Del Pezzo Surfaces of degree 6 over an arbitrary field

DEL PEZZO SURF A CES OF DEGREE 6 O VER AN ARBITR AR Y FIELD MARK BLUNK Abstract. W e give a characterization o f all del Pezzo surfaces o f degree 6 ov er an arbitrary field F . A surface is deter mined by a pair of separa ble algebras . These algebras are used to compute the Quillen K -theory of the surface. As a consequence, w e o btain a n index reduction formula for the function field of the surface. 1. Introduction If X is an algebraic v ariet y defined o v er an arbitrary field F , a common metho d (cf. the in tro duction of [13]) f or learning v arious prop erties of X is to first study X := X × Spec F Sp ec( F ), the extension of scalars of X to a separable closure F of F , and then to study the action of the G alois group Gal( F /F ) on a lgebraic groups and other algebraic ob jects asso ciated to X . This is particularly useful when dealing with a class of v a r ieties that all b e- come isomorphic ov er F , e.g. Sev eri-Brauer v arieties or inv olution v arieties. A Sev eri-Brauer v a riet y is determined b y a cen tral simple F -algebra A , and an inv olution v ariet y is determined by a ce n tral sim ple F -algebra A with an orthogonal in v olution of the first k ind ( A, σ ). In either case this algebraic data determines geometrical and top ological information ab out the corresponding v ariet y . In pa r ticular the Quillen K -groups of the v ariet y a re determined the algebra in the Sev eri-Brauer example, and the algebra with in v olutio n in the in volution v ariet y example. This w as prov ed for Sev eri- Brauer v arieties and in volution v arieties in [8] and [11], resp ectiv ely . P anin pro ve d in [7] a more general theorem computing the K -theory of pro jectiv e homogeneous v arieties, whic h contains b oth examples as sp ecial cases. In all of these examples, as in this pap er, the action of algebraic groups plays a s ignifican t role. An imme- diate consequence of this computation of the K -theory is an index reduction form ula, whic h determines how extending scalars of a division F -algebra to the function field of t he v ariety reduces the index of the algebra. In this pap er w e will study del P ezzo surfaces of degree 6 ov er F , obtaining similar results. A del P ezzo surface S is a smo oth pro jectiv e surface o ver a field F suc h that the an t i- canonical bundle ω − 1 S is ample. The degree (the self-inters ection n umber of ω S ) of any suc h surface can b e any integer b et w een 1 and 9. Such v arieties w ere discussed in [1 ], [2], and [12]. As men t io ned in some o f these references, a del Pez zo surface of degree 6 is a toric v a r iety for a particular t w o dimensional torus, whic h w e will describ e b elow. W e explore this toric struc- ture in Section 2. The result is Theorem 2.4, a classification of all suc h surfa ces 1 2 MARK BLUNK up t o isomorphism preserving the action o f the torus. Se ction 3 con ta ins the main r esult of the pap er, Theorem 3.5 , where it is pro ved that a del P ezzo surface of degree 6 is determined by a pair B and Q o f separable F - algebras, with cen ters K and L ´ etale quadratic and cubic ov er F resp ectiv ely , and b oth con taining K ⊗ F L as a subalgebra. Moreo v er, cor K/F ( B ) and cor L/F ( Q ) m ust b e split. As an immediate corollary of Theorems 2 .4 and 3.5, w e give a nec- essary and sufficien t condition in terms of B and Q for determining when the corresp onding surface will ha v e a rationa l p oint. In Section 4, we relate the algebras B and Q to the endomorphism rings of lo cally free shea ve s on the a sso ciated del P ezzo surfa ce S . These sheav es are used in Theorem 4.2 to relate the Quillen K -theory of S to that o f B and Q , b y sho wing 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 corolla r y w e obtain a n index reduction form ula for the function field of S . I w ould lik e to thank m y a dvisor Alexander Merkurjev, who p o sed this question to me, and answ ered sev eral o f m y questions whic h dev elop ed along the w a y . W e use the following notations and conv en tions: An F -v ariet y is a separated sc heme of finite type o v er Sp ec( F ). F will denote a separable closure of F . An F -algebra A is separable if A ⊗ F L is semisimple for ev ery field extension L of F . Suc h a n algebra is Azuma ya o ver its cen ter, whic h is an ´ e tale extension of F . Γ will denote the group G al( F /F ). F or an y F -v ariety X and an y field extension E of F , w e will denote X × Spec F Sp ec( E ) (resp. X × Spec F Sp ec( F )) b y X E (resp. X ). F or an y separable F -algebra A and an y ´ etale extension E of F , w e will denote A ⊗ F E (resp. A ⊗ F F ) by A E (resp. A ). If D is a Cartier divisor on a v ariet y X , L ( D ) will denote the corresp o nding in vertible sheaf on X . F or any v ariet y X and an y separable algebra A , P ( X ; A ) will denote t he exact category of left A ⊗ F O X -mo dules whic h are lo cally free O X -mo dules. W e will denote P ( X ; F ) (resp. P (Sp ec F ; A )) b y P ( X ) (resp. P ( A )). F or any in teger n , K n ( X ; A ) will denote the Quillen group K n ( P ( X ; A )). As ab o v e, we will denote K n ( X ; F ) by K n ( X ) and K n (Sp ec F ; A ) b y K n ( A ). F or an y algebraic torus T , b T will denote the Γ- mo dule of c ha racters Ho m F ( T , G m, F ). 2. Toric V arieties W e first recall f rom [4], [5], and [12] some basic prop erties of the v ariet y e S , the blow up of P 2 at the 3 non-collinear p oints [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1]. The v ariety e S can be realized as a closed s ub v ariety of P 2 × P 2 , de fined DEL PEZZO SUR F A CES OF DEGREE 6 OVER AN ARBITRAR Y FIELD 3 b y the equations x 0 y 0 = x 1 y 1 = x 2 y 2 . The pro jection on to the first factor o f P 2 is the blow do wn 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 pro jection on to the second f actor o f P 2 is the blow do wn 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 } . Prop osition 2.1. L et e S b e the blow up of P 2 at the thr e e p oints [1 : 0 : 0 ] , [0 : 1 : 0] , and [0 : 0 : 1 ] . i. T he variety e S is a del Pezzo surfac e of de gr e e 6 over F , and if F is sep- ar ably close d, any del Pezzo surfac e S of de gr e e 6 over F is isomorphi c to e S . ii. T he gr oup CH 1 ( e S ) is gener ate d by the lines l 0 , l 1 , l 2 , m 0 , m 1 , and m 2 . iii. T he interse ction p airing on CH 1 ( e S ) is de termi n e d by the fol lowing r e- lations: l 2 i = − 1 , m 2 i = − 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 gr oup CH 2 ( e S ) is cyclic, gene r ate d by the c l a ss of any r ational p oin t. As men tioned in [1], there is an action of the torus e T = G 3 m / G m on P 2 , described b y: ( 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 − 1 0 y 0 , t − 1 1 y 1 , t − 1 2 y 2 ] . Here G m em b eds in to G 3 m diagonally . This action sends e S to itself, and is faithful and transitiv e on the o p en subset e U of e S , the complemen t of the sub v ariet y defined b y the equation x 0 x 1 x 2 y 0 y 1 y 2 = 0. This closed sub v ariet y has 6 irreducible comp onents , the lines l 0 , l 1 , l 2 , m 0 , m 1 , and m 2 , whic h b y the prop osition are arranged in a hexagon. Th us e S is a e T -toric v ariet y , with fa n dual to the hexagon o f lines. There is also an a ction of the symm etric groups S 2 and S 3 on e S . The nontrivial elemen t of S 2 acts on P 2 × P 2 b y inte rc hanging the x i and y i , and the S 3 action on P 2 × P 2 arises from the diagona l action of S 3 on the co ordinates x 0 , x 1 , x 2 and y 0 , y 1 , y 2 . The S 2 and S 3 actions comm ut e with eac h other, and b oth groups send e S ⊂ P 2 × P 2 to itself. Therefore they induce an action of S 2 × S 3 on e S , preserving the set o f lines l 0 , l 1 , l 2 , m 0 , m 1 , and m 2 , and th us inducing an isomorphism from S 2 × S 3 on t o the auto morphism group of the hexagon of lines. The torus e T is the connected comp onent of the iden tity of the algebraic gro up Aut F ( e S ) of automorphisms of e S , and S 2 × S 3 is the group of connected components. The action of S 2 × S 3 on e S define a section S 2 × S 3 → Aut F ( e S ), so w e ha v e the following split exact sequence of algebraic g r o ups: 1 → e T → Aut F ( e S ) → S 2 × S 3 → 1 . 