On the equations for universal torsors over del Pezzo surfaces
aa r X i v : . [ m a t h . AG ] J un On the equations for universal torsors overdel Pezzo surfaces
Vera Serganova and Alexei SkorobogatovNovember 10, 2018 `a Jean-Louis Colliot-Th´el`ene
Introduction
Universal torsors were invented by Jean-Louis Colliot-Th´el`ene and Jean-JacquesSansuc; for smooth projective varieties X with H ( X, O ) = 0 they play the rolesimilar to that of n -coverings of elliptic curves. The foundations of the theory ofdescent on torsors were laid in a series of notes in Comptes Rendus de l’Acad´emiedes Sciences de Paris in the second half of the 1970’s, and a detailed account waspublished in [4]. The theory has strong number theoretic applications if the torsorscan be described by explicit equations, and if the resulting system of equations canbe treated using some other methods, whether algebraic or analytic. Such is thecase for surfaces fibred into conics over P : the universal torsors over these surfacesare closely related to complete intersections of quadrics of a rather special kind. Todescribe them we use the following terminology. If Z ⊂ A mk is a closed subset ofan affine space with a coordinate system over a field k , then the variety obtainedfrom Z by multiplying coordinates by non-zero numbers will be called a dilatationof Z . If exactly n geometric fibres of the conic bundle X → P are singular, thenthere is a non-degenerate quadric Q ⊂ A nk such that the universal torsors over X are stably birationally equivalent to the product of a complete intersection of n − Q , and a Severi–Brauer variety (see [4], Thm. 2.6.1). This descriptionwas key for a plethora of applications of the descent theory to the Hasse princi-ple, weak approximation, zero-cycles, R-equivalence and rationality problems (theZariski conjecture), see, e.g., [6] and [2]. A stumbling block for a similar treatmentof cubic and more general smooth del Pezzo surfaces without a pencil of rationalcurves is, possibly, the absence of a satisfactory presentation of their universal tor-sors. Known descriptions of universal torsors over diagonal cubic surfaces ([4], 2.5,[5], 10) lack the simplicity and the symmetry manifest in the conic bundle case.1ne way to look at a non-degenerate quadric in A n is to think of it as a homo-geneous space of the semisimple Lie group G associated with the root system D n which naturally appears in connection with conic bundles with n singular fibres, seee.g. [9]. Indeed, over an algebraically closed field we can identify Q with the orbitof the highest weight vector of the fundamental 2 n -dimensional representation V of G . Then an ‘essential part’ of the torsor is the intersection of n − V ).The aim of this paper is to generalize this description from the case of conicbundles to that of del Pezzo surfaces. (Recall that these two families exhaustall minimal smooth projective rational surfaces, according to the classification ofEnriques–Manin–Iskovskih.) We build on the results of our previous paper [13],where we studied split del Pezzo surfaces, i.e. the case when the Galois action onthe set of exceptional curves is trivial. The main result of [13] is a construction of anembedding of a universal torsor over a split del Pezzo surface X of degree 5, 4, 3 or2 into the orbit of the highest weight vector of a fundamental representation of thesemisimple simply connected Lie group G which has the same root system as X , i.e.A , D , E or E . This orbit is the punctured affine cone over G/P , where P ⊂ G is a maximal parabolic subgroup. The embedding is equivariant with respect to theaction of the N´eron–Severi torus T of X , identified with a split maximal torus of G extended by G m . In Theorem 2.5 we describe universal torsors over split del Pezzosurfaces of degree d as intersections of 6 − d dilatations of the affine cone over G/P by k -points of a maximal torus of GL( V ) which is the centralizer of T ⊂ GL( V ).This gives a more conceptual approach to the equations appeared previously in thework of Popov [11] and Derenthal [7]. This approach can be called a global descrip-tion of torsors compared to their local description obtained by Colliot-Th´el`ene andSansuc in [4], 2.3.For a general del Pezzo surface X of degree 4, 3 or 2 with a rational point weconstruct an embedding of a universal torsor over X into the same homogeneousspace as in the split case, but this time equivariantly with respect to the actionof a (possibly, non-split) maximal torus of G , see Theorem 4.4. The case of delPezzo surfaces of degree 5, where a rational point comes for free by a theorem ofEnriques and Swinnerton-Dyer, was already known ([15], Thm. 3.1.4). The proofof Theorem 4.4 uses a recent result of Philippe Gille [8] and M.S. Raghunathan [12]which describes possible Galois actions on the character group of a maximal torusin a quasi-split algebraic group. This result implies that the N´eron–Severi torus T of X embeds into the same split group G extended by G m , exactly as in the case ofa split del Pezzo surface.The condition on the existence of a rational point on X is not a restriction inthe case of degree 5, but is clearly a restriction for smaller degrees, limiting thescope of possible applications. However, if X is a del Pezzo surface of degree 4, thiscondition is necessary as well as sufficient for our construction: if X can be realized2nside a twisted form of the quotient of G/P by a maximal torus, then X has arational point, see Corollary 4.5 (i). Finally, in Corollary 4.5 (ii) we show that anydel Pezzo surface of degree 4 with a k -point has a universal torsor which is a denseopen subset of the intersection of the affine cone over G/P with its dilatation by a k -point of the centralizer of T in GL( V ).We recall the construction of [13] in Section 1 alongside with all necessary notation.In Section 2 we describe torsors over split del Pezzo surfaces as intersections ofdilatations of the affine cone over G/P . In Section 3 we prove a uniqueness propertyused in the proof of the main results in the non-split case in Section 4.The ideas developed in this paper originate in the second author’s discussionswith Victor Batyrev, to whom we are deeply grateful.
