Geometries arising from trilinear forms on low-dimensional vector spaces
aa r X i v : . [ m a t h . AG ] M a r Geometries arising from trilinear forms onlow-dimensional vector spaces
Ilaria Cardinali, Luca GiuzziJuly 21, 2018
Abstract
Let G k ( V ) be the k -Grassmannian of a vector space V with dim V = n .Given a hyperplane H of G k ( V ), we define in [3] a point-line subgeometry ofPG( V ) called the geometry of poles of H . In the present paper, exploitingthe classification of alternating trilinear forms in low dimension, we char-acterize the possible geometries of poles arising for k = 3 and n ≤ , K ) arising from hyperplanes of G ( V ) . Keywords : Grassmann Geometry; Hyperplanes; Multilinear forms.
MSC : 15A75; 14M15; 15A69.
Denote by V a n -dimensional vector space over a field K . For any fixed 1 ≤ k < n the k -Grassmannian G k ( V ) of V is the point-line geometry whose points are the k –dimensional vector subspaces of V and whose lines are the sets ℓ Y,Z := { X : Y ⊂ X ⊂ Z, dim X = k } where Y and Z are subspaces of V with Y ⊂ Z, dim Y = k − Z = k + 1. Incidence is containment.It is well known that the geometry G k ( V ) affords a full projective embedding,called Grassmann (or Pl¨ucker) embedding and denoted by ε k , sending every k -subspace h v , . . . , v k i of V to the point [ v ∧ · · · ∧ v k ] of PG( V k V ), where we adoptthe notation [ u ] to refer to the projective point represented by the vector u . Also,for X ⊆ V we put [ X ] := { [ x ] : x ∈ h X i} for the points of the projective spaceinduced by h X i .A hyperplane H of G k ( V ) is a proper subspace of G k ( V ) such that any line of G k ( V ) is either contained in H or it intersects H in just one point.1t is well known, see Shult [13] and also Havlicek [10], Havlicek and Zanella[11] and De Bruyn [5], that the hyperplanes of G k ( V ) all arise from the Pl¨uckerembedding of G k ( V ) in PG( V k V ), i.e. they bijectively correspond to propor-tionality classes of non-zero linear functionals on V k V . More in detail, for anyhyperplane H of G k ( V ) there is a non-null linear functional h on V k V such that H = ε − k ([ker( h )] ∩ ε k ( G k ( V ))). Equivalently, if χ h : V × . . . × V → K is thealternating k -linear form on V associated to the linear functional h , defined by theclause χ h ( v , . . . , v k ) := h ( v ∧ · · · ∧ v k ), then the hyperplane H is the set of the k -subspaces of V where χ h identically vanishes. So, the hyperplanes of G k ( V ) bijec-tively correspond also to proportionality classes of non-trivial alternating k -linearforms of V .In a recent paper [3] we introduced the notion of i -radical of a hyperplane H of G k ( V ). In the present work we shall just consider the case of the lower radical R ↓ ( H ), for i = 1, and that of the upper radical R ↑ ( H ), for i = k −
1. The lowerradical R ↓ ( H ) of H is the set of points [ p ] ∈ PG( V ) such that all k -spaces X with p ∈ X belong to H ; the upper radical R ↑ ( H ) of H is the set of ( k − Y of V such that all k -spaces through Y belong to H .In the same paper [3] we investigated the problem of determining under whichconditions the upper radical of a given hyperplane might be empty. Working inthe case k = 3, we also defined a point-line subgeometry P ( H ) = ( P ( H ) , R ↑ ( H ))of PG( V ) called the geometry of poles of H , whose points are called H -poles, apoint [ p ] ∈ P ( H ) and a line ℓ ∈ R ↑ ( H ) being incident when p ∈ ℓ (see Section 1.1).Some aspects of this geometry had already been studied, under slightly differentsettings, in [6, 7]. It has been shown in [6] that the set P ( H ) is actually eitherPG( V ) or an algebraic hypersurface in PG( V ); for more details see Section 1.1.In this paper we shall focus on the geometry P ( H ), providing explicit equationsfor its points and lines and also a geometric description in the cases where acomplete classification of trilinear forms on a vector space V is available, namelydim( V ) ≤ K arbitrary (see [12]) and dim( V ) = 7 with K a perfect fieldwith cohomological dimension at most 1 (see [4]).We shall briefly recall the definition of geometry of poles in the next sectionand we will state our main results in Section 1.2. The organization of the paperis outlined in Section 1.3. Assume k = 3 and let H be a given hyperplane of G ( V ). For any (possibly empty)projective space [ X ] we shall denote by dim( X ) the vector dimension of X. Let [ p ] be a point of PG( V ) and consider the point-line geometry S p ( H ) havingas points, the set of lines of PG( V ) through [ p ] and as lines, the set of planes [ π ]of PG( V ) through [ p ] with π ∈ H . It is easy to see (see [3]) that S p ( H ) is a polar2pace of symplectic type (possibly a trivial one). Let R p ( H ) := Rad ( S p ( H )) bethe radical of S p ( H ) and put δ ( p ) := dim(Rad ( S p ( H ))) . We call δ ( p ) the degree of [ p ] (relative to H ). If δ ( p ) = 0 then we say that [ p ]is smooth , otherwise we call [ p ] a pole of H or, also, a H - pole for short. Clearly, apoint is a pole if and only if it belongs to a line of the upper radical R ↑ ( H ) of H .So, R ↑ ( H ) = ∅ if and only if all points are smooth.As the vector space underlying the symplectic polar space S p ( H ) has dimension n − δ ( p ) is even when n is odd and it is odd if n is even. In particular, when n is even all the points are poles of degree at least 1. If they all have degree 1, thenwe say that H is spread-like .We shall provide in Theorem 2 a direct geometric proof of the result of [3,Theorem 4], stating that for n = 6 there are spread-like hyperplanes if and onlyif the field K is not quadratically closed, extending a result of [7]; this is alsoimplicit in the classification of [12]; observe that for n = 8 we prove in [3] that forquasi-algebraically closed fields there are no spread-like hyperplanes.More in general, when all points of PG( V ) are poles of the same degree δ and( δ + 1) | n , the set { π p : [ p ] ∈ PG( V ) } of all subspaces π p := { [ u ] ∈ [ ℓ ] : p ∈ ℓ and ℓ ∈ R ↑ ( H ) } as [ p ] varies in PG( V ) might in some case possibly be a spreadof PG( V ) in spaces of projective dimension δ . We are not currently aware of anycase where this happens for δ > Before stating our main results we need to fix a terminology for linear function-als on V V and to recall what is currently known from the literature regardingtheir classification. As already pointed out, a classification of alternating trilinearforms of V would determine a classification of the geometries of poles defined byhyperplanes of G ( V ) . We will first introduce the notion of isomorphism for hyperplanes and thenotions of equivalence, near equivalence and geometrical equivalence for k -linearforms in general.We say that two hyperplanes H and H ′ of G k ( V ) are isomorphic , and we write H ∼ = H ′ , when H ′ = g ( H ) := { g ( X ) } X ∈ H for some g ∈ GL( V ), where g ( X )is the natural action of g on the subspace X , i.e. g ( X ) = h g ( v ) , . . . , g ( v k ) i for X = h v , . . . , v k i . Recall that two alternating k -linear forms χ and χ ′ on V are said to be (linearly)equivalent when χ ′ ( x , . . . , x k ) = χ ( g ( x ) , . . . , g ( x k )) , ∀ x , . . . , x k ∈ V g ∈ GL( V ). Accordingly, if H and H ′ are the hyperplanes associated to χ and χ ′ , we have H ∼ = H ′ if and only if χ ′ is proportional to a form equivalent to χ . Note that if χ ′ = λ · χ for a scalar λ = 0 then χ and χ ′ are equivalent if andonly if λ is a k -th power in K .We say that two forms χ and χ ′ are nearly equivalent , and we write χ ∼ χ ′ ,when each of them is equivalent to a non-zero scalar multiple of the other. Hence H ∼ = H ′ if and only if χ ∼ χ ′ .We extend the above terminology to linear functionals of V k V in a naturalway, saying that two linear functionals h, h ′ ∈ ( V k V ) ∗ are nearly equivalent andwriting h ∼ h ′ when their corresponding k -alternating