4 MARK BLUNK ✠ ❘ P 2 x 1 = 0 x 2 = 0 x 0 = 0 P 2 y 0 = 0 y 1 = 0 y 2 = 0 e S l 0 l 1 l 2 m 0 m 1 m 2 p 1 p 2 Figure 1. The Hexagon of Lines. W e ha ve another w ay to realize e S as a closed sub v ariety of a pro duct of pro jectiv e spaces. D efine f i : e S : → P 1 for i = 0 , 1 , 2 b y 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 ] . Eac h f i is w ell defined, as the t w o definitions agree on the ov erlap, and thus is a morphism of v arieties. These morphisms define a morphism f : e S → P 1 × P 1 × P 1 . If we denote the bi-homogeneous co ordinates of P 1 × P 1 × P 1 b y X 0 , X 1 , Y 0 , Y 1 , Z 0 , and Z 1 , it can b e show n that f maps e S isomorphically onto the hypersurface of P 1 × P 1 × P 1 defined b y the equation X 0 Y 0 Z 0 = X 1 Y 1 Z 1 . DEL PEZZO SUR F A CES OF DEGREE 6 OVER AN ARBITRAR Y FIELD 5 The morphism f sends e T to the torus Ker( G 2 m / G m × G 2 m / G m × G 2 m / G m m − → G 2 m / 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 ). No w let S b e a del Pezz o surface of degree 6 o v er an arbitrary field F . Then S is a del P ezzo surface o f degree 6 o ver F , and thus b y Prop o sition 2.1 is isomorphic o v er F t o e S . So S is an F - f orm of e S . As the six lines of the hexagon form a full set of exce ptional curves in S , the action of Γ on S is globally stable on the set of lines of the hexagon. Therefore, there is an op en sub v ariet y U whose complemen t Z is isomorphic ov er F to the hexagon of lines. The action of Γ on Z p ermutes its irreducible comp onen ts, inducing an action o f Γ on the hexagon. Let T denote the connected comp onent of the iden tity of Aut F ( S ). The group of connected compo nen ts G of Aut F ( S ) is an ´ e tale gr o up sc heme: it is the group sc heme determined (as in Prop osition 20.16 of [3]) by the automorphism group of the hexagon of lines, with contin uous Γ-action on this finite group as in the previous paragraph. So T is a torus, S is a T -toric v ariet y , with an op en set U whic h is a T -torsor, and Γ-action on t he fan determined b y the ´ etale group sc heme G . This Γ-action on the hexagon determines a homo mo r phism γ : Γ → S 2 × S 3 . Pro jecting on to either factor yields co cycles with v alues in S 2 and S 3 , and th us γ determines a pair ( K, L ), where K and L a re ´ etale quadrat ic and cubic extensions o f F , resp ectiv ely . Note that while the fa n, dual to t he hexagon of lines, is the same fo r all del P ezzo surfaces of degree 6 ov er F , the p o ssible Γ-actions o n the fan ar e in a one-to-one corresp o ndence with pairs ( K , L ). F or a fixed co cyc le γ (i.e. a fixed pair ( K , L )), we will classify all del P ezzo surfaces S of degree 6 where the Γ- action on Z ⊂ S is determined by γ . W e hav e f rom [12] the following short exact sequence of Γ-mo dules: 0 → b T → Z [ K L/F ] → Pic( S ) → 0 . Here K L denotes the a lg ebra K ⊗ F L , a nd Z [ K L/F ] is the lattice of t he six lines of Z . The homomorphism Z [ K L/F ] → Pic( S ) takes a line to the corre- sp onding Cartier divisor on S . As described in [1], this short exact sequence can b e extended into the exact sequence (1) 0 → b T → Z [ K L/F ] → Z [ K /F ] ⊕ Z [ L/F ] → Z → 0 . Here Z [ L/F ] is the la t tice of pairs of opp o site lines, and Z [ K/F ] is the lattice of triangles, where eac h triangle is a triple of sk ew lines. The homomo r phism Z [ K L/F ] → Z [ L/F ] sends eac h line to t he pair con taining it, and the ho- momorphism Z [ K L/F ] → Z [ K/ F ] sends eac h line to the triangle con ta ining it. The homomorphism Z [ K /F ] ⊕ Z [ L/F ] → Z is the differenc e of the a ug- men tation maps. This sequence induces the follo wing short exact sequence of Γ-mo dules: (2) 0 → b T → Z [ K L/F ] / Z → Z [ K /F ] / Z ⊕ Z [ L/F ] / Z → 0 . 6 MARK BLUNK where Z em b eds in to Z [ K /F ], Z [ L/F ], and Z [ K L/F ] diagonally . In ana lo gy with R (1) K/F ( G m ) := Ker( N K/F : R K/F ( G m ) → G m ), we define the follo wing alg ebraic F -groups: G L := Ker( N K L/L : R K L/F ( G m ) → R L/F ( G m )) G K := Ker( N K L/K : R K L/F ( G m ) → R K/F ( G m )) These groups are F -tori, dual to the Γ- mo dules Z [ K L/F ] / Z [ L/F ] and Z [ K L/F ] / Z [ K /F ], where Z [ K /F ] a nd Z [ L/F ] are dia g onally em b edde d in Z [ K L/F ]. The em- b eddings of R K/F ( G m ) and R L/F ( G m ) in to R K L/F ( G m ) induce em b eddings R (1) K/F ( G m ) → G L and R (1) L/F ( G m ) → G K . The description of e T ⊂ e S ⊂ P 2 × P 2 ab ov e descends to t he following exact sequence: (3) 1 → R (1) K/F ( G m ) → G L → T → 1 . Similarly , the description of f ( e T ) ⊂ P 1 × P 1 × P 1 ab ov e descends to 1 → R (1) L/F ( G m ) → G K → T → 1 . W e will use these sequences in Section 4. Finally , f r o m (1) a nd (2), w e hav e corr esp o nding sequences of F -tor i: (4) 1 → G m → R K/F ( G m ) × R L/F ( G m ) → R K L/F ( G m ) → T → 1 , and 1 → R (1) K/F ( G m ) × R (1) L/F ( G m ) φ − → R (1) K L/F ( G m ) → T → 1 . Recall that for E = K , L , and K L , H 1 ( F , R (1) E /F ( G m )) = F × / N E /F ( E × ) H 2 ( F , R (1) E /F ( G m )) = Ker(cor E /F : Br( E ) → Br( F )) . Moreo ver, as N K L/F  ( K L ) ×  is a subgroup of N K/F (K × ) and N L/F ( L × ), it follo ws that the restriction of the homomorphism of H 1 groups induced by φ to either factor is just f actoring out the correspo nding subgroup of the quotien t, and th us φ will b e surjectiv e. Therefore, by the induced long exact sequence in cohomolog y , we obtain the follow ing exact sequence: 1 → H 1 ( F , T ) → Ker(cor K/F ) × Ker(cor L/F ) → Ker(cor K L/F ) . where t he la st homomor phism sends a pair ( x, y ) to res K L/K ( x ) − res K L/L ( y ) ∈ Br( K L ). Let C 1 b e the set of K -algebra isomor phism classes of Azuma y a K -a lg ebras B of rank 9 suc h that B L = B ⊗ K K L and cor K/F ( B ) are split, C 2 the set of L -algebra isomorphism classes of Azumay a L -a lg ebras Q of rank 4 suc h that Q K = Q ⊗ L K L and cor L/F ( Q ) are split, and set C = C 1 × C 2 . Then C is a p ointe d set with distinguished elemen t ( M 3 ( K ) , M 2 ( L )), and the map ψ : C → Ker(cor K/F ) × Ker(cor L/F ) sending a pair ( B , Q ) t o ([ B ] , [ Q ]) is a morphism of p ointed sets. Moreo v er, res K L/K ([ B ]) = [ B ⊗ K K L ] a nd res K L/L ([ Q ]) = [ Q ⊗ L K L ] are trivial, so it follows that