Preliminary remarks
Let k be a field of characteristic 0 with an algebraic closure k .Let V be a vector space over k , and let T ⊂ GL( V ) be a split torus, i.e. T ≃ G nm for some n . Let Λ ⊂ ˆ T be the set of weights of T in V , and let V λ ⊂ V be thesubspace of weight λ . We have V = ⊕ λ ∈ Λ V λ . Let S be the centralizer of T in GL( V ),i.e. S = Y λ ∈ Λ GL( V λ ) ⊂ GL( V ) . In what follows we always assume that dim V λ = 1 for all λ ∈ Λ; then S is a maximaltorus in GL( V ). Let π λ : V → V λ be the natural projection. For A ⊂ V we write A × for the set of points of A outside ∪ π − λ (0).Let r = 4 , , X of degree d = 9 − r is theblowing-up of P in r k -points in general position (i.e., no three points are on a lineand no six are on a conic). The Picard group Pic X is a free abelian group of rank r + 1, generated by the classes of exceptional curves on X . Let T = G r +1 m . Oncean isomorphism ˆ T ˜ −→ Pic X is fixed, T is called the N´eron–Severi torus of X . Auniversal torsor f : T → X is an X -torsor with structure group T , whose type isthe isomorphism ˆ T ≃ Pic X (see [15], p. 25). We call a divisor in T an exceptionaldivisor if it is the inverse image of an exceptional curve in X .Now suppose that dim V equals the number of exceptional curves on X . We canmake an obvious but useful observation. Lemma 1.1
Let
T → X be a universal torsor over a split del Pezzo surface X .Let φ and ψ be T -equivariant embeddings T → V such that for each weight λ ∈ Λ the divisors of functions π λ φ and π λ ψ are equal to the same exceptional divisor withmultiplicity . Then ψ = s ◦ φ for some s ∈ S ( k ) .Proof Since T is a universal torsor we have k [ T ] ∗ = k ∗ , hence two regular functionswith equal divisors differ by a non-zero multiplicative constant. QED3 onstruction in the split case Let the pair consisting of a root system R of rank r and a simple root α be one of the pairs in the list(A , α ) , (D , α ) , (E , α ) , (E , α ) . (1)Here and elsewhere in this paper we enumerate roots as in [3]. Let G be the splitsimply connected simple group with split maximal torus H and root system R. Let ω be the fundamental weight dual to α , and let V = V ( ω ) be the irreducible G -module with the highest weight ω . It is known that V is faithful and minuscule, see[3]. Let P ⊂ G be the maximal parabolic subgroup such that G/P ⊂ P ( V ) is theorbit of the highest weight vector. The affine cone over G/P is denoted by (
G/P ) a .It is easy to check that the G -module S ( V ) is the direct sum of two irreduciblesubmodules V ( ω ) ⊕ V (2 ω ). For r ≤ V ( ω ) is a non-trivial irreducible G -moduleof least dimension; it is a minuscule representation of G . If r = 7, then V ( ω ) is theadjoint representation; it is quasi-minuscule, that is, all the non-zero weights havemultiplicity 1 and form one orbit of W . If pr is the natural projection V → V ( ω ),and Ver : V → S ( V ) is the Veronese map x x , then it is well known that( G/P ) a = ( pr ◦ Ver) − (0) (see [1], Prop. 4.2 and references there).Since the eigenspaces of H in V are 1-dimensional, V has a natural coordinatesystem with respect to which S is the ‘diagonal’ torus. Let the torus T ⊂ S be theextension of H by the scalar matrices G m ⊂ GL( V ). Note that an eigenspace of H in V is also an eigenspace of T , so that there is a natural bijection between thecorresponding sets of characters.Let V sf be the dense open subset of V consisting of the points whose H -orbits areclosed and whose stabilizers in T are trivial. Let ( G/P ) sf a = ( G/P ) a ∩ V sf . In [13]we constructed a T -equivariant closed embedding of T into ( G/P ) sf a such that eachweight hyperplane section T ∩ π − λ (0) is an exceptional divisor with multiplicity 1.Then X × = f ( T × ) is the complement to the union of exceptional curves on X .We need to recall the details of this construction. It starts with the case (R , α ) =(A , α ) where the torsor T is the set of stable points of ( G/P ) a which is the affinecone over the Grassmannian Gr(2 , T is open and dense in( G/P ) a in this case. As in [13] we use dashes to denote the previous pair in (1);the previous pair of (A , α ) is (A × A , α (1)1 + α (2)2 ), though it will not be used.For r ≥ T ′ ⊂ ( G ′ /P ′ ) sf a over a split del Pezzo surface ofdegree 10 − r is already constructed, and proceed to construct T as follows.Let Λ n ⊂ Λ be the set of weights λ such that n is the coefficient of α in thedecomposition of ω − λ into a linear combination of simple roots. Let V n = ⊕ λ ∈ Λ n V λ ,then V = M n ≥ V n . (2)The subspaces V n are G ′ -invariant. In fact, V n = 0 for n > V = V ⊕ V ⊕ V ⊕ V , V = 0 unless r = 7. The degree 0 component V ≃ k is the highest weightsubspace, and the degree 1 component V is isomorphic to V ′ as a G ′ -module. The G ′ -module V is irreducible with highest weight ω . For r = 7 we have V ≃ k .Recall from [13] that g t = ( t, , t − , t − ) is an element of T , for any t ∈ k ∗ . Let U ⊂ ( G/P ) a be the set of points of ( G/P ) a outside ( V ⊕ V ) ∪ ( V ⊕ V ). The naturalprojection π : V → V defines a morphism U → V \ { } which is the compositionof a torsor under G m = { g t | t ∈ k ∗ } and the morphism inverse to the blowing-up of( G ′ /P ′ ) a \ { } in V \ { } ([13], Cor. 4.2). There is a G ′ -equivariant affine morphismexp : V → ( G/P ) a such that π ◦ exp = id, and the affine cone over exp( V ) isdense in ( G/P ) a . As in [13] we write exp( x ) = (1 , x, p ( x ) , q ( x )). The map p can beidentified, up to a non-zero constant, with the composition of the second Veronesemap and the natural projection S ( V ) → V , so that ( G ′ /P ′ ) a = p − (0). Since V isthe direct sum of 1-dimensional weight spaces, it has a natural coordinate system.The weight coordinates of p ( x ) will be written as p µ ( x ), where µ ∈ W ω .The choice of a point in V × defines an isomorphism V × ≃ S compatible with theaction of S . Using this isomorphism we define a multiplication on V × , and thenextend it to V . However, none of our formulae will depend of this isomorphism.Suppose that T ′ ⊂ ( G ′ /P ′ ) a ⊂ V ′ = V is such that f ′ : T ′ → X ′ = T ′ /T ′ is auniversal torsor over