forms are nearly equivalent.We say that two hyperplanes H and H ′ are geometrically equivalent if theincidence graphs of their geometries of poles are isomorphic; the forms defininggeometrically equivalent hyperplanes are called geometrically equivalent as well.Note that nearly equivalent forms are always geometrically equivalent but theconverse does not hold in general. For example, let V be a vector space over a field K which is not quadratically closed and suppose dim( V ) = 6. To any quadraticextension of K there correspond a Desarguesian line-spread S of PG( V ), and thegeometry (PG( V ) , S ) is a geometry of poles associated to a trilinear form. Allhyperplanes inducing line-spreads are geometrically equivalent. If K is a finitefield or K = R , it is easy to see that the hyperplanes inducing S must also beisomorphic. However, this is not the case when K = Q or when K is a field ofcharacteristic 2 which is not perfect. In particular, in the latter case hyperplanesarising from forms of type T (1)10 ,λ and T (2)10 ,λ , see Table 1, are geometrically equivalentbut not isomorphic.Clearly, nearly equivalent or isomorphic hyperplanes are always geometricallyequivalent. V V Given a non-trivial linear functional h ∈ ( V V ) ∗ , let χ h and H h be respectivelythe alternating trilinear form and the hyperplane of G ( V ) associated to it. Whenno ambiguity might arise, we shall feel free to drop the subscript h in our notation.By definition, R ↓ ( H ) = [Rad ( χ )], where Rad ( χ ) = { v ∈ V : χ ( x, y, v ) =0 , ∀ x, y ∈ V } . Define the rank of h as rank ( h ) := cod V (Rad ( χ )) = dim( V /
Rad ( χ )).Obviously, functionals of different rank can never be nearly equivalent.It is known that if h is a non-trivial trilinear form, then rank ( h ) ≥ h ) = 4 (see [3, Proposition 19] for the latter result).Fix now a basis E := ( e i ) ni =1 of V . The dual basis of E in V ∗ is E ∗ :=( e i ) ni =1 , where e i ∈ V ∗ is the linear functional such that e i ( e j ) = δ i,j (Kroneckersymbol). The set ( e i ∧ e j ∧ e k ) ≤ i 6, Revoy [12] proves that all trilinear forms, up to equivalence,are of type T i , 1 ≤ i ≤ T (1)10 ,λ , T (2)10 ,λ . In particular, if the field is quadraticallyclosed, all forms of rank 6 are either of type T or of type T . If the field is notquadratically closed, it is possible to distinguish two families of classes of formslinearly equivalent among themselves according as they are equivalent to T or to T over the quadratic closure K (cid:3) of K . More in detail, if char ( K ) is odd, T (1)10 ,λ = T (2)10 ,λ and each form of a type in T (1)10 ,λ is equivalent to T over K (cid:3) . If char ( K ) = 2, thenthe classes T (2)10 ,λ and T (1)10 ,λ are in correspondence with respectively the separableand the inseparable quadratic extensions of the field K ; furthermore, any form ofa type in T (2)10 ,λ is equivalent to a form of type T in K (cid:3) , while any form of type T (1)10 ,λ is equivalent in K (cid:3) to a form of type T .If n = 7 and K is a perfect field of cohomological dimension at most 1, Cohenand Helmick [4] show that all trilinear forms, up to equivalence, are of a typedescribed in Table 1.Under the conditions of Table 1, we shall provide a classification for the ge-ometries of poles.We now present our main results; for the notions of extension, expansion andblock decomposition as well as some of the notation, see Section 3. By the symbol | x, y | ij we mean the ( i, j )-Pl¨ucker coordinate of the line [ x, y ] spanned by thevectors x = ( x i ) ni =1 and y = ( y i ) ni =1 written in coordinates with respect to the basis5 , i.e. | x, y | ij := x i y j − x j y i is the ij -coordinate of e i ∧ e j with respect to the basis( e i ∧ e j ) ≤ i Suppose dim( V ) ≤ and let h be a non-trivial linear functional on V V having type as described in Table 1, with associated alternating trilinear form χ . Denote by H the hyperplane of G ( V ) defined by h . Then one of the followingoccurs:1. h