ψ maps in to H 1 ( F , T ). DEL PEZZO SUR F A CES OF DEGREE 6 OVER AN ARBITRAR Y FIELD 7 Theorem 2.2. ψ : C → H 1 ( F , T ) is a n isomorp hism of p ointe d s ets. Pr o of. If ψ ( B , Q ) = ψ ( B ′ , Q ′ ), then [ B ] = [ B ′ ] ∈ Ker(cor K/F ) ⊂ Br( K ) . Then B and B ′ are similar Azuma y a K -algebras of the same rank, and thus m ust b e isomorphic as K -alg ebras. Similarly , Q and Q ′ are isomorphic, so that ψ is injectiv e. No w let ( x, y ) ∈ H 1 ( F , T ), so tha t ( x, y ) ∈ Ker(cor K/F ) × Ker(c or L/F ), and res K L/K ( x ) = res K L/L ( y ). This implies that 3 x = cor K L/K (res K L/K ( x )) = cor K L/K (res K L/L ( y )) = res K/F (cor L/F ( y )) = 0 . Similarly , 2y = 0. Th us res K L/K ( x ) = res K L/L ( y ) has order divisible b y 2 and 3, and therefore is trivial. If L is not a field, then L = F × E , where E is a n ´ etale quadratic extension of F . Then res K L/K ( x ) = ( x, res E ⊗ F K/K ( x )), and so res K L/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 b y K L if and only if x 1 and x 2 are split b y L ). Th us for all p o ssible K and L , there is an Azuma y a K - algebra B o f rank 9 that represen ts x in Br( K ). Since res K L/K ( x ) and cor K/F ( x ) a re trivial, B ⊗ K K L a nd cor K/F ( B ) are split. Similarly , there is an Azuma ya L -algebra Q of rank 4 whic h represen ts y suc h that Q ⊗ L K L and cor L/F ( Q ) are split. Then ( B , Q ) ∈ A , and φ ( B , Q ) = ( x, y ), so ψ is surjectiv e.  Remark 2.3. As K L is an ´ etale algebra of degree 3 ov er K , if K L splits B , then K L can b e em b edded as a subalgebra of B . Similarly , K L can b e em b edded as a subalgebra of Q . If ( B , Q ) = ( B ′ , Q ′ ) in C = H 1 ( F , T ), then an y K -isomorphism from B to B ′ sends K L ⊂ B to a subalgebra o f B ′ isomorphic to K L . Moreo v er, if w e c ho ose a fixed embedding of K L into b oth B and B ′ , b y applying Sk olem-No ether t o B ′ w e can find a n isomorphism from B to B ′ whic h restricts to t he iden tit y on K L . Similarly , w e may assume tha t Q to Q ′ are isomorphic via a n isomorphism whic h is the iden tity on K L . It f ollo ws that if K and L are ´ etale quadratic and cubic extensions of F resp ectiv ely , and T is the t wo dimensional torus induced fr o m K and L as in the exact sequence (4), t hen elemen ts of H 1 ( F , T ) are dete rmined b y triples ( B , Q, K L ), where B is an Azuma ya K -algebra of rank 9 suc h that cor K/F ( B ) is split, Q is an Azuma y a algebra ov er L of rank 4 suc h that cor L/F ( Q ) is split, and w e hav e a fixed embedding of K L as a subalgebra in to b oth B and Q . Tw o triples ( B , Q, K L ) and ( B ′ , Q ′ , K L ) will determine the same elem en t of H 1 ( F , T ) if there are K L - a lgebra isomorphisms from B to B ′ and Q to Q ′ . If S is a del P ezzo surface of degree 6, and if T is the connected comp onen t of the iden tit y of Aut F ( S ), S is a T -toric v ariet y , with Γ-action o n the fan induced by the Γ-action γ on the connected comp onen ts of Z , the hexagon of lines. The T -to rsors U ⊂ S is determined b y an elemen t of the p oin ted set H 1 ( F , T ). Tw o surfaces S a nd S ′ will b e isomorphic as toric v arieties if and only if T and T ′ are isomorphic as algebraic gro ups, and there is an 8 MARK BLUNK isomorphism from S to S ′ whic h preserv es the action of T ∼ = T ′ on S and S ′ , thus inducing isomorphisms Γ- actions o n the fan and isomorphisms of the T -to r sors determining S and S ′ . Th us we hav e prov ed the following: Theorem 2.4. L et S b e a de l Pezzo surfac e of de gr e e 6, and T b e the c on- ne cte d c omp onent of the identity of the gr oup Aut F ( S ) . Then S is a T -toric variety with Γ -invaria n t fan determine d by a p air ( K , L ) and t he T -torsor U determine d by a triple ( B , Q, K L ) . Two triples ( B , Q, K L ) and ( B ′ , Q ′ , K ′ L ′ ) wil l describ e isomorphic toric varieties if and only if K and L ar e isom orphic to K ′ and L ′ as F -algebr as, (so that T ∼ = T ′ ), an d ther e exist K L -algebr a isomorphisms fr om B to B ′ and Q to Q ′ . 3. The Main Theorem W e w ould no w like to classify these surfaces up to isomorphism as abstract v arieties. This is less r estrictiv e t ha n isomorphism as t oric v arieties. W e will see t ha t a del P ezzo surface is still determined by a triple ( B , Q, K L ), but now w e will a llo w F -algebra isomorphisms on B and Q , i.e. algebra isomorphisms whic h may not fix K L . Let S b e a del P ezzo surface of degree 6 o v er F . Then S is a T -toric v ariety , where T is the connected comp onen t of Aut F ( S ), with the Γ-action on the fan determining a pair ( K, L ), a nd t he T -torsor U ⊂ S determining a pair ( B , Q, K L ). Let G b e the group of connected comp onen ts of Aut F ( S ). As w e ha ve sho wn ab ov e, G is an ´ etale group sc heme, determined b y the action of Γ on the hexagon of lines . In particular, G ( F ) = Aut F ( K L ) ∼ = Aut F ( K ) × Aut F ( L ). Consider the follo wing action of G ( F ) on H 1 ( F , T ): if ( g , h ) ∈ Aut L ( K L ) × Aut K ( K L ) and ( B , Q, K L ) ∈ H 1 ( F , T ) then g nontrivial sends B to B op , and sends the em b edding i : K L → B to i g : K L → B op , where i g ( z ) = i ( g ( z )) op ∈ B op . As K L is a cyclic extension of L , Q is a cyclic L -alg ebra, so t here is an elemen t l ∈ L × suc h t ha t Q is g enerated by K L and an elemen t y , sub ject to the relat io ns y 2 = l and z y = y σ ( z ) for ev ery z ∈ K L , where σ ∈ Aut L ( K L ) is the non trivial automo r phism of K L ov er L . Suc h an a lg ebra is denoted ( K L/L, l ). Let h act o n Q = ( K L/L, L ) b y h · ( K L/L, l ) = ( K L/L, h ( l )), and send the em b edding K L → Q t o K L h − → K L → ( K L, h ( l )). As l ∈ L × determines Q = ( K L/L, l ) up to multiplication b y an elemen t of the subgroup N KL/L  (KL) ×  ⊂ L × , and a s Aut F ( L ) take s N KL/L  (KL) ×  to itself, we see that t his action is well defined. The orbits of this group action can b e describ ed in terms of F - algebra iso- morphisms on B and Q , as w e will sho w b elow. W e will use the follo wing prop osition sev eral times: W e will need the following prop osition: Prop osition 3.1 (Prop osition 4.18 o f [3]) . L et ( B , τ ) b e a c en tr al simple F - algebr a of de gr e e n with unitary involution, and le t K b e the c enter of B . F or every F -sub algebr a L of B which is ´ etale of dimension n over F , ther e ex i s ts a unitary i n volution of B fixing L . DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 9 Prop osition 3.2. ( B ′ , Q ′ ) = g · ( B , Q ) f o r some g ∈ G ( F ) if and only if ther e ar e F -automorphisms φ B : B → B ′ and φ Q : Q → Q ′ such that φ B | K L = φ Q | K L = g . Pr o of. Assume that ( B ′ , Q ′ , K L ) = g · ( B , Q, K L ) for some g ∈ G ( F ). An y F -aut o morphism g of K L can b e expressed as the comp osition o f t w o automor- phisms, o ne fixing K and one fixing L . So it suffices to consider the separate cases where K and L are fixed b y the automorphism. W e first consider the case where g fixes K , so that B ′ = B . By Sk olem- No ether, there is a K -auto morphism φ B of B suc h that φ B | K L = g . No w, Q = ( K L/L, l ), Q ′ = ( K L/L, g ( l )) and as σ and g comm ute, g exten ds to a n F -aut o morphism φ Q from Q to Q ′ , by sending K L to K L