a del Pezzo surface X ′ , moreover, the T ′ -invariant hyperplanesections of T ′ are the exceptional divisors. In [13] we proved that for any k -point x in T ′× there exists a non-empty open subset Ω( x ) ⊂ ( G ′ /P ′ ) × a , whose definitionis recalled in the beginning of the next section, such that for any y in Ω( x ) theorbit T ′ y is the scheme-theoretic intersection x − y T ′ ∩ ( G ′ /P ′ ) a (see Cor. 6.5 of[13]). Therefore, if T is the proper transform of x − y T ′ in U , then X = T /T is theblowing-up of X ′ at the image of x . Consequently one proves that T ⊂ ( G/P ) sf a .Equivalently, T can be defined as the affine cone (without zero) over the Zariskiclosure of exp( x − y T ′ \ T ′ y ) in ( G/P ) sf a .The construction of an embedding of a universal torsor over X into ( G/P ) sf a is amain result of [13] (Thm. 6.1). The following corollary to this theorem complementsit by showing that our embedding is in a sense unique. Corollary 1.2
Let
T ⊂ V sf be a closed T -invariant subvariety such that T /T isa split del Pezzo surface and the weight hyperplane sections of T are exceptionaldivisors with multiplicity . Then for some s ∈ S ( k ) the torsor s T is a subset of ( G/P ) a obtained by our construction (for some choice of a basis of simple roots ofour root system R ).Proof The construction of [13] recalled above produces a universal torsor ˜ T overthe same split del Pezzo surface X inside ( G/P ) a , satisfying the condition that theweight hyperplane sections are the exceptional divisors with multiplicity 1. Theidentifications of the exceptional curves on X with the weights of V coming from5 and ˜ T may be different, however the permutation that links them is an auto-morphism of the incidence graph of the exceptional curves on X . It is well known(see [10]) that the group of automorphisms of this graph is the Weyl group W ofR. Thus replacing ˜ T by its image under the action of an appropriate element of W (that is, a representative of this element in the normalizer of H in G ), we ensurethat the identification of the weights with the exceptional curves is the same forboth embeddings. (The choice of this element in W is equivalent to the choice ofa basis of simple roots in our construction.) The multiplicity 1 condition in theconstruction of [13] is easily checked by induction from the case r = 4 where weconsider the Pl¨ucker coordinate hyperplane sections of Gr(2 , µ ∈ W ω ⊂ ˆ H we write S µ ( V ) for the H -eigenspace of S ( V ) of weight µ , and S µ ( V ) ∗ for the dual space. Let Ver µ be theVeronese map V → S ( V ) followed by the projection to S µ ( V ). For r = 6 we write S ( V ) for the zero weight H -eigenspace in S ( V ), and Ver : V → S ( V ) for thecorresponding natural map.As in the previous corollary, we denote by T ⊂ V sf a closed T -invariant subvarietysuch that T /T is a split del Pezzo surface and the weight hyperplane sections of T are exceptional divisors with multiplicity 1. Let I ⊂ k [ V ∗ ] be the ideal of T , I µ = I ∩ S µ ( V ) ∗ , and, for r = 6, let I = I ∩ S ( V ) ∗ .Let ˜ µ be the character by which T acts on S µ ( V ). The T -invariant hypersurfacein T cut by the zeros of a form from S µ ( V ) ∗ \ I µ is mapped by f : T → X to a conicon X . The class of this conic in Pic X , up to sign, is ˜ µ ∈ ˆ T under the isomorphismˆ T ≃ Pic X given by the type of the torsor f : T → X (see the comments beforeProp. 6.2 in [13]). All conics on X in a given class are obtained in this way; theyform a 2-dimensional linear system, hence the codimension of I µ in S µ ( V ) ∗ is 2.Let I ⊥ µ ⊂ S µ ( V ) be the 2-dimensional zero set of I µ . The corresponding projectivesystem defines a morphism f µ : X → P = P ( I ⊥ µ ) whose fibres are the conics of theclass ˜ µ . The link between Ver µ and f µ is described in the following commutativediagram: V ⊃ ( G/P ) a ⊃ T → X Ver µ y y y f µ y S µ ( V ) ⊃ p ⊥ µ ⊃ I ⊥ µ \ { } → P (3)Here p ⊥ µ is the zero set of p µ ∈ S µ ( V ) ∗ . Lemma 1.3
For µ ∈ W ω the vertical maps in ( ) are surjective, and dim S µ ( V ) = r − .Proof For the two right hand maps the statement is clear. The map V → S µ ( V )is surjective because all eigenspaces of T in V are 1-dimensional. Since dim S µ ( V )6oes not change if we replace µ by wµ for any w ∈ W , to calculate dim S µ ( V ) wecan assume that µ = ω . But ω is a weight of H ′ in V , so we have S ω ( V ) = S ω ( V ) ⊕ ( V ) ω , where dim ( V ) ω = 1. Starting with the case of the Pl¨uckercoordinates for r = 4, one shows by induction that dim S ω ( V ) = r −
1. Hencedim S µ ( V ) = r − p ⊥ µ = r − µ .To compute Ver µ (( G/P ) a ) we can continue to assume that µ = ω . If x ∈ V ,then Ver ω sends exp( x ) ∈ ( G/P ) a to Ver ω ( x ) + p ω ( x ). Thus the projection ofVer ω (( G/P ) a ) to S ω ( V ) = Ver ω ( V ), which is a vector space of dimension r − ω maps ( G/P ) a surjectively onto p ⊥ ω . QEDArguing by induction as in this proof, it is easy to show starting with the case ofthe Pl¨ucker coordinates on Gr(2 , p µ ( x ) is the sum of all the monomials ofweight µ with non-zero coefficients. Unless stated otherwise we assume that r ≥
5, so that G ′ is of type A , D or E .Recall that we use dashes to denote objects related to the ‘previous’ root system.Let x be a k -point of T ′× . We define the dense open subset Ω( x ) ⊂ ( G ′ /P ′ ) × a as the set of k -points x such that exp( x − x T ′× ) is not contained in V \ V × , thatis, in the union of weight hyperplanes of V . For r = 5 or 6 the set Ω( x ) is thecomplement to the union of the closed subsets Z µ ( x ) = { x ∈ ( G ′ /P ′ ) a | p µ ( x − xu ) ∈ I ′ µ } for all weights µ of V ; for r = 7 one also removes the closed subset Z ( x ) = { x ∈ ( G ′ /P ′ ) a | q ( x − xu ) ∈ I ′ } . The condition y ∈ Ω( x ) implies that for all µ the vectors Ver µ ( x ) and Ver µ ( y ) arenot proportional. Since dim S µ ( V ) ∗ = 2+dim I ′ µ , we see that for any y ∈ T ′ ∩ Ω( x )the subspace I ′ µ ⊂ S µ ( V ) ∗ consists of the forms vanishing at x and y . The ideal I ′ is generated by the I ′ µ , so T ′ is uniquely determined by any two of its pointssatisfying a certain open condition.Recall that π : V → V is the natural projection, cf. (2). Lemma 2.1