has type T (rank ). In this case H is the trivial hyperplane centeredat Rad ( χ ) and R ↑ ( H ) is the set of the lines of PG( V ) that meet [Rad ( χ )] non-trivially.2. h has type T (rank ), namely Rad ( χ ) is -dimensional. In this case H isa trivial extension Ext Rad ( χ ) (Exp( H )) of a symplectic hyperplane Exp( H ) ,constructed in a complement V of Rad ( χ ) in V starting from the line-set H of a symplectic generalized quadrangle. The elements of R ↑ ( H ) are thelines of PG( V ) that either belong to H or pass through the point [Rad ( χ )] = R ↓ ( H ) or such that their projection onto V is in H . h has type T (rank ). Then H is a decomposable hyperplane Dec( H , H ) arising form the hyperplanes H and H of G ( V ) and G ( V ) for a suitabledecomposition V = V ⊕ V with dim( V ) = dim( V ) = 3 . Then R ↑ ( H ) = { [ x, y ] : x ∈ V \ { } , y ∈ V \ { }} . h has type T (rank ). Then R ↑ ( H ) = { [ x, y ] = [ a + b, ω ( a )] : a ∈ V \{ } , b ∈ V }∪{ [ x, y ] ⊆ V } for a decomposition V = V ⊕ V with dim( V ) = dim( V ) =3 and ω an isomorphism of V interchanging V and V .5. h has type T (1)10 ,λ or T (2)10 ,λ (rank ). Then R ↑ ( H ) is a Desarguesian line spreadof PG( V ) corresponding to the field extension K [ µ ] with µ a root of p λ ( t ) . Theorem 2. Let V := V (6 , K ) . Line-spreads of PG( V ) induced by hyperplanes of G ( V ) exist if and only if K is a non-quadratically closed field. Draisma and Shaw prove that when K is a finite field there always exist hy-perplanes of G ( V ) with dim V = 6 having a Desarguesian spread as upper radical[7, § § K is algebraicallyclosed with characteristic 0, [7, Remark 9].Our Theorem 2 generalizes their results to arbitrary fields K and providesnecessary and sufficient conditions for the existence of spread-like hyperplanes for n = 6; its statement correspond to Theorem 20, point 5 in [3] (and also to Theorem4 in [3]).It also further clarifies the result of [12] linking the forms T ( i )10 ,λ with quadraticextensions of the field K . 6 heorem 3. Suppose dim( V ) = 7 and let h be a non-trivial linear functional on V V having type as described in Table 1. Denote by χ the associated alternatingtrilinear form and by H the hyperplane of G ( V ) defined by h .If rank ( h ) ≤ then H is a trivial extension Ext Rad ( h ) ( H ′ ) where H ′ is ahyperplane of G ( V ′ ) with V = Rad ( h ) ⊕ V ′ , dim( V ′ ) ≤ , and H ′ is defined by atrilinear form h ′ of type as in Theorem 1.If rank ( h ) = 7 then one of the following occurs:1) h has type T . Then, there are two non-degenerate symplectic polar spaces S and S , embedded as distinct hyperplanes in PG( V ) such that both determinethe same polar space S on their intersection. The radical of S is a point,say p . There are also two totally isotropic planes A and A of S such that A ∩ A = { p } . The poles of H are the points of S ∪ S , the poles of degree being the points of A ∪ A . The lines of P ( H ) are the totally isotropic linesof S i that meet A i non-trivially, for i = 1 , .2) h has type T . The poles of H lie in a hyperplane S of PG( V ) . A non-degeneratepolar space of symplectic type is defined in S and a totally isotropic plane A of S is given. The lines of P ( H ) are the totally isotropic lines of S that meet A non-trivially. The points of A are the poles of H of degree .3) h has type T . The poles of H are the points of a cone of PG( V ) having asvertex a plane A and as basis a hyperbolic quadric Q . A conic C is given in A such that the elements of C are the poles of degree . There is a correspondencemapping each point [ x ] ∈ C to a line ℓ x contained in a regulus of Q . For each [ x ] ∈ C it is possible to define a line spread S x of h A, ℓ x i / h x i ∼ = PG(3 , K ) suchthat R ↑ ( H ) = { ℓ ⊆ h x, s i : [ x ] ∈ C , s ∈ S x } .4) h has type T . In this case H = Exp( H ) is a symplectic hyperplane. Inparticular, the geometry P ( H ) is a non-degenerate polar space of symplectictype and rank , naturally embedded in a hyperplane [ V ] of PG( V ) . All polesof H have degree .5) h has type T . Then P ( H ) is a split-Cayley hexagon naturally embedded in anon-singular quadric of PG( V ) . All poles of H have degree .6) h has type T ( i )11 ,λ , i = 1 , . The poles of H are the points of a subspace S of codimension in V . There is only one point [ p ] ∈ S which is a pole ofdegree . Furthermore, there is a line-spread F of S / h p i ∼ = PG(3 , K ) such that R ↑ ( H ) := { ℓ ⊆ [ π, p ] : π ∈ F } . Theorems 1 and 3 correspond to Theorems 20 and 21 of [3], where they werepresented without a detailed proof. In the present paper we have chosen to re-fine the results announced [3], by providing a fully geometric description of the7eometries of poles arising for n ≤ 7, without having to recourse to coordinates.In any case, the original statements for cases 3,5 and 7 of [3, Theorem 21] can beimmediately deduced from Theorem 3 in light of the equations of Table 5. In Section 2 we shall explain how to algebraically describe points and lines of a ge-ometry of poles. Draisma and Shaw [6] have shown that either the set of H -poles isall of PG( V ) or it determines an algebraic hypersurface in PG( V ) described by anequation of degree ( n − / 2. In Section 2 we shall study such varieties. In partic-ular, in Section 2.2 we shall explicitly determine their equations as determinantalvarieties and in Section 2.3 describe some hyperplanes whose variety of poles isreducible in the product of distinct linear factors. In Section 3 we will presentthree families of hyperplanes of G k ( V ) obtained by extension , expansion and blockdecomposable construction . Our main theorems will be proved in Section 4. Forthe ease of the reader, all the tables are collected in Appendix A. Throughout this section we take E = ( e i ) ≤ i ≤ n as a given basis of V and thecoordinates of vectors of V will be given with respect to E. Let h : V V → K be a linear functional associated to a given hyperplane H of G ( V ), where V is a n -dimensional vector space over a field K . For any u ∈ V consider the bilinear alternating form h u : V / h u i × V / h u i → K , h u ( x + h u i , y + h u i ) := h ( u ∧ x ∧ y ) . By definition of H -pole, a point [ u ] ∈ PG( V ) is a H -pole if and only if the radicalof h u is not trivial. Consider also the bilinear alternating form on Vχ u : V × V → K , χ u ( x, y ) = h ( u ∧ x ∧ y ) = x T M u y (1)where M u is the matrix associated to χ u with respect to the basis E of V . Clearly,Rad ( h u ) = (Rad ( χ u )) / h u i ; thus the rank of the matrix of h u with respect to anybasis of V / h u i and the rank of the matrix M u are exactly the same. Proposition 2.1. Let [ u ] be a point of PG( V ) with u = ( u i ) ni =1 and let M u be the n × n -matrix associated to the alternating bilinear form χ u . If u i = 0 then the i -thcolumn (row) of M u is a linear combination of the other columns (rows) of M u . roof. Denote by C , . . . C n the columns of the matrix M u and let x = ( x i ) ni =1 and u = ( u i ) ni =1 . Let M ( i ) u be the ( n − × ( n − M u obtained bydeleting its i -th column and i -th row. For any x ∈ V, the condition x T M u u = 0 isequivalent to( x T · C ) u + ( x T · C ) u + · · · + ( x T · C i ) u i + · · · + ( x T · C n ) u n = 0 , where x T · C i := P nj =1 x j c ji with C i = ( c ji ) nj =1 . Since u i = 0, we have( x T · C ) u u i + ( x T · C ) u u i + · · · + x T · C i + · · · + ( x T · C n ) u n u i = 0 , i.e. x T · ( u u i C + · · · + C i + · · · + u n u i C n ) = 0 . The previous condition holds for any x ∈ V, hence C i = − n X j = 1 j = i u j u i C j . As M u is antisymmetric, the same argument can be applied also to the i -throw of M u . The following corollaries are straightforward. Corollary 2.2. rank ( M u ) ≤ n − . Corollary 2.3. The point [ u ] is a H -pole if and only if rank ( M u ) ≤ n − . Note that the matrix M u is antisymmetric; hence its rank must be an