via g , and y to y ′ . Then φ B and φ Q agree on K L . No w assume t ha t g = σ is t he non- t r ivial L -automorphism of K L , so that Q ′ = Q and B ′ = B op . As in the previous paragraph, b y Skole m-No ether there is an L -automorphism φ Q of Q suc h that φ Q | K L = g . Moreo ver, as cor K/F ( B ) is split and L is an ´ etale cubic extension of F , w e kno w b y Prop o sition 3.1 that B has a unitary in v olution τ whic h is the iden tity on L . The in v olution τ defines an F -isomorphism φ B from B to B op , suc h that φ B | K L = σ = g = φ Q | K L . Con ve rsely , assume that φ B : B → B ′ and φ Q : Q → Q ′ are F -isomorphisms suc h that φ B | K L = φ Q | K L = g ∈ Aut F ( K L ). As in the arg umen ts a b o v e, 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 -a lgebras, so that ( B ′ , Q, K L ) = ( B ′ , Q ′ , K L ) in H 1 ( F , T ). If w e restrict φ B to the cente r of B , w e get an F - isomorphism of K . If φ | K is the iden tity (i.e. g is trivial), then B and B ′ are isomorphic as K -algebras. If φ | K is not the iden tity , then b y pre-comp osing φ w ith the isomorphism from B op to B induced b y an y unitary in v o lut io n τ fixing L on B , w e see that B op and B ′ are isomorphic as K -algebras. In either case, ( B ′ , Q ′ , K L ) = ( B ′ , Q, K L ) = g · ( B , Q, K L ) in H 1 ( F , T ). No w assume that g fixes K , so that φ B : B → B ′ is an isomorphism of K - algebras, a nd t hen ( B ′ , Q ′ , K L ) = ( B , Q ′ , K L ) in H 1 ( F , T ). If Q = ( K L/L, l ) , and if l ′ = φ Q ( l ) = g ( l ), then Q is isomorphic o ve r L to ( K L/L, l ′ ), and so ( B ′ , Q ′ , K L ) = ( B , Q ′ , K L ) = g · ( B , Q ) in H 1 ( F , T ). Finally , assume that g do es not fix K or L . If σ is the non trivial L - automorphism of K L , σ g do es fix K . Moreo ve r, b y p ost-compo sing φ B with the F -isomorphism φ B ′ : B ′ → ( B ′ ) op induced b y any unitary inv olution τ o f B ′ fixing L (whic h exist by Prop osition 3.1), w e get isomorphisms φ B ′ ◦ φ B : B → ( B ′ ) op and φ Q : Q → Q ′ suc h that ( φ B ′ ◦ φ B ) | K L = φ Q | K L = σ g . Therefore b y our argumen t in the previous paragraph, σ · ( B ′ , Q ′ , K L ) = (( B ′ ) op , Q ′ , K L ) = σ g · ( B , Q, K L ) in H 1 ( F , T ). Acting on b oth sides of this equation b y σ , w e get ( B ′ , Q ′ , K L ) = g · ( B , Q, K L ).  The ne xt theorem relates isomorphism classes of de l P ezzo Surfaces of degree 6 with G ( F )-o rbits of H 1 ( F , T ). W e will need the following s tandard result from Ga lois cohomology: (cf. Corollary (28.1 0) of [3] or Chapter I, Section 5.5, Corollary 2 of [9].) 10 MARK BLUNK Prop osition 3.3. L et Γ b e a pr ofinite gr oup, A a nd B b e Γ -gr oups with A a normal sub gr oup of B , and set C = B / A . If β ∈ H 1 (Γ , B ) , and b a c o cycle r epr esenting β , then the elements of H 1 (Γ , B ) with the same image as β in H 1 (Γ , C ) c o rr esp onding bije ctively with ( C b ) Γ -orbits of the set H 1 (Γ , A b ) . Theorem 3.4. The isomorphism class of S c orr esp onds to a G ( F ) -orbit of H 1 ( F , T ) . Pr o of. As men tioned in Section 2, w e hav e the f ollo wing split exact sequence of algebraic groups: (5) 1 → e T → Aut F ( e S ) → S 2 × S 3 → 1 , where e T is the connected comp onent of the identit y of Aut F ( e S ), a nd S 2 × S 3 is the group of connected comp onents . This sequence induces the split exact sequence of p ointed sets: 1 → H 1 ( F , e T ) → H 1 ( F , Aut F ( e S )) → H 1 ( F , S 2 × S 3 ) → 1 . The elemen ts of H 1 ( F , Aut F ( e S )) are in a o ne-to-one corresp ondence with the set o f isomorphisms classes o f F -forms of e S , whic h by Prop osition 2.1 are del P ezzo surfaces o f degree 6. Let β ∈ Z 1 ( F , Aut F ( e S )) b e a co cycle whose cohomology class is determined b y the isomorphism class of S , and let γ b e the image of β in Z 1 ( F , S 2 × S 3 ). The co cycle γ is determined by the action of the Galois g roup Γ o n Z ⊂ S , and induces a pa ir ( K , L ). The t wist of e T b y γ is the torus T , determined b y ( K , L ) as in the seque nce (4), and the t wist of S 2 × S 3 is the ´ etale group sc heme G . The result follows b y Prop osition 3.3.  Note that as H 1 ( F , Aut F ( e S )) → H 1 ( F , S 2 × S 3 ) is surjectiv e, we see that all p ossible pairs ( K , L ) are realized b y the action of Γ on Z , for Z con tained in some del P ezzo surface S of de gree 6. So let S 1 and S 2 b e t wo del P ezzo surfaces of degree 6, and let ( B i , Q i , K i L i ) b e an e lemen t of the the G i ( F )-o rbit 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 line s (so that ( K 1 , L 1 ) ∼ = ( K 2 , L 2 ), T 1 ∼ = T 2 , and G 1 ∼ = G 2 ; w e denote thes e algebraic ob jects ( K, L ), T , and G , resp ectiv ely). It follow s from Prop osition 3.2 that t wo pairs ( B 1 , Q 1 , K L ) and ( B 2 , Q 2 , K L ) 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 suc h that φ B 1 | K L = φ Q 1 | K L . W e ha v e pro v ed the follo wing theorem: Theorem 3.5. Ther e ar e bije ctions, inverse to e ac h other, b etwe en the fol low- ing two sets: • The s e t of isomorphism classe s of del Pezzo surfac es of de gr e e 6. • The set of triples ( B , Q, K L ) , mo dulo the r elation: ( B , Q, K L ) ∼ ( B ′ , Q ′ , K ′ L ′ ) if ther e ar e F -alge b r a isomorphisms φ B : B → B ′ and φ Q : Q → Q ′ such that φ B | K L = φ Q | K L . F or the rest o f this pap er, S ( B , Q, K L ) will denote the del P ezzo surface of degree 6 determined b y the triple ( B , Q, K L ). DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 11 Corollary 3.6. The surfac e S ( B , Q, K L ) c ontains a r a tion a l p oint i f and only if B and Q ar e split. Pr o of. Since S ( B , Q, K L ) is a T -toric v ariety for a tw o dimensional torus T , S has a ratio nal p oint if and only if the corresp onding T -torsor U is a trivial torsor (cf. Prop osition 4 of [14 ]) . By Theorem 2 .4, this o cc urs precisely when B a nd Q are split.  Remark 3.7. If S 0 is a T -toric model (i.e. the T -t orsor U ⊂ S 0 is trivial), then the map H 1 ( F , T ) → H 1 ( F , Aut F ( S )) induced b y T ֒ → Aut F ( S ) tak es a T - torsor U to the surface determined b y U a nd the Γ-action on the fan determined b y the pair ( K , L ). Thus for an y surface S , the elemen ts of H 1 ( F , T ) in the fib er of the isomorphism class of S determine the p ossible non-isomorphic T - toric structures on S , where T is the connected comp onen t o f the iden tit y of the alg ebraic 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 K L -a lgebra structure of B and Q , preserving only the F - algebra structure and the embedding of K L in to B a nd Q . The g roup Aut F ( K ) alw ays has order 2, but the group Aut F ( L ) can ha ve order 1, 2, 3, or 6. If Aut F ( L ) has order less than 6, then the orbit of ( B , Q, K L ) in H 1 ( F , T ) con ta ins at most 6 elemen ts. It Aut F ( L ) ha s o r der 6 , then L = F 3 is not a field, and th us B is necessarily split. If B is split, then the pa ir ( B , Q, K L ) ∈ H 1 ( F , T ) is fixed by the subgroup Aut F ( K ) of G ( F ), and so again the G ( F )- o rbit of ( B , Q, K L ) in H 1 ( F , T ) has at most 6 elemen ts. Thu s for a del P ezzo surface S of degree 6, there at most 6 non-isomorphic T -tor ic structures on S . Remark 3.8. W e w ould lik e to relate this characterization of del P