Let x be a k -point of T ′× , let y be a k -point of Ω( x ) ∩ T ′ , and let T ⊂ ( G/P ) sf a be the torsor defined by the triple ( T ′ , x , y ) as described above. Thenwe have the following statements. (i) The closed set Z µ ( x ) ⊂ ( G ′ /P ′ ) a consists of the points x ∈ ( G ′ /P ′ ) a ( k ) suchthat p µ ( x − y x ) = 0 . For r = 7 the closed set Z ( x ) consists of the points x ∈ ( G ′ /P ′ ) a ( k ) such that q ( x − y x ) = 0 . The open set Ω( x ) ∩ T ′ is the inverse image of the complement to all ex-ceptional curves on X ′ and to all conics on X ′ passing through f ′ ( x ) . For r = 7 one also removes from the cubic surface X ′ ⊂ P the nodal curve cut by the tangentplane to X ′ at f ′ ( x ) . We have T × = π − (Ω( x ) ∩ T ′ ) . (iii) We have t = exp( x − y ) ∈ T × .Proof (i) The inclusion of Z µ ( x ) into the hypersurface given by p µ ( x − y x ) = 0is clear: assigning the variable u the value y ∈ T ′ we see that p µ ( x − xu ) ∈ I ′ µ implies that p µ ( x − xy ) = 0. Conversely, let us prove that every point x of ( G ′ /P ′ ) a satisfying the condition p µ ( x − y x ) = 0, is in Z µ ( x ). Using Lemma 1.3 we see thatthe set of quadratic forms p µ ( x − yu ) on V for a fixed x and arbitrary y ∈ ( G ′ /P ′ ) a is a vector subspace L ⊂ S µ ( V ) ∗ of codimension 1, in fact this is the space of formsvanishing at x . As was pointed out before the statement of the lemma, I ′ µ is thesubspace of L of codimension 1 consisting of the forms vanishing at y . This provesthe desired inclusion.Now let r = 7. The inclusion of Z ( x ) into the hypersurface q ( x − y x ) = 0is clear for the same reason as above. Conversely, let x ∈ ( G ′ /P ′ ) a ( k ) be suchthat q ( x − y x ) = 0. We need to prove that q ( x − xu ) vanishes for any k -point u of T ′ . In the end of the proof of Prop. 6.2 of [13] we showed that the dual spaceH ( X ′ , O ( − K X ′ )) ∗ is a 4-dimensional vector subspace of S ( V ), so that we have acommutative diagram similar to (3): V ⊃ T ′ → X ′ Ver y y ϕ y S ( V ) ⊃ H ( X ′ , O ( − K X ′ )) ∗ \ { } → P (H ( X ′ , O ( − K X ′ )) ∗ )where ϕ is the anticanonical embedding X ′ ֒ → P . In loc. cit. we also showed thatfor any x ∈ ( G ′ /P ′ ) a ( k ) the cubic form q ( x − xu ), considered as a linear form on S ( V ), vanishes on the tangent space T x ≃ P to ϕ ( X ′ ) ⊂ P at ϕf ′ ( x ). It is thusobvious that if q ( x − xu ) vanishes at any point of ϕ ( X ′ ) outside of T x , then q ( x − xu )vanishes at any k -point u of T ′ . But ϕf ′ ( y ) / ∈ T x , otherwise q ( x − y z ) = 0 forany k -point z of ( G ′ /P ′ ) a contradicting the assumption that y is in Ω( x ). Thus q ( x − xy ) = 0 implies that q ( x − xu ) ∈ I ′ .(ii) The geometric description of Ω( x ) ∩ T ′ follows from [13], Cor. 6.3. HenceΩ( x ) ∩ T ′ is obtained from T ′ by removing the images π ( E ) of all exceptionaldivisors E ⊂ T , so that π ( T × ) = Ω( x ) ∩ T ′ .(iii) Recall that exp( x ) gives a section of the natural morphism π : T → x − y T ′ over the complement to the fibre T ′ y . Thus t ∈ T . Since y is in Ω( x ) ∩ T ′ we seefrom (ii) that t is in T × . QEDLet T ⊂ V sf be a closed T -invariant subvariety such that T /T is a split delPezzo surface and the weight hyperplane sections of T are exceptional divisors with8ultiplicity 1. The torsor T defines an important subset of the torus S . Namely,let Z be the closed subset of S consisting of the points s such that s T ⊂ ( G/P ) a .Equivalently, Z = T x ∈T × ( k ) x − ( G/P ) × a . The set Z is T -invariant, since such are( G/P ) a and T . In the case when T ⊂ ( G/P ) sf a , the variety Z contains the identityelement 1 ∈ S ( k ). Lemma 2.2
Under the assumptions of Lemma 2.1 for r = 4 we have Z = T , andfor r ≥ we have π ( Z ) = y − Ω( x ) which is dense and open in y − ( G ′ /P ′ ) × a . Theclosed subvariety Z ⊂ S is the affine cone (without zero) over t − exp( x − y Ω( x )) ;in particular, Z is geometrically integral, and t − T × ⊂ Z . For r = 5 this inclusionis an equality.Proof The statement in the case r = 4 is clear since T is dense in ( G/P ) a , andthe only elements of S that leave G/P = Gr(2 ,
5) invariant are the elements of T . (Indeed, it is well known that the group of relations among the classes of 10exceptional curves on X is generated by the quadratic relations given by degenerateelements of conic pencils on X . These quadratic relations are in a natural bijectionwith the quadratic equations among the Pl¨ucker coordinates of Gr(2 , r ≥
5. For a fixed x , in order to construct an embedding T ⊂ ( G/P ) a we can choose any y in the dense open subset Ω( x ) ⊂ ( G ′ /P ′ ) a . The embeddingsdefined by ( x , y ) and ( x , y ) satisfy the conditions of Lemma 1.1. We obtain anelement s ∈ Z such that π ( s ) = y − y . Thus π ( Z ) contains y − Ω( x ).Let us prove that π ( Z ) ⊂ y − ( G ′ /P ′ ) × a . Let π : V → V ≃ k be the naturalprojection. Choose y ∈ T ⊂ ( G/P ) a such that π ( y ) = y ∈ Ω( x ) ⊂ ( G ′ /P ′ ) × a .By Lemma 4.1 of [13] we have π ( y ) = 0. Thus π ( sy ) = 0 for any s ∈ Z . Butsince sy ∈ ( G/P ) a , an inspection of cases in Lemma 4.1 of [13] shows that π ( sy ) = π ( s ) y ∈ ( G ′ /P ′ ) × a . Therefore, π ( Z ) ⊂ y − ( G ′ /P ′ ) × a . Next, we note that st ∈ ( G/P ) × a (since t ∈ T × by Lemma 2.1). The coordinates of the projection of st to V equal p µ ( π ( s ) x − y ), up to a non-zero constant, hence p µ ( π ( s ) x − y ) = 0 for all µ .But for r ≤ y − Ω( x ) ⊂ y − ( G ′ /P ′ ) × a is given by p µ ( x − y u ) = 0, byLemma 2.1 (i). For r = 7 a similar argument shows that q ( π ( s ) x − y ) = 0. Thuswe obtain the equality π ( Z ) = y − Ω( x ).By Lemma 2.1 (iii), t = exp( x − y ) is in T × so we have t Z ⊂ ( G/P ) × a . Since Z isinvariant under the action of G m = { g t | t ∈ k ∗ } , we see from Lemma 4.1 of [13] that Z is a G m -torsor over π ( Z ) = y − Ω( x ). Moreover, t − exp( x − y x ) is a section ofthis torsor. This proves that Z is the affine cone over t − exp( x − y Ω( x )).If r = 5, then Ω( x ) is a dense open subset of T ′ as both sets are Zariski open inGr(2 , Z = 2 + dim G ′ /P ′ which equals 8 , ,
18 for r =5 , ,
7, respectively.