evennumber. By Corollary 2.3 it is clear that if n is even then every point of PG( V ) isa H -pole and this holds for any hyperplane H of G ( V ) . More precisely, the degreeof the point [ u ] is δ ( u ) = ( n − − rank ( M u ) . So, it is straightforward to see thatthe set of all the H -poles of degree at least t is either the whole of PG( V ) or thedeterminantal variety of PG( V ) described by the condition rank ( M u ) ≤ ( n − − t ≤ n − 2; see [9, Lecture 9] for some properties of these varieties. Furthermore allentries of M u are linear homogeneous polynomials in the coordinates of u ; so thecondition rank ( M u ) ≤ n − u for [ u ] to be a H -pole.In Table 2 and Table 3 of Appendix A we have explicitly written the matrices M u associated to the trilinear forms h of type T i of Table 1 where u = ( u i ) ni =1 . Tosimplify the notation, when rank ( h ) < dim( V ), we have just written the rank ( h ) × rank ( h )-matrix associated to h u | V/ Rad ( h ) . 9o determine the elements of the upper radical of H , namely the lines ℓ = [ x, y ]of PG( V ) with the property that any plane through them is in H , we need todetermine conditions on x and y such that the linear functional˜ h xy : V → K , ˜ h xy ( u ) = h ( u ∧ x ∧ y ) (2)is null. To do this, it is sufficient to require that ˜ h xy annihilates on the basis vectorsof V , i.e. ˜ h xy ( e i ) = 0 , for every i = 1 , . . . , n . Let h be a trilinear form associated to the hyperplane H and for any u ∈ V , let χ u be the alternating bilinear form as in Equation (1) whose representative matrix is M u . With 1 ≤ i ≤ n , denote by M ( i ) u the principal submatrix of M u obtained bydeleting its i -th row and its i -th column. The matrix M ( i ) u is a ( n − × ( n − K ; so, itsdeterminant is a polynomial of degree n − u , . . . , u n which isa square in the ring K [ u , . . . , u n ], that is there exists a polynomial d i ( u , . . . , u n )with deg d i ( u , . . . , u n ) = ( n − / M ( i ) u = ( d i ( u , . . . , u n )) . (3)Define g i ( u , . . . , u n ) to be the polynomial in K [ u , . . . , u n ] such that d i ( u , . . . , u n ) = u α i i g i ( u , . . . , u n )where α ∈ N and u α i +1 i does not divide d i ( u , . . . , u n ). Theorem 2.4. The set P ( H ) of H -poles is either the whole pointset of PG( V ) orthere exists an index i , ≤ i ≤ n , such that P ( H ) is an algebraic hypersurface of PG( V ) with equation g i ( u , . . . , u n ) = 0 .Proof. Suppose P ( H ) = PG( V ). By [6], P ( H ) is an algebraic hypersurface admit-ting an equation of degree ( n − / 2; so P ( H ) contains at most ( n − / j : u j = 0. Then, there exists i with 1 ≤ i ≤ n suchthat Π i P ( H ). By Corollary 2.3, [ u ] ∈ PG( V ) is a H -pole if and only ifrank ( M u ) ≤ n − . If n is even, this always happens. So, assume n odd. We shallwork over the algebraic closure K of K . Take u = ( u , . . . , u n ) with u j ∈ K . Wecan regard u as a vector in coordinates with respect to the basis induced by E on V := V ⊗ K . In any case, the matrix M u is antisymmetric and its entries arehomogeneous linear functionals in u , . . . , u n defined over the field K .Consider the points [ u ] with u i = 0 . By Proposition 2.1, the i -th row/columnof M u is a linear combination of the remaining n − M u =rank M ( i ) u . 10et d i ( u , . . . , u n ) be as in Equation (3). We now show that d i ( u , . . . , u i − , , u i +1 , . . . , u n ) = 0for all u , . . . , u i − , u i +1 , . . . , u n ∈ K . Indeed, when u i = 0, by Proposition 2.1,there exists a column C j of M u , u j = 0, such that C j = − n X k = 1 k = j u k u j C k . So, the j -th column of M ( i ) u is also a linear combination of the other columns of M ( i ) u . Hence det M ( i ) u = 0 = ( d i ( u , . . . , , . . . , u n )) .Since K is algebraically closed, we have d i ( u , . . . , u n ) = u i d ′ i ( u , . . . , u n ) , with d ′ i ( u , . . . , u n ) a polynomial in K [ u , . . . , u n ] with deg d ′ i = ( n − / 