ezzo sur- faces of degree 6 b y triples ( B , Q, K L ) with the c haracterization b y triples ( B , τ , L ) found in [1]. A triple ( B , τ , L ) is an Azuma y a K -alg ebra B o f rank 9, a unitary in v olutio n τ on B , and a cubic ´ etale F -a lg ebra L suc h that L ⊂ Sym( B , τ ). Tw o triples ( B , τ , L ) and ( B ′ , τ ′ , L ′ ) a re isomorphic if there is an F -algebra isomorphism φ : B → B ′ suc h that τ ′ φ = φτ and φ ( L ) = L ′ . So let ( B , Q, K L ) b e a triple as in Theorem 3.5. Then B is an Azuma ya K -alg ebra of rank 9, and is classified up to isomorphism as an F -algebra. As B L is split, so B contains K L , and hence L , as a subalgebra. Since cor K/F ( B ) is split, w e kno w that B has a unitar y in volution. Moreo v er, since L is a n ´ etale cubic extension of F con tained in B , there is some unitary inv olution τ suc h that L ⊂ Sym( B , τ ), by Prop osition 3.1. So the B and L in o ur c haracterization matc h with the B and L described in [1]. The rest of the remark seeks to relate Q and the inv olution τ . That is, w e w an t 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 p ossible Q . As K /F and K L/L are cyclic, Br( K /F ) ∼ = F × / N K/F (K × ) 12 MARK BLUNK and Br( K L/L ) ∼ = L × / N KL/L  (KL) ×  . The restriction homomorphism res L/F : Br( F ) → Br( L ) sends the subgroup Br( K /F ) to Br( K L/L ). As K and L ha v e coprime degrees , res L/F | Br( K/F ) : Br( K /F ) → Br( K L/L ) is injec tiv e, and the c yclic algebra Q corresp onds to an elemen t of Br( K L/L ) / res L/F (Br( K /F )) , i.e. an elemen t 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 in volution on B which is the identit y on L , and u ∈ L × , then τ u := In t ( u ) ◦ τ is also a unitary inv olution on B fixing L . Moreov er, τ u is conjugate to τ v if uv − 1 ∈ F × N KL/L  (KL) ×  . Th us, after a c hoice of a particular in volution τ , w e ha v e a morphism of p oin ted sets from L × /F × N KL/L  (KL) ×  to the set of conjugacy classes of unitary in v o lutions of B whic h are the iden tity on L , sending u to τ u . By Corollary 19.3 of [3], this map is a surjection. So w e ha ve a surjectiv e map fro m L × /F × N KL/L  (KL) ×  to the set of isomorphism classes o f triples ( B , τ , L ). Theorem 3.5 should say that the fibers of this surjection should b e the o rbit of the group Aut F ( L ) in L × /F × N KL/L  (KL) ×  , corresp onding to the Aut F ( L )- orbit of ( B , Q ). How ev er, to make this statemen t cor r ect, we need to c ho ose a particular in volution τ on B . It is not clear in general what this in volution should be. The in volution τ should be c hosen so tha t the surface S ( B , τ , L ) should corresp ond 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 hav e a rational p oin t, b y Corollar y 3.6. The next r emark constructs the inv olution in this case. Remark 3.9. Giv en K and L , w e w ill find a triple ( B , τ , L ), (i.e. a cen tral sim- ple algebra of degree 3 ov er K with an inv olution τ suc h that L ⊂ Sym( B , τ ),) so that the corresponding del P ezzo surface S ( B , τ , L ) constructed in [1] has a rational p oin t. As K L is a t hr ee dimensional ve ctor space ov er K , B := End K ( K L ) is a Azuma y a K -algebra of rank 9. Left m ultiplicatio n b y an elemen t of K L deter- mines an em b edding of K L into B . If σ is the nontrivial L -automorphism of K L , h ( x, y ) = T r K L/K ( σ ( x ) y ) defines a hermitian form on K L . This hermit- ian f orm on K L induc es an in v olution of the second kind τ on B , suc h that L ⊂ Sym( B , τ ). So w e ha v e a triple ( B , τ , L ). Let S denote the corresp onding del P ezzo surface o f 6, constructed in [1]. I claim that S contains a rational p oint. According to [1], it suffices to sho w that there is a righ t ideal I of B of reduced dimension 1 such that ( I · τ ( I ) ) ∩ Sym( B , τ ) ⊂ F ⊕ L ⊥ , where L ⊥ = { x ∈ Sym( B , τ ) | T rd( l x ) = 0 , fo r all l ∈ L } . DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 13 Let W = span K (1) ⊂ LK , so that W is a o ne dimensional K -subspace of K L . and then I = Hom K ( K L, W ) is a rig h t ideal of B of reduced dimension 1, gene rated b y the linear map t = T r K L/K : K L → K ֒ → K L . W e w an t to sho w that t ∈ Sym( B , τ ). First, note that for an y x ∈ K L , T r K L/K ( σ ( x )) = σ (T r LK/L ( x )). If x, y ∈ K L , h ( x, t ( y )) = T r K L/K ( σ ( x ) T r K L/K ( y )) = T r K L/K ( σ ( x )) T r K L/K ( y ) = T r K L/K ( σ (T r K L/K ( x )) y ) = h ( t ( x ) , y ) . So t ∈ Sym( B , τ ), which implies that τ ( I ) is a left ideal o f B , also generated b y t . Th us, I · τ ( I ) = tB t . In order to pro v e I · τ ( I ) ∩ Sym( B , τ ) ⊂ F ⊕ L ⊥ , it suffices to consider the case where L = F 3 is split. So w e can c ho ose a basis e 1 , e 2 , e 3 of idemp oten ts for K L o ver K . In t his basis, τ is the standard adjoint in volution, Sym( B , τ ) is the set o f hermitian matrices, and F ⊕ L ⊥ is the set o f hermitian matrices where the diagonal en tries agree. Moreo v er, t is the ma t rix with ones in ev ery en try , so t ∈ F ⊕ L ⊥ . A direct calculation show s 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 Surf aces Let S = S ( B , Q, K L ) b e a del P ezzo surface o f degree 6, a T -toric v ariet y for a t w o dimensional torus T . Let Z ⊂ S b e the closed v ariety suc h that 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 [ K L/F ] is the lattice of connected comp onen t s of Z ⊂ S , the lines l i and m i , and the homomorphism Z [ K L/F ] → Pic( S ) sends eac h line to the corresp onding in ve rtible sheaf on S . F rom the exact sequence (1), we see that b T is the subgroup of Z [ K L/F ] generated b y 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 an y one of these 3 generators can b e expressed a s a linear combination of the other 2 . So Pic( S ) is generated b y the inv ertible shea v es L ( − l i ), L ( − m j ), and w e hav e that the in vertible sheav es L ( − l i − m j ) and L ( − l j − m i ) are isomorphic f or i, j = 0 , 1 , 2. There is another wa y to reco ver these generators and r elat io ns, whic h do es not dep end on the theory of toric v arieties. Recall that there is a morphism p 1 : S → P 2 , o btained b y blowing do wn the lines m 0 , m 1 , and m 2 . If x 0 , x 1 , x 2 are the ho mogeneous co o rdinates 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 equiv alen t divisors on P 2 , and th us 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 equiv alent divisors on S . Therefore the corresp onding in v ertible she a v es L ( − m 1 − l 0 − m 2 ), L ( − m 0 − l 1 − m 2 ), and L ( − m 0 − l 2 − m 1 ) on S a re isomorphic. F rom this w e can conclude that Pic( S ) is generated b y the in v ertible shea v es 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 w e hav e that L ( − l i − m j − l k ) and L ( − l j − m i − l k ) are isomorphic for a n y 14 MARK BLUNK i, j = 0 , 1 , 2 . This presen tation is equiv alen t to that in the previous para graph. Similarly , this pr esen tation can b e obtained by considering the morphism p 2 : S → P 2 obtained by blo wing down the lines l 0 l 1 , and l 2 . W e define the following lo cally free shea ves on 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 lo cally free shea v es I 1 ⊕ I 2 and J 1 ⊕ J 2 ⊕ J 3 , compatible with the action o n S . Therefore I 1 ⊕ I 2 and J 1 ⊕ J 2 ⊕ J 3 descend to sheav es I and J o n S . W e will consider the followin g endomorphism rings: B ′ = End O S ( I ) op , and Q ′ = End O S ( I ) op . As S is pro jectiv e, End O S ( O S ) op = F , and since I and J are O S -mo dules, it follo ws that B ′ and Q ′ are F - algebras. F o r i , j , and k not equal, End O S ( L ( − m i − l j − m k )) = End O S ( L ( − l i − m j − l k )) = F , so w e see that F 6 em b eds diagonally into End O S ( I 1 ⊕ I 2 ). Moreo ver, since L ( − l i − m j ) and L ( − l j − m i ) ar e isomorphic for an y i , j , I 1 ⊕ I 2 = ( L ( − m 1 − l 0 − m 2 ) ⊕ L ( − l 1 − m 0 − l 2 )) ⊗ F V , where V is an F -v ector space of dimension 3. An elemen t of Hom O S ( L ( − m 1 − l 0 − m 2 ) , L ( − l 1 − m 0 − l 2 )) is g iv en by a glo bal section of L ( m 2 − l 2 ). An y non-zero globa l section of L ( m 2 − l 2 ) w ould give a function defined on a neighborho o d of l 2 ⊂ S with v anishing set l 2 . Blo wing do wn the lines l i , this function w o uld then correspo nd to a function defined on an op en subset of P 2 with v anishing set a p o int, whic h is imp ossible, since a p oint is a co dimension 2 sub v ariety of P 2 . Th us L ( m 2 − l 2 ) has no nonzero global sections. Similarly Hom O S ( L ( − l 1 − m 0 − l 2 ) , L ( − m 1 − l 0 − m 2 )) = 0, and so End O S ( I 1 ⊕ I 2 ) = End O S ( L ( − m i − l j − m k ) × L ( − l i − m j − l k )) ⊗ F End F ( V ) = End F 2 ( V F 2 ). So the cen ter of End O S ( I 1 ⊕ I 2 ) is a cop y of F 2 , contained in F 6 . This c hain F 2 ⊂ F 6 ⊂ End O S ( I 1 ⊕ I 2 ) descends to K ⊂ K L ⊂ End O S ( I ), with K the cen ter of End O S ( I ). No w let E b e an y separable field extension o f F o v er whic h the lines l i , m j are defined. This is equiv alen t to E splitting b oth K and L . the ab ov e argumen ts sho w t hat End O S ( I ) ⊗ F E ≈ M 3 ( E 2 ), where E 2 ≈ K ⊗ F E . Therefore, w e conclude that B ′ is an Azumay a K -alg ebra of rank 9 whic h con t a ins K L as a subalgebra. A similar a rgumen t sho ws that Q ′ is an Azuma y a L -algebra of rank 4 whic h a lso con tains a copy of K L . Theorem 4.1. B ′ = End O S ( I ) op and B ar e isomorphic as K -algebr as. Sim i - larly, Q ′ = End O S ( J ) op and Q ar e isomorphi c as L -alg ebr as. Pr o of. Let e S b e as in Proposition 2.1, a nd let e I and e J b e the sheav es a sso ciated to e S as ab o v e. Twisting e I and e J b y the Γ-a ctio n corresponding to the pair DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 15 ( K , L ), w e get shea v es I 0 and J 0 asso ciated to the T -toric mo del S 0 . Let B ′ 0 = End O S ( I 0 ) op and Q ′ 0 = End O S ( J 0 ) op . The em b eddings of K L in to B ′ 0 and Q ′ 0 described ab ov e induce the fo llo wing commutativ e diagrams (cf. (3)): 1 / / R (1) K/F ( G m ) / /   G L / /   T / /   1 1 / / R K/F ( G m ) / / R K/F ( GL ( B ′ 0 )) / / R K/F ( PGL ( B ′ 0 )) / / 1 , and 1 / / R (1) L/F ( G m ) / /   G K / /   T / /   1 1 / / R L/F ( G m ) / / R L/F ( GL ( Q ′ 0 )) / / R L/F ( PGL ( Q ′ 0 )) / / 1 . The firs t diagram induce s the follo wing comm utativ e diagram of cohomolo g y sets: H 1 ( F , T ) / /   Ker(cor K/F : Br( K ) → Br( F ))   1 / / H 1 ( K , P GL ( B ′ 0 )) / / Br( K ) . The left v ertical arrow sends the triple ( B , Q, K L ) to t he endomorphism ring B ′ = End( I ) op , where I is the sheaf asso ciated to the surface S ( B , Q, K L ) constructed a b o v e. The upper horizon ta l arro w sends the triple ( B , Q, K L ) to the class of [ B ] ∈ Br( K ), whic h lands in the subgroup of elemen ts of tr ivial norm. The righ t v ertical map is the inclusion homomorphism. The lo w er horizon t a l arro w sends a K -a lg ebra to its corresp o nding elemen t in Br( K ). By the comm utativity of the diagram, we see that [ B ] = [ B ′ ] in Br( K ). But these a lg ebras hav e the same rank, so they mus t be isomorphic as K -a lgebras. A similar arg umen t show s that Q ′ = End O S ( J ) op and Q are isomorphic as L -algebras.  Let A b e the separable F -a lg ebra 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 ) b y M 1 7→ O S ⊗ F M 1 , u B from P ( B ) to P ( S ) b y M 2 7→ I ⊗ B M 2 , and u Q from P ( Q ) to P ( S ) b y M 3 7→ J ⊗ Q M 3 . If w e set P = O S ⊕ I ⊕ J , then the resp ectiv e righ t actions of F , B , and Q on O S , I , a nd J com bine to giv e a righ t action o f A = F × B × Q on P . Therefore, w e can define an exact functor from P ( A ) to P ( S ) b y 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 -v ariet y , then w e ha v e an exact functor f r o m 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 pro jection morphisms. This induces a 16 MARK BLUNK homomorphism φ Y : K 0 ( Y ; A ) → K 0 ( Y × S ). F urthermore, if E is an y field extension of F , then φ na t urally 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 isom orphism. W e will pro v e this in sev eral stages. Let us first consider t he case where F is separably closed. By Prop o sition 2.1, S is isomorphic t o the blo w up of the pro jectiv e plane at t he 3 non-collinear p oin ts [1 : 0 : 0 ], [0 : 1 : 0], and [0 : 0 : 1]. Recall tha t w e hav e the filtration 0 = K 0 ( S ) (3) ⊂ K 0 ( S ) (2) ⊂ K 0 ( S ) (1) ⊂ K 0 ( S ) (0) = K 0 ( S ) b y co dimension of support, and homomorphisms C H i ( S ) → K 0 ( S ) ( i/i +1) , whic h send the class of a subv ariety V to the equiv a- lence class [ O V ]. These homomorphisms are isomorphisms f or i = 0 , 1 , 2. So by Prop osition 2.1, if P is a rational p oint 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 ]. Moreov er, as CH 0 ( S ), CH 1 ( S ), and CH 2 ( S ) ar e free ab elian groups with ranks 1, 4, and 1, resp ective ly , K 0 ( S ) is free ab elian with rank 6. Since F is separably closed, K , L , B , and Q are split, and th us K 0 ( A ) is also free ab elian of rank 6. Therefore φ will b e an isomorphism pro vided it is s urjectiv e. So it s uffices 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 o f 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 - v ector space o f 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 ). No w Hom F 2 ( V F 2 , F × 0) is a righ t End O S ( I )-mo dule, and thus a left A - mo dule, where the F and Q comp onen t of A = F × B × Q act trivially . Therefore, φ  Hom F 2 ( V F 2 , F × 0)  = h I ⊗ B Hom F 2 ( V F 2 , F × 0) i = "  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 Mor it a equiv a lence in the second line. A mirror ar gumen t sho ws that φ  Hom F 2 ( V F 2 , 0 × F )  = [ L ( − l 1 − m 0 − l 2 )], and a similar argumen t applied to J a nd Q sho ws that [ L ( − l 0 − m 1 )], [ L ( − l 0 − m 2 )], and [ L ( − l 1 − m 2 )] are in the image of φ . No w let i, j ∈ { 0 , 1 , 2 } and not equal. By Prop o sition 2.1, the lines l i and m j ha ve in tersection a rat io nal p o int P of S , with m ultiplicit y 1, the lines m i and m j are sk ew, and the lines l i and l j are sk ew. Thus w e hav e the follo wing resolutions o f O P and O S : DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 17 0 → L ( − l i − m j ) 0 @ ⊗L ( m j ) ⊗L ( l i ) 1 A − − − − − − − − → L ( − l i ) ⊕L ( − m j ) ( ⊗L ( l i ) , −⊗L ( m j )) − − − − − − − − − − → O S → O P → 0 , 0 → L ( − m i − m j ) 0 @ ⊗L ( m i ) ⊗L ( m j ) 1 A − − − − − − − − → L ( − m i ) ⊕ L ( − m j ) ( ⊗L ( m i ) , −⊗L ( m j )) − − − − − − − − − − − → O