Definition 2.3 r − points z , . . . , z r − in Z ( k ) are in general position if for anyweight µ ∈ W ω the vectors Ver µ ( z i ) , i = 0 , . . . , r − , are linearly independent. emma 2.4 Let
T ⊂ V sf be a closed T -invariant subvariety such that T /T isa split del Pezzo surface and the weight hyperplane sections of T are exceptionaldivisors with multiplicity . Then Z contains r − k -points in general position.More precisely, for any k -point z of Z the points ( z , . . . , z r − ) ∈ Z ( k ) r − such that z , z , . . . , z r − are in general position, form a dense open subset of Z r − .Proof We first note that Ver µ ( Z ) is dense in a vector subspace of S µ ( V ) of dimension r −
3. Indeed, assume without loss of generality that µ = ω . Then, as in the proofof Lemma 1.3, we have S ω ( V ) = S ω ( V ) ⊕ ( V ) ω . The image of t Z consists of thepoints Ver ω ( x − y u ) + p ω ( x − y u ), where u is in Ω( x ), by Lemma 2.2. Since Ver ω sends ( G ′ /P ′ ) a to a vector space of dimension r −
3, by Lemma 1.3, we see thatVer ω ( t Z ) is a dense subset of a vector space of this dimension. Hence the same istrue for Z .We can choose the points z , . . . , z r − in Z ( k ) one by one, in such a way that z n is in the complement to the union of the inverse images under Ver µ of the linearspan of Ver µ ( z i ), i = 0 , . . . , n −
1. This complement is non-empty since Ver µ ( Z ) isa Zariski dense subset of a vector space of dimension r −
3. QEDEquations for T have been given by Popov [11] and Derenthal [7]. The followingresult gives a concise natural description of these equations, in terms of the wellknown equations of ( G/P ) a ⊂ V . Theorem 2.5
Let r = 4 , , or . Every split del Pezzo surface X of degree − r has a universal torsor T which is an open subset of the intersection of r − dilatations of ( G/P ) a by k -points of the diagonal torus S . In the above notation wehave T × = \ z ∈Z ( k ) z − ( G/P ) × a = r − \ i =0 z − i ( G/P ) × a , (4) where z = 1 , z , . . . , z r − are k -points of Z in general position.Proof By Corollary 1.2 it is enough to prove the theorem for T which satisfies theassumptions of Lemma 2.1. The torsor T is clearly contained in the closed set S = ∩ s ∈Z ( k ) s − ( G/P ) a ⊂ V . Since T × is closed in V × , the density of T in S implies T × = S × . To prove this density it is enough to show that x − y T ′ is dense in π ( S ).For v ∈ V ⊗ k we write v = ( v , v , v , v ), where v i ∈ V i ⊗ k . Similarly, we write s ∈ S ( k ) as ( s , s , s , s ), where s i ∈ GL( V i ⊗ k ). In this notation the set \ s ∈Z ( k ) { ( s − t, s − tx, s − tp ( x ) , s − tq ( x )) | x ∈ V ⊗ k, t ∈ k ∗ } is dense in S . This set can also be written as \ s ∈Z ( k ) { ( t, x, ( ts ) − s − p ( s x ) , ( ts ) − s − q ( s x )) | x ∈ V ⊗ k, t ∈ k ∗ } . , , , ∈ Z , we see that π ( S ) is contained in the set of x ∈ V ⊗ k suchthat for all s ∈ Z ( k ) we have s − s − p ( s x ) = p ( x ) . Let J ⊂ k [ V ∗ ] be the ideal of x − y T ′ , and J µ = J ∩ S µ ( V ) ∗ . In the same wayas I ′ µ , the ideal J µ has codimension 2 in S µ ( V ) ∗ . Lemma 1.3 implies that thelinear span L of the quadratic forms p µ ( yy − x ) on V for a fixed y ∈ ( G ′ /P ′ ) × a and arbitrary y ∈ ( G ′ /P ′ ) × a has codimension 1 in S µ ( V ) ∗ (in fact, L is the spaceof forms vanishing at y ). Lemma 2.2 implies that L coincides with the linear spanof the quadratic forms p µ ( s x ), for all s ∈ Z . Hence the linear span of the forms s − s − ,µ p µ ( s x ) − p µ ( x ), for all s ∈ Z , has codimension at most 2 in S µ ( V ) ∗ . However,the inclusion x − y T ′ ⊂ π ( S ) implies that this space is in J µ , and thus coincideswith J µ . This holds for every µ , and the ideal J is generated by the J µ (since thesame is true for I ′ ), therefore x − y T ′ is dense in π ( S ).Let us prove the second equality in (4). It is well known that the intersection ofthe ideal of ( G/P ) a with S µ ( V ∗ ) is 1-dimensional; let P µ ( u ) be a non-zero elementin this intersection. Then P µ ( z i u ), where z , . . . , z r − are in general position, span avector space of dimension r − T with S µ ( V ∗ ), which has the same dimension. Thus P µ ( z i u ), i = 0 , . . . , r −
4, is a completesystem of equations of T of weight µ . This completes the proof. QED Remark
In the case r = 5 the general position condition has a clear geometricinterpretation. By the last claim of Lemma 2.2 we have T × = s Z for some s ∈ S ( k )well defined up to T ( k ). If T ⊂ ( G/P ) sf a , then Z contains 1, to that s is a k -pointof T × . Then the previous theorem implies T × = s Z = ( G/P ) × a ∩ r − s ( G/P ) × a , (5)where r is a k -point in T × such that f ( s ) and f ( r ) are points in X × not containedin a conic on X , cf. diagram (3). Here f ( s ) is uniquely determined by T , whereas r can be any point in the open subset of X given by this condition.This remark can be seen as a particular case of the following description of Z .For any g and h in T × ( k ) such that Ver µ ( h ) and Ver µ ( g ) are not proportional forany µ ∈ W ω , we have Z = g − ( G/P ) × a ∩ h − ( G/P ) × a . The proof is similar to thatof Theorem 2.5; we omit it here since we shall not need this fact.To construct r − Z in general position is not hard, because the pointsof Z are parameterized by polynomials. Indeed, decompose V = V , ⊕ V , ⊕ V , similarly to (2), and consider the points t − exp( x − y exp( v i )), where v , . . . , v r − in V , satisfy certain open conditions which are easy to write down using Lemma 2.2.11 A uniqueness result
