} = K_{0}^{V} (u) for any V. In particular, K_{0}(u) = ϕ.

Theorem 5.1. u : A → S is an isomorphism in C.

Proof. Let V be any F -variety. Equating K_{0}(V; A) (resp. K_{0}(V × S)) with Mor_{C}(V, A) (resp. Mor_{C}(V, S)), K_{0}^{V} (u) : K_{0}(V; A) → K_{0}(V × S) is just postcomposition in C with u. By Corollary 4.6, K_{0}^{V} (u) = ϕ_{V} is surjective for any variety V. In particular, if V = S, there is an element v ∈ K_{0}(S; A) such that uv = [O_{Δ}], i.e. u has a right inverse v in C.

We want to show that v is also a left inverse to u in C, i.e. vu = [A] ∈

K_{0}(A^{op} ⊗_{F} A). As K_{0}(A^{op} ⊗_{F} A) ↪ K_{0}((A^{op} ⊗_{F} A)\overline{F}) = K_{0}(Aop ; \overline{F} ⊗\overline{F} A\overline{F}), it

suffices to consider the case where K, L, B and Q are split. So A = F ×M_{3}(F × F)×M_{2}(F ×F ×F), and thus A is isomorphic in C to F ×(F ×F)×(F ×F ×F) (cf. example 1.6 of [6]). So K_{0}(A) ≅ ℤ^{6}, and Mor_{C}(A, A) = K_{0}(A^{op} ⊗_{F} A) ≅ M_{6}(ℤ). Moreover, under this isomorphism [A] ∈ K_{0}(A^{op} ⊗_{F} A) corresponds to the identity matrix.

So vu ∈ K_{0}(A^{op} ⊗_{F} A) is represented by a matrix M with integer entries. It follows that the corresponding homomorphism K_{0}(vu) from K_{0}(A) ≅ ℤ^{6} to itself is multiplication by this matrix M. Now, as v is a right inverse to u in C, K_{0}(v) is a right inverse to K_{0}(u). However, K_{0}(u) = ϕ is an isomorphism by Theorem 4.2, so in fact K_{0}(v) = K_{0}(u)^{−1}. Thus K_{0}(vu) = K_{0}(v)K_{0}(u) = id_{K_{0}(A)}, which forces M to be the identity matrix. Thus vu = [A] ∈ K_{0}(A^{op} ⊗ A), i.e. vu = id_{A} in C. □

Corollary 5.2. For any integer n, any central simple F -algebra D, and any F -variety V,

K_{n}(V; A ⊗_{F} D) ≅ K_{n}(V × S; D).

In particular, K_{n}(F) ⊕ K_{n}(B) ⊕ K_{n}(Q) = K_{n}(A) ≅ K_{n}(S).

Proof. For any central simple F -algebra D, Morita Equivalence gives a natural isomorphism K_{0}(S; A^{op}⊗_{F} D^{op}⊗_{F} D) = K_{0}(S; A^{op}). Thus the isomorphism u :

A → S in C also defines an isomorphism from A⊗_{F} D to (S, D) in C. Applying the realization functor K_{n}^{V} yields K_{n}(V; A ⊗_{F} D) ≅ K_{n}(V × S; D). □

We conclude the paper with an Index Reduction Formula for the function field of the surface S(B, Q, KL). We will need the following lemma:

Lemma 5.3 ([10]). Let X be a irreducible F -variety, and D a central simple F -algebra. The restriction homomorphism K_{0}(X; D) → K_{0}(D_{F (X)}) induced by the inclusion Spec(F (X)) → X is surjective.

Lemma 5.4. Let X be an irreducible F -variety, and D a central simple F - algebra.

ind D_{F (X)} = \frac{1}{deg D} g. c. d.\{rank(P), ∀P ∈ P(X; D)\}.

Proof. We recall that for any field E and any central simple E-algebra D′, K_{0}(D′) is cyclic, generated by the class of a simple D′-module M′. Moreover, dim_{E}(M′) = deg(D′) ind(D′).

The rank homomorphism rank : K_{0}(X; D) → K_{0}(F (X)) has the following decomposition:

K_{0}(X; D) → K_{0}(D_{F (X)}) → K_{0}(F (X)),

where the first map is induced by the inclusion Spec(F (X)) → X, and the second map takes the class of a D_{F (X)}-module to the class of the corresponding F (X)-vector space.