S → 0 , and 0 → L ( − l i − l j ) 0 @ ⊗L ( l i ) ⊗L ( l j ) 1 A − − − − − − − → 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 ha ve 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 w e tak e k ∈ { 0 , 1 , 2 } not equal to i or j and m ultiply this equation by [ L ( − l k )], w e 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 argumen t with l and m in t erchanged sho ws that [ O m k ] is in the image of φ f o r k ∈ { 0 , 1 , 2 } . Fina lly , [ O P ] = [ O S ] − [ L ( − l 0 )] − [ L ( − m 1 )] + [ L ( − l 0 − m 1 )] is in the image of φ , and th us φ is surjectiv e when F is separably closed. Prop osition 4.3. φ : K 0 ( A ) → K 0 ( S ) is an iso m orphism if B and Q ar e split. Pr o of. By the preceding argument, φ F : K 0 ( A ) → K 0 ( S ) is an isomorphism. Moreo ver, φ F comm utes with the action of Γ on b o th K 0 ( A ) and K 0 ( S ), and th us it descends to a n isomorphism on the Γ-in v arian t subgroups. Therefore, w e hav e t he following comm uta t ive diagram: K 0 ( A )   φ / / K 0 ( S )   K 0 ( A ) Γ φ F / / K 0 ( S ) Γ . As φ F and the left v ertical map K 0 ( A ) → K 0 ( A ) Γ are isomorphisms, φ m ust b e injectiv e. Moreo v er, if the right v ertical map is injectiv e, then φ is surjectiv e, 18 MARK BLUNK and hence an isomorphism. So it suffices to sho w that K 0 ( S ) → K 0 ( S ) Γ is injectiv e. T o s ee this, note tha t the rank and w edge ho mo mor phisms rank : K 0 ( S ) → Z and ∧ : K 0 ( S ) (1) → Pic( S ) comm ute with the action of Γ. Th us w e hav e the follo wing short exact sequences of Γ-mo dules: 0 → K 0 ( S ) (2) → K 0 ( S ) (1) ∧ − → Pic( S ) → 0 and 0 → K 0 ( S ) (1) → K 0 ( S ) rank − − → Z → 0 . These sequences of Γ-mo dules induce the follow ing long exact sequences: 0 → ( K 0 ( S ) (2) ) Γ → ( K 0 ( S ) (1) ) Γ ∧ − → Pic( S ) Γ → H 1 ( F , K 0 ( S ) (2) ) and 0 → ( K 0 ( S ) (1) ) Γ → ( K 0 ( S )) Γ rank − − → Z → H 1 ( F , K 0 ( S ) (1) ) . The map K 0 ( S ) → K 0 ( S ) Γ induces the following comm utativ e diagrams: 0 / / K 0 ( S ) (2) / /   K 0 ( S ) (1) ∧ / /   Pic( S ) / /   0 0 / / ( K 0 ( S ) (2) ) Γ / / ( K 0 ( S ) (1) ) Γ ∧ / / Pic( S ) Γ / / H 1 ( F , K 0 ( S ) (2) ) , and 0 / / K 0 ( S ) (1) / /   K 0 ( S ) rank / /   Z / / =   0 0 / / ( K 0 ( S ) (1) ) Γ / / K 0 ( S ) Γ rank / / Z / / H 1 ( F , K 0 ( S ) (1) ) . As B and Q are split, S has a rational p o in t by Corollary 3.6. Th us the homomorphism K 0 ( S ) 2 → ( K 0 ( S ) (2) ) Γ is a surjectiv e homomorphism of free ab elian groups of rank 1, and therefore an isomorphism. Moreo ver, the homo- morphism Pic( S ) → Pic( S ) Γ is injectiv e. Thus, b y applying the Snake Lemma to the first diagram and then to the second, w e see that K 0 ( S ) → ( K 0 ( S )) Γ is injectiv e.  Remark 4.4. If P and P ′ are rational p oints of S , they define equal classes in CH 2 ( S ) by Prop osition 2.1, a nd hence [ O P ] = [ O P ′ ] in K 0 ( S ) (2) . So K 0 ( S ) (2) is generated b y the Γ-inv aria nt elemen t [ O P ], and th us is a trivial Γ-mo dule. Similarly , our arguments a b ov e sho w that t he se t { [ 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-inv aria n t ba - sis of K 0 ( S ) (1) , and thus K 0 ( S ) (1) is a p erm utation mo dule. It follows that H 1 ( F , K 0 ( S ) (2) ) = H 1 ( F , K 0 ( S ) (1) ) = 0 . W e will need the follow ing pr o p osition (cf. Prop osition 6.1 o f [6]). DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 19 Prop osition 4.5. I f Y is a variety such t hat t he homomorphism φ F ( y ) : K 0 ( A F ( y ) ) → K 0 ( S F ( y ) ) is an isomo rphism for every y ∈ Y , then φ Y : K 0 ( Y ; A ) → K 0 ( S × Y ) is surje ctive. Pr o of. W e do this b y double induction on the dimension of Y and the n umber of irreducible comp o nents of Y . If Y has a proper irreducib le comp onen t Y ′ , with comple men t U , w e hav e the following lo calization exact sequence (cf. [8 ]) : K 0 ( Y ′ ; A ) / / φ Y ′   K 0 ( Y ; A ) / / φ Y   K 0 ( U ; A ) / / φ U   0 K 0 ( Y ′ × S ) / / K 0 ( Y × S ) / / K 0 ( U × S ) / / 0 By our inductiv e assumption, the v ertical maps on the righ t and on the left are surjectiv e. This implies that the middle vertic al map is surjectiv e as well. So w e ma y assume that Y is ir r educible. If Y is not reduced, then w e the natural Y red → Y induces the comm utativ e diagram K 0 ( Y red ; A ) / / φ Y red   K 0 ( Y ; A ) φ Y   K 0 ( Y red × S ) / / K 0 ( Y × S ) , where the horizon tal arrow s are isomorphisms. Th us w e ma y also assume tha t Y is reduced. No w 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 op en set U in Y s uc h the image of x in K 0 ( U × S ) is in the image of φ U . W e again consider the lo calization exact sequence : K 0 ( Z ; A ) / / φ Z   K 0 ( Y ; A ) / / φ Y   K 0 ( U ; A ) / / φ U   0 K 0 ( Z × S ) / / K 0 ( Y × S ) / / K 0 ( U × S ) / / 0 , where Z is the complemen t o f U in Y . By assumption, Z has a strictly smaller dimension than Y , and so b y our inductiv e hypothesis, φ Z is surjectiv e. A standard diag r am chase sho ws that x ∈ Im( φ Y ).  Pr o of of The or em 4.2. Let S B ( B ) b e the Sev eri-Brauer K -v ariet y ass o ciated to B , S B ( Q ) b e the Se v eri-Bra uer L -v ariet y asso ciated to Q , and Y = R K/F ( S B ( B )) × R L/F ( S B ( Q )) b e the pro duct of the restriction o f scalars of b oth v arieties. Then for an y field extension E of F , Y ( E ) is no nempty if and only if S B ( B )( K ⊗ F E ) and S B ( 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. 20 MARK BLUNK The pro jection p : Y → Sp ec( F ) induce the following diagram: K 0 ( A ) φ / / p ∗   K 0 ( S ) ( p Y ) ∗   K 0 ( Y ; A ) φ Y / / p ∗ O O K 0 ( Y × S ) , ( p Y ) ∗ O O where p Y : Y × S → S is the pro jection induced b y p . Bo th sq uares comm ute, and p ∗ p ∗ is the iden tity homomorphism , as Y is a geometrically rational v a- riet y . F or ev ery y ∈ Y , Y ( F ( y )) 6 = ∅ , 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 b y Prop osition 4.3. So b y Prop osition 4 .5, φ Y : K 0 ( Y ; A ) → K 0 ( Y × S ) is surjectiv e. A dia g ram c hase sho ws the top horizontal map φ is also surjectiv e. No w let E b e an y field extension of F suc h that S ( E ) 6 = ∅ . Then, B E and Q E are split, and so φ E : K 0 ( A E ) → K 0 ( S E ) is an isomorphism, again b y Prop o- sition 4.3. The homomorphisms φ a nd φ E fit into the follow ing commutativ e diagram: K 0 ( A ) φ / /   K 0 ( S )   K 0 ( A E ) φ E / / K 0 ( S E ) , where the v ertical homomorphisms are induced b y the inclusion F ⊂ E . The b ottom horizon tal map is an isomorphism, a nd the left v ertical map is injectiv e. It follows that φ is injective , and hence an isomorphism.  As φ is an isomorphism, the h yp othesis on the v ariet y V in Proposition 4.5 is alwa ys true, we obtain the following corollary . Corollary 4.6. φ V : K 0 ( V ; A ) → K 0 ( V × S ) is surje ctive for any F -variety V .  