The choice of y plays the role of a ‘normalization’ for the embedding of a torsorinto ( G/P ) a . It is convenient to choose these normalizations in a coherent way. Let M , . . . , M r be k -points in general position in P , and let X r be the blowing-up of P in M , . . . , M r . The complement to the union of exceptional curves X × r ⊂ X r can be identified with an open subset U ⊂ P . Choose u ∈ U ( k ). At every step ofour inductive process we can choose the points y in the fibre of T ′ → X ′ over u .Thus we get a compatible family of the y (more precisely, of torus orbits) that aremapped to each other by the surjective maps T → T ′ . In our previous notation,the point t = exp( x − y ) must be taken for the point y of the next step.If A is a subset of the torus S , then we denote by P n ( A ) ⊂ S the set of productsof n elements of A in S . We define P ( A ) = T . Proposition 3.1
Let r and n be integers satisfying ≤ r ≤ , ≤ n ≤ r − .Under the assumptions of Lemma 2.1, if at every step of our construction we choosethe points y over a fixed point of U , then we have the following statements: (i) P n +1 ( t − T × ) ⊂ t − ( G/P ) × a , (ii) P n ( t − T × ) ⊂ Z .Proof (i) and (ii) are clearly equivalent. For n = 0 the inclusion (i) is the maintheorem of [13], and this also covers the case r = 4. Let n ≥
1. Recall that theprojection π maps t − T onto y − T ′ . Assume that we have the desired inclusions for n − T ′ and T , namelyP n ( t − T × ) ⊂ t − ( G/P ) × a , P n ( y − T ′× ) ⊂ y − ( G/P ) × a . By Lemma 4.1 of [13] every k -point of ( G/P ) × a can be written as g x · exp( v ), where x ∈ k ∗ , and v ∈ V ⊗ k . By the first inclusion in induction assumption this is also truefor elements of t P n ( t − T × ). Since T is g x -invariant, we have exp( v ) ∈ t P n ( t − T × ).On applying π to both sides we deduce v ∈ x − y P n ( y − T ′× ). Applying thesecond inclusion in induction assumption we obtain v ∈ x − y ( G ′ /P ′ ) × a . There-fore, P n ( t − T × ) is contained in the affine cone over t − exp( x − y ( G ′ /P ′ ) × a ). Butthis implies P n ( t − T × ) ⊂ Z , since t Z is the intersection of the affine cone overexp( x − y ( G ′ /P ′ ) × a ) with V × , by the last statement of Lemma 2.2. This proves (ii),and hence also (i). QED Proposition 3.2
Let r = 4 , , or , and let T ⊂ V sf be a closed T -invariantsubvariety such that X = T /T is a split del Pezzo surface of degree − r , andthe weight hyperplane sections of T are exceptional divisors with multiplicity . Let Z ⊂ S be the closed subset of points z such that z T ⊂ ( G/P ) a . Then there is aunique s ∈ S ( k ) defined up to an element of T ( k ) , such that P r − ( T × ) ⊂ s Z . roof By Corollary 1.2, up to translating T by an element of S ( k ), we can assumethat T ⊂ ( G/P ) a is obtained by our construction. Thus the existence of s followsfrom Proposition 3.1. We prove the uniqueness by induction in r . For r = 4 thestatement is clear, since the only elements of S that leave Gr(2 ,
5) invariant are theelements of T (see the proof of Lemma 2.2).Assume r ≥
5. By Lemma 2.2, P r − ( t − T × ) ⊂ s Z implies P r − ( y − T ′× ) ⊂ π ( s ) y − ( G ′ /P ′ ) × a , from which it follows that P r − ( y − T ′× ) ⊂ π ( s ) Z ′ . By inductionassumption π ( s ) is unique up to an element of T ′ ( k ). Therefore, s is unique up toan element of T ( k ). QED Remark
For r = 5 the inclusion P r − ( T × ) ⊂ s Z is an equality by the last claim ofLemma 2.2, but this is no longer so for r = 6 or 7, for dimension reasons. Let Γ = Gal( k/k ). Let G be a split simply connected semisimple group over k witha split maximal k -torus H and the root system R. There is a natural exact sequenceof algebraic k -groups 1 → H → N → W → , (6)where N is the normalizer of H in G , and W is the Weyl group of R. The actionof N by conjugation gives rise to an action of W on the torus H . Since H issplit, the Galois group Γ acts trivially on W . Thus the continuous 1-cocycles of Γwith values in W are homomorphisms Γ → W , and the elements of H ( k, W ) arehomomorphisms Γ → W considered up to conjugation in W . Theorem 4.1 (Gille–Raghunathan)
For any σ ∈ Hom(Γ , W ) the twisted torus H σ is isomorphic to a maximal torus of G .Proof See [8], Thm. 5.1 (b), or [12], Thm. 1.1. QEDRecall from [14], I.5.4, that (6) gives rise to the exact sequence of pointed sets1 → N ( k ) → G ( k ) → ( G/N )( k ) ϕ −→ H ( k, N ) → H ( k, G ) . (Note by the way that the last map here is surjective.) The homogeneous space G/N is the variety of maximal tori of G , so that an equivalent form of the Gille–Raghunathan theorem is the surjectivity of the composite map( G/N )( k ) → H ( k, N ) → H ( k, W ) = Hom(Γ , W ) / Inn( W ) , where Inn( W ) is the group of inner automorphisms of W . We fix an embeddingof H σ as a maximal torus of G , this produces a k -point [ H σ ] in G/N . The choiceof a k -point g in G such that g Hg − = H σ defines a 1-cocycle ρ : Γ → N ( k ),13 ( γ ) = g − · γ g , which is a lifting of σ ∈ Z ( k, W ) = Hom(Γ , W ). We have[ ρ ] = ϕ [ H σ ] ([14], ibidem ), moreover, the image of [ ρ ] in H ( k, G ) is trivial.Let G → GL( V ) be an irreducible representation of G . Define T ⊂ GL( V ) as atorus generated by H and the scalar matrices G m . The group N acts by conjugationon T . The twisted torus T σ is just the extension of H σ by scalar matrices.Let ( G/P ) a ⊂ V be the orbit of the highest weight vector (with zero added toit); P ⊂ G is a parabolic subgroup, and ( G/P ) a is the affine cone over G/P . Themaximal torus H σ ⊂ G acts on ( G/P ) a , and so does T σ . Define U σ to be the denseopen subset of ( G/P ) a consisting of the points with closed H σ -orbits and trivialstabilizers in T σ .The group N ⊂ G acts on V preserving V sf and V × , thus giving rise to theaction of W on V sf /T and on V × /T by automorphisms of algebraic varieties (notnecessarily preserving some group structure on V × /T ). The action of N preserves( G/P ) sf a ⊂ V , thus W acts on Y = ( G/P ) sf a /T . Hence we define the twisted forms( V sf /T ) σ , ( V × /T ) σ and Y σ . The variety ( V × /T ) σ is an open subset of the quasi-projective toric variety ( V sf /T ) σ , which contains Y σ as a closed subset. Lemma 4.2