As K_{0}(F (X)) is cyclic, the image of the rank homomorphism is n[F (X)], where n is the greatest common divisor of the numbers rank(P), for all P ∈ P(X; D). By the previous lemma, the homomorphism K_{0}(X; D) → K_{0}(D_{F (X)}) is surjective. Thus if M is a simple D_{F (X)}-module,

n = dim_{F (X)}(M) = deg(D_{F (X)}) ind(D_{F (X)}) = deg(D) ind(D_{F (X)}),

and the result follows. □

Corollary 5.5 (Index Reduction Formula). Let S = S(B, Q, KL) be a del Pezzo surface of degree 6. For any central simple F -algebra D, ind D_{F (S)} is equal to:

i. g. c. d.{ind(D), 2 ind(D ⊗_{F} B), 3 ind(D ⊗_{F} Q)}, if K and L are fields. ii. g. c. d.{ind(D), ind(D ⊗_{F} B_{1}), ind(D ⊗_{F} B_{2})}, if K = F × F and L is

a field. Here B = B_{1} × B_{2}. iii. g. c. d.{ind(D), ind(D ⊗_{F} Q_{1}), 2 ind(D ⊗_{F} Q_{2})}, if K is a field, and

L = F × E. Here Q = Q_{1} × Q_{2}. iv. g. c. d.{ind(D), ind(D ⊗_{F} Q_{1}), ind(D ⊗_{F} Q_{2}), ind(D ⊗_{F} Q_{3})}, if K is a

field, and L = F × F × F. Here Q = Q_{1} × Q_{2} × Q_{3}. v. ind D, when K and L are not fields.

Remark 5.6. In case ii., Q = M_{2}(L) is necessarily split, as K is not a field. Then ind(D ⊗_{F} M_{2}(L)) = ind(D_{L}), and as ind(D) divides [L : F] ind(D_{L}) = 3 ind(D_{L}), the greatest common divisor will not change if we remove the term 3 ind(D ⊗_{F} Q). Similarly, in cases iii., iv., and v., we can remove the term with the split B or Q when computing greatest common divisors.

Proof. As u : A → S is an isomorphism in C, it defines an isomorphism K_{0}(u) from K_{0}(A ⊗_{F} D) to K_{0}(S; D). Moreover, as A = F × B × Q, K_{0}(A ⊗_{F} D) ≅ K_{0}(D) ⊕ K_{0}(B ⊗_{F} D) ⊕ K_{0}(Q ⊗_{F} D).

We will consider the case where K and L are fields. The proof of the other cases are similar. As D, B ⊗_{F} D, and Q ⊗_{F} D are central simple algebras (with centers F, K, and L, respectively), their K_{0} groups are cyclic, generated by the class of a simple module. Therefore by Lemma 5.4, deg(D) ind(D_{F (S)}) will equal the greatest common divisor of the ranks of the images of simple D, B ⊗_{F} D and Q ⊗_{F} D modules under the image of K_{0}(u) : K_{0}(A ⊗_{F} D) → K_{0}(S; D).

So let M_{B} be a simple B ⊗_{F} D-module. Then dim_{K}(M_{B}) = deg(B ⊗_{F} D) ind(B ⊗_{F} D), and thus

rank(K_{0}(u)(M_{B})) = rank(M_{B} ⊗_{B} I)

= \frac{dim_{F} (M_{B}) rank(I)}{dim_{F} (B)}

= \frac{dim_{K}(M_{B}) rank(I)}{dim_{K}(B)}

= \frac{deg(B ⊗_{F} D) ind(B ⊗_{F} D) rank(I)}{dim_{K}(B)}

= 2 deg(D) ind(B ⊗_{F} D).

Similarly, if M_{Q} (resp. M_{F}) is a simple Q ⊗_{F} D-module (resp. D-module), rank(K_{0}(u)(M_{Q})) = 3 deg(D) ind(D⊗_{F} Q) (resp. rank(K_{0}(u)(M_{F})) = deg(D) ind(D)), and the result follows. □

References

[1] J.-L. Colliot-Thélène, N. Karpenko, and A. Merkurjev, Rational Surfaces and Canonical Dimension of PGL_{6}, Algebra i Analiz, 19 (2007), no. 5, 159–178.

[2] Corn, Patrick, Del Pezzo surfaces of degree 6, Math. Res. Lett. 12 (2005), no. 1, 7584

[3] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits.

[4] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.

[5] Y. I. Manin, Cubic Forms, North-Holland, Amsterdam, 1974. Translated from The Russian by M. Hazewinkel.

[6] A. S. Merkujev, and I. A. Panin, K-Theory of Algebraic Tori and Toric Varieties., K-Theory 12 (1997), no 2, 101–143.

[7] I. A. Panin, On the Algebraic K-Theory of twisted flag varieties, K-Theory 8 (1994), no. 6, 541–585.

[8] D. Quillen, Higher Algebraic K-Theory I, Lecture Notes in Mathematics, no. 341, Springer-Verlag, Berlin, 1972, 85–147.

[9] J.-P. Serre, Galois Cohomology, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion.

[10] A. Schofield, and M. Van den Bergh, The index of a Brauer class on a Brauer-Severi variety, Transactions of the American Mathematical Society 333 (1992), 729–739

[11] D. Tao, A variety associated to an algebra with involution, J. Algebra 168 (1994), no. 2, 479-520.

[12] V. E. Voskresenskiĭ, Projective invariant Demazure models, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 195–210.

[13] V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavskiĭ.

[14] V. E. Voskresenskiĭ, and A.A. Klaychko, Toric Fano Varieties and systems of roots, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 2, 237–263.

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

  E-mail address: mblunk@math.ucla.edu