5. Highe r K-theor y W e ha ve shown that the K 0 groups of S and A coincide. W e will sho w that this is also true for t he higher Quillen K -gro ups. F or a ny F - v arieties X , Y , and Z , and separable F - algebras A , B , a nd 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 pro j ections of X × Y × Z on to its factors. This functor is bi-exact, and th us induces a pro duct 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 ) . W e will denote the image of u ⊗ x under this map by u • B x . W e recall t he K -Motivic Category C and some of its prop erties. The details can b e found in [6 ] and [7]. Ob jects of C are pairs ( X , A ), where X is an DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 21 F -v ariety and A is a separable F -algebra. F or t wo pairs ( X , A ) and ( Y , B ) in C , w e set Mor C (( X , A ) , ( Y , B )) := K 0 ( X × Y ; A op ⊗ F B ). The comp osition la w is g ◦ f = g • B f , fo r f : ( X , A ) → ( Y , B ) and g : ( Y , B ) → ( Z , C ) in C . F or any pair ( X , A ) with X smo oth, the iden tity elemen t 1 ( X,A ) ∈ K 0 ( X × X ; A op ⊗ F A ) is the elemen t [ O ∆ ⊗ F A ], where ∆ ⊂ X × X is the diagonal. F or an y F -v ariety X and any separable F - algebra A , w e will write X for the pair ( X , F ) and A for the pair ( Sp ec F , A ). Finally , for an y F -v ariet y V and an y nonnegative in teger n , w e ha ve a realization functor K V n , whic h sends an ob ject ( X , A ) to K n ( V × X ; A ), and K V n ( 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 ). W e will denote K Spec F n b y K n . As w e mentioned in the beginning o f Section 4, there is a left action of A op = End O S ( O S ) × End O S ( I ) × End O S ( J ) o n the lo cally free sheaf P = O S ⊕ I ⊕ J . So P ∈ P ( X ; A op ). Th e corresp onding elemen t [ P ] ∈ K 0 ( S ; A op ) defines a morphism u : A → S in C . It follo ws from the construction of the realization functor tha t φ V = K V 0 ( u ) for an y V . In particular, K 0 ( u ) = φ . Theorem 5.1. u : A → S is an iso m orphism in C . Pr o of. Let V b e an y F -v ariet y . Equating K 0 ( V ; A ) (resp. K 0 ( V × S )) with Mor C ( V , A ) (resp. Mor C ( V , S )), K V 0 ( u ) : K 0 ( V ; A ) → K 0 ( V × S ) is just p ost- comp osition in C with u . By Corollary 4.6, K V 0 ( u ) = φ V is surjectiv e f o r any v ariet y V . In particular, if V = S , there is an elemen t v ∈ K 0 ( S ; A ) suc h that uv = [ O ∆ ], i.e. u has a right in ve rse v in C . W e w a n t to sho w that v is also a left in v erse to u in C , i.e. v u = [ A ] ∈ K 0 ( A op ⊗ F A ). As K 0 ( A op ⊗ F A ) ֒ → K 0 (( A op ⊗ F A ) F ) = K 0 ( A op F ⊗ F A F ), it suffices to consider the case wh ere K , L , B and Q are split. So A = F × M 3 ( F × F ) × M 2 ( F × F × F ) , and th us 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 ). Moreo ver, under this isomorphism [ A ] ∈ K 0 ( A op ⊗ F A ) corresp onds to the iden tit y matrix. So v u ∈ K 0 ( A op ⊗ F A ) is repres en ted by a matrix M with in teger en tries. It follo ws that the corresp onding homomorphism K 0 ( v u ) fro m K 0 ( A ) ∼ = Z 6 to itself is multiplication by this matrix M . No w, as v is a rig h t in v erse to u in C , K 0 ( v ) is a rig h t in v erse to K 0 ( u ). Ho wev er, K 0 ( u ) = φ is an isomorphism by The orem 4.2, so in fact K 0 ( v ) = K 0 ( u ) − 1 . Th us K 0 ( v u ) = K 0 ( v ) K 0 ( u ) = id K 0 ( A ) , whic h forces M to b e the iden t ity mat rix. Th us v u = [ A ] ∈ K 0 ( A op ⊗ A ), i.e. v u = id A in C .  Corollary 5.2. F or any in te ger n , any c en tr al simple F -algebr a D , and any F -variety V , K n ( V ; A ⊗ F D ) ∼ = K n ( V × S ; D ) . In p a rticular, K n ( F ) ⊕ K n ( B ) ⊕ K n ( Q ) = K n ( A ) ∼ = K n ( S ) . Pr o of. F or an y cen tral simple F -algebra D , Morita Equiv alence g ives a natural isomorphism K 0 ( S ; A op ⊗ F D op ⊗ F D ) = K 0 ( S ; A op ). Th us the isomorphism u : 22 MARK BLUNK A → S in C also defines an isomorphism fro m A ⊗ F D to ( S, D ) in C . Applying the realization functor K V n yields K n ( V ; A ⊗ F D ) ∼ = K n ( V × S ; D ) .  W e conclude the paper with an Index Reduction F ormu la for t he function field o f the surface S ( B , Q, K L ). W e will need the following lemma: Lemma 5.3 ([10]) . L et X b e a irr e ducible F -variety, and D a c e n tr al simple F -algeb r a. The r estriction homomorphism K 0 ( X ; D ) → K 0 ( D F ( X ) ) induc e d by the inclusion Sp ec( F ( X )) → X is surje ctive. Lemma 5.4. L et X b e an irr e ducible F -variety, and D a c en tr al simple F - algebr a. ind D F ( X ) = 1 deg D g . c . d . { rank( P ) , ∀ P ∈ P ( X ; D ) } . Pr o of. W e recall that for any field E and an y cen tral simple E -alg ebra D ′ , K 0 ( D ′ ) is cyclic, generated by the class of a simple D ′ -mo dule M ′ . Moreov er, dim E ( M ′ ) = deg ( D ′ ) ind( D ′ ). The rank homomo r phism rank : K 0 ( X ; D ) → K 0 ( F ( X )) has the follow ing decomp osition: K 0 ( X ; D ) → K 0 ( D F ( X ) ) → K 0 ( F ( X )) , where the first map is induced by the inclusion Sp ec( F ( X )) → X , and the second map takes the class of a D F ( X ) -mo dule to the class of the corresp onding F ( X )- v ector space. As K 0 ( F ( X )) is cyclic, the image of the rank homomorphism is n [ F ( X )], where n is the greatest common divis or of the num b ers rank( P ), fo r all P ∈ P ( X ; D ). By the previous lemma, the homomorphism K 0 ( X ; D ) → K 0 ( D F ( X ) ) is surjectiv e. Thus if M is a simple D F ( X ) -mo dule, n = dim F ( X ) ( M ) = deg ( D F ( X ) ) ind( D F ( X ) ) = deg ( D ) ind( D F ( X ) ) , and the result follow s.  Corollary 5.5 (Index Reduction F ormula) . L et S = S ( B , Q, K L ) b e a del Pezzo surfac e of de gr e e 6. F or any c entr al simple F -alge b r a D , ind D F ( S ) is e qual to: i. g . c . d . { ind( D ) , 2 ind( D ⊗ F B ) , 3 ind( D ⊗ F Q ) } , if K and L ar e fields. ii. g . c . d . { ind( D ) , ind( D ⊗ F B 1 ) , ind( D ⊗ F B 2 ) } , if K = F × F and L i s a field. Her e 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 . Her e 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 . Her e Q = Q 1 × Q 2 × Q 3 . v. ind D , whe n K and L ar e not fi e lds. Remark 5.6. In case ii., Q = M 2 ( L ) is necessarily split, a s 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 w e remo v e the term DEL PEZZO SUR F ACES OF DEGREE 6 OVER AN A RBITRAR Y FIELD 23 3 ind( D ⊗ F Q ). Similarly , in cases iii., iv., and v., w e can remov e the term with the split B o r Q when computing greatest common divisors. Pr o of. 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 ). Moreo v er, as A = F × B × Q , K 0 ( A ⊗ F D ) ∼ = K 0 ( D ) ⊕ K 0 ( B ⊗ F D ) ⊕ K 0 ( Q ⊗ F D ). W e will consider the case where K and L are fields. The pro of of the other cases are similar. As D , B ⊗ F D , a nd Q ⊗ F D a re central simple algebras (with cen ters F , K , and L , resp ectiv ely), their K 0 groups are cyclic, generated b y the class of a simple mo dule. Therefore b y 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 mo dules under the image of K 0 ( u ) : K 0 ( A ⊗ F D ) → K 0 ( S ; D ). So let M B b e a simple B ⊗ F D -mo dule. Then dim K ( M B ) = deg( B ⊗ F D ) ind( B ⊗ F D ), and th us rank( K 0 ( u )( M B )) = rank( M B ⊗ B I ) = dim F ( M B ) r a nk( I ) dim F ( B ) = dim K ( M B ) r a nk( I ) dim K ( B ) = 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 -mo dule (res p. D -mo dule), 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 follow s.  Reference s [1] J.- L. Collio t-Th´ el` ene, N. Karp enko, and A. Merk ur jev, R ational Sur fac es and Canonic al Dimension of PGL 6 , Algebra i Analiz, 19 (2007 ), no. 5, 1 59–1 78. [2] Cor n, Patrick, Del Pezzo surfac es of de gr e e 6 , Ma th. 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