The k -varieties Y σ and U σ /T σ are isomorphic.Proof Recall that g ∈ G ( k ) is a point such that ρ ( γ ) = g − · γ g ∈ Z ( k, N )is a cocycle that lifts σ ∈ Z ( k, W ) = Hom(Γ , W ). The group N acts on thehomogeneous space ( G/P ) a as a subgroup of G , so we can define ( G/P ) a,ρ as thetwist of ( G/P ) a by ρ . It is immediate to check that the map x g x on k -points of( G/P ) a gives rise to an isomorphism of k -varieties ( G/P ) a,ρ ˜ −→ ( G/P ) a . If G ρ is theinner form of G defined by ρ , then G ρ acts on ( G/P ) a,ρ on the left. The embedding H ֒ → G gives rise to an embedding H σ ֒ → G ρ , so that T σ acts on ( G/P ) a,ρ on theleft. On the other hand, T σ also acts on ( G/P ) a on the left. It is straightforward tocheck that the isomorphism ( G/P ) a,ρ ˜ −→ ( G/P ) a is T σ -equivariant.Let ( G/P ) sf a,ρ be the subset of ( G/P ) a,ρ consisting of the points with closed H σ -orbits with trivial stabilizers in T σ . The closedness of orbits and the triviality of sta-bilizers are conditions on k -points, hence we obtain a T σ -equivariant k -isomorphism( G/P ) sf a,ρ ˜ −→ U σ . It descends to an isomorphism Y σ ˜ −→ U σ /T σ . This proves thelemma. QED Corollary 4.3
For any homomorphism σ : Γ → W the twisted variety Y σ has a k -point, and so does ( V × /T ) σ .Proof Since k is an infinite field, any dense open subset of ( G/P ) a contains k -points.Thus Y × σ ( k ) = ∅ , but this is a subset of ( V × /T ) σ , so that this variety has a k -point.QED Remark
This approach via the Gille–Raghunathan theorem generalizes a key ingre-dient in the second author’s proof of the Enriques–Swinnerton-Dyer theorem that14very del Pezzo surface of degree 5 has a k -point, from quotients of Grassmanniansby the action of a maximal torus to quotients of arbitrary homogeneous spaces ofquasi-split semisimple groups. We plan to return to this more general statement inanother publication.We now assume that R is the root systems of rank r in (1), and that the highestweight of the G -module V is the fundamental weight dual to the root indicated in(1). Then V is minuscule, so that the centralizer S of H in GL( V ) is a torus. Let R = S/T . We obtain an exact sequence of k -tori:1 → T → S → R → . (7)The group N acts by conjugation on T and hence also on S and R . The connectedcomponent of 1 acts trivially, so we obtain an action of W on these tori (preservingthe group structure). On twisting T , S and R by σ we obtain an exact sequence of k -tori: 0 → T σ → S σ → R σ → . (8)Note in passing that the character group ˆ S has an obvious W -invariant basis, whichgives rise to a Galois invariant basis of ˆ S σ . In other words, S σ is a quasi-trivial torus;in particular, H ( k, S σ ) = 0 as follows from Hilbert’s theorem 90. Note also that V × /T is a torsor under R , so that ( V × /T ) σ is a torsor under R σ . By Corollary 4.3this torsor is trivial, that is, there is a (non-canonical) isomorphism ( V × /T ) σ ≃ R σ .Let X be a del Pezzo surface over k , not necessarily split, of degree 9 − r , where r isthe rank of the root system R. Let X be the surface obtained from X by extendingthe ground field from k to k . We write X × for the complement to the union ofexceptional curves on X . Our construction identifies ˆ S with the free abelian groupDiv X \ X × X generated by the exceptional curves on X , and ˆ T with Pic X (via thetype of the universal torsor T → X ). The Galois group permutes the exceptionalcurves on X , thus defining a homomorphism σ X : Γ → W , where W is the Weylgroup of R. This homomorphism is well defined up to conjugation in W , so we havea well defined class [ σ X ] ∈ H (Γ , W ), where Γ acts trivially on W .We now assume σ = σ X . Then we get isomorphisms of Γ-modulesˆ S σ = Div X \ X × X, ˆ T σ = Pic X, thus T σ is the N´eron–Severi torus of X . The kernel of the obvious surjective mapDiv X \ X × X → Pic X is k [ X × ] ∗ /k ∗ , hence the dual sequence of (8) coincides with thenatural exact sequence of Γ-modules0 → k [ X × ] ∗ /k ∗ → Div X \ X × X → Pic X → . (9)There is a natural bijection between the morphisms X × → R σ and the homomor-phisms of Γ-modules ˆ R σ → k [ X × ] ∗ . Universal torsors on X exist if and only if the15xact sequence of Γ-modules1 → k ∗ → k [ X × ] ∗ → k [ X × ] ∗ /k ∗ → R σ = k [ R σ ] ∗ /k ∗ = k [ X × ] ∗ /k ∗ → k [ X × ] ∗ , and hence defines a morphism φ : X × → R σ . By the ‘local description of torsors’(see [4], 2.3 or [15], Thm. 4.3.1) the restriction of a universal X -torsor to X × isthe pull-back of the torsor S σ → R σ to X × via φ . Moreover, this gives a bijectionbetween the splittings of (10) and the universal X -torsors. In our case it is easyto see that φ is an embedding. The isomorphism ˆ R σ = k [ X × ] ∗ /k ∗ comes from ourconstruction, thus after extending the ground field to k , the morphism φ coincides,up to translation by a k -point of R , with the embedding of X × into ( V ⊗ k k ) × /T obtained from the embedding T × ⊂ V × . Theorem 4.4
Let r = 4 , , or . Let X be a del Pezzo surface of degree − r with a k -point, and let σ ∈ H (Γ , W ) be the class defined by the action of the Galoisgroup on the exceptional curves of X . There exists an embedding X ֒ → Y σ suchthat the divisors in Y σ \ Y × σ cut the exceptional curves on X with multiplicity .The restriction of U σ → Y σ to X ⊂ Y σ is a universal X -torsor whose type is theisomorphism ˆ T σ = Pic X .Proof From Corollary 4.3 we get an embedding Y σ ֒ → R σ , which becomes unique ifwe further assume that a given k -point of Y σ goes to the identity element of R σ .Since X ( k ) = ∅ , there is a unique embedding φ : X × → R σ such that the inducedmap φ ∗ : ˆ R σ → k [ X × ] ∗ is a lifting of the isomorphism ˆ R σ = k [ R σ ] ∗ /k = k [ X × ] ∗ /k ∗ ,and φ sends a given k -point of X × to 1.Let L be the k -subvariety of the torus R σ whose points are c ∈ R σ ( k ) such that cX × ⊂ Y × σ , where the multiplication is the group law of R σ . To prove the firststatement we need to show that L ( k ) = ∅ . Let P n ( X × ) be the k -subvariety of R σ whose k -points are products of n elements of X × ( k ) in R σ ( k ). The surface X issplit, hence it follows from Proposition 3.2 that there exists a unique c ∈ R σ ( k )such that P r − ( X × )( k ) ⊂ c L ( k ). But since P r − ( X × ) and L are subvarieties of R σ defined over k we conclude that c is a k -point. If m is k -point of X × , then c − m r − is a k -point of L , as required.To check that the restriction of U σ → Y σ to X ⊂ Y σ is a universal torsor we cango over to k where it follows from our main theorem in the split case. QED Remark
Let X be a del Pezzo surface with a k -point, of degree 5, 4, 3 or 2. Although( G/P ) a contains some universal X -torsor, other universal X -torsors of the sametype are naturally embedded into certain twists of ( G/P ) a . Indeed, all torsors16f the same type are obtained from any of them by twisting by the cocycles in Z ( k, T σ ). The natural map H σ → T σ gives a surjection H ( k, H σ ) → H ( k, T σ )since H ( k, G m ) = 0 by Hilbert’s theorem 90. Therefore, if T ⊂ ( G/P ) a and θ ∈ Z ( k, H σ ), then T θ is contained in the twist of ( G/P ) a by the 1-cocycle in Z ( k, G ) coming from θ ∈ Z ( k, H σ ). By general theory ([14], I.5) this twist is a lefthomogeneous space of the inner form of G defined by θ . (In the case of Gr(2 ,
5) weonly obtain twists that are isomorphic to Gr(2 ,
5) because H ( k, SL( n )) = 0.) Wenote that by Steinberg’s theorem every class in H ( k, G ) comes from H ( k, H σ ) forsome maximal torus H σ ⊂ G . Corollary 4.5
Let X be a del Pezzo surface of degree such that universal X -torsors exist. Let σ ∈ H (Γ , W ) be the class defined by the action of the Galoisgroup on the exceptional curves of X . (i) X × and Y × σ are k -subvarieties of R σ . Moreover, Y σ contains cX for some c ∈ R σ ( k ) if and only if X has a k -point. (ii) If X ⊂ Y σ , then X = Y σ ∩ mY σ for some m ∈ R σ ( k ) .Proof. (i) In view of Theorem 4.4 it remains to prove the ‘only if’ part. We notethat Y σ embeds into R σ by Corollary 4.3. The existence of universal X -torsorsimplies that X embeds into R σ , as was discussed before Theorem 4.4. For r = 5 theinclusion X × ⊂ c L from the proof of Theorem 4.4 is an equality by the remark inthe end of Section 3. Hence if L has a k -point, then so does X × .(ii) A del Pezzo surface of degree 4 with a k -point is known to be unirational (i.e.,is dominated by a k -rational variety, see [10]). Thus X , and hence also L = c − X × contains a Zariski dense set of k -points. The variety X is contained in Y σ ∩ h − c Y σ for any h ∈ X × ( k ), and this inclusion is an equality for any h in a dense open subsetof X , see the remark after the proof of Theorem 2.5. QED Remark
By the previous remark an arbitrary universal torsor over a del Pezzo surface X of degree 4 with a k -point embeds into a twisted form of ( G/P ) a by a cocyclecoming from θ ∈ Z ( k, H σ ). This twisted form ( G/P ) a,θ is naturally a subset of avector space (non-canonically isomorphic to V ) acted on by S σ . Thus any universaltorsor over X is an open subset of the intersection of two k -dilatations of ( G/P ) a,θ . References [1] V.V. Batyrev and O.N. Popov. The Cox ring of a del Pezzo surface. In:
Arith-metic of higher-dimensional algebraic varieties (Palo Alto, 2002), Progr. Math.
Birkh¨auser, 2004, 85–103.[2] A. Beauville, J-L. Colliot-Th´el`ene, J-J. Sansuc et P. Swinnerton-Dyer. Vari´et´esstablement rationnelles non rationnelles.
Ann. Math. (1985) 283–318.173] N. Bourbaki.
Groupes et alg`ebres de Lie.
Chapitres IV-VIII. Masson, Paris,1975, 1981.[4] J-L. Colliot-Th´el`ene et J-J. Sansuc. La descente sur les vari´et´es rationnelles, II.
Duke Math. J. (1987) 375–492.[5] J-L. Colliot-Th´el`ene, D. Kanevsky et J-J. Sansuc. Arithm´etique des surfacescubiques diagonales. In: Diophantine approximation and transcendence theory (Bonn, 1985) Lecture Notes in Math.
Springer-Verlag, 1987, 1–108.[6] J-L. Colliot-Th´el`ene, J-J. Sansuc and P. Swinnerton-Dyer. Intersections of twoquadrics and Chˆatelet surfaces. I, II.
J. reine angew. Math. (1987) 37–168.[7] U. Derenthal. Universal torsors of Del Pezzo surfaces and homogeneous spaces.
Adv. Math. (2007) 849–864.[8] Ph. Gille. Type des tores maximaux des groupes semi-simples.
J. RamanujanMath. Soc. (2004) 213–230.[9] B.E. Kunyavski˘ı and M.A. Tsfasman. Zero-cycles on rational surfaces andN´eron–Severi tori. Izv. Akad. Nauk SSSR Ser. Mat. (1984) 631–654.[10] Yu.I. Manin. Cubic forms.
Del Pezzo surfaces and algebraic groups.
Diplomarbeit, Universit¨atT¨ubingen, 2001.[12] M.S. Raghunathan. Tori in quasi-split groups.
J. Ramanujan Math. Soc. (2004) 281–287.[13] V.V. Serganova and A.N. Skorobogatov. Del Pezzo surfaces and representationtheory. J. Algebra and Number Theory (2007) 393–419.[14] J-P. Serre. Cohomologie galoisienne.