2. Weremark that the unknowns of the polynomials may assume their values in K butthe coefficients are all in K . By Corollary 2.3, a point [ u ] ∈ PG( V ) \ Π i (i.e. u i = 0)is a H -pole if and only if det M ( i ) u = 0, i.e. d ′ i ( u , . . . , u n ) = 0 . Denote by Γ i the algebraic variety (over K ) of equation d ′ i ( u , . . . , u n ) = 0.Since we are assuming Π i P ( H ) and P ( H ) \ Π i = Γ i \ Π i , we have P ( H ) = C ( P ( H ) \ Π i ) = C (Γ i \ Π i ) , where C ( X ) denotes the projective closure of X in PG( V ) with Π i regarded as thehyperplane at infinity. Note that since P ( H ) is an algebraic variety of PG( V ) whichdoes not contain Π i , the points of P ( H ) ∩ Π i are exactly those of the projectiveclosure C ( P ( H ) \ Π i ) ∩ Π i . The same applies to the points at infinity of the affinevariety Γ i \ Π i .Suppose u βi | d ′ i ( u , . . . , u n ) and u β +1 i d ′ i ( u , . . . , u n ) for β ∈ N .Then, d ′ i ( u , . . . , u n ) = u βi g i ( u , . . . , u n ) and g i ( u , . . . , u n ) = 0 is an equationfor C (Γ i \ Π i ) = P ( H ). This completes the proof.Observe that if α i = 1, then g i ( u , . . . , u n ) = 0 is exactly an equation of degree( n − / P ( H ).If n is even then each point of PG( V ) is a H -pole. Likewise, when n is oddand H is defined by a trilinear form χ of rank less than n , then also each pointof PG( V ) is a H -pole. Indeed, whenever R ↓ ( H ) = Rad ( χ ) is non-trivial, for any[ u ] ∈ PG( V ) there exists a line ℓ through [ u ] in R ↑ ( H ) which meets Rad ( χ ) andso [ u ] is a H -pole. In any case the above conditions, while sufficient, are not ingeneral necessary; in fact, in Section 3.3 we shall provide a construction which11ight lead to alternating forms of rank n with n odd and R ↓ ( H ) = ∅ such that allpoints of PG( V ) are H -poles.Note that Theorem 2.4 shows that the variety of the H -poles in PG( V ) admitsat least one equation over K of degree at most ( n − / 2; this does not meanthat ( n − / g i ( u , . . . , u n ) generates the radical ideal of such variety (even over K ). Forinstance, in the case of symplectic hyperplanes, see Section 3.2.1, the variety ofpoles is always a hyperplane of PG( V ) and so it admits an equation of degree 1for any odd n . It is an interesting problem to investigate which algebraic varieties might ariseas set of H -poles; as noted before such varieties will always be skew-symmetricdeterminantal varieties [8]. We leave the development of this study to a furtherpaper. However, with this aim in mind, we give here a construction for hyperplaneswhose variety of poles are reducible. Theorem 2.5. Suppose V = V + V with ( e , . . . , e n ) and ( e n , . . . , e n ) bases of V and V respectively. For i = 1 , let H i be a hyperplane of G ( V i ) whose H i -poles in PG( V i ) satisfy respectively the equations f ( u , . . . , u n ) = 0 and f ( u n , . . . , u n ) =0 . Then there exists a hyperplane H of G ( V ) whose set of H -poles defines avariety of PG( V ) with equation f ( u , . . . , u n ) = 0 where f ( u , . . . , u n ) := u n · f ( u , . . . , u n ) f ( u n , . . . , u n ) . Proof. For i = 1 , h i the trilinear form on V i defining H i and considerthe extension h i : V × V × V → K given by h i ( x, y, z ) = h i ( π i ( x ) , π i ( y ) , π i ( z ))where π : V → V is the projection on V along h e n +1 , . . . e n i and π : V → V isthe projection on V along h e , . . . e n − i . Put h := h + h and let H be the hyperplane of G ( V ) defined by h. For any[ u ] ∈ PG( V ), let χ u as in Equation (1) be the bilinear alternating form induced by h and represented by the matrix M u with respect to the basis ( e , . . . , e n , . . . , e n )of V. By construction, the matrix M u has the following structure: M u = M ( v n ) 0 − ( v n ) T − ( v n ) T v n ) M M := (cid:18) M ( v n ) − ( v n ) T (cid:19) is the n × n -matrix representing h and ¯ M = (cid:18) − ( v n ) T ( v n ) M (cid:19) is the ( n − n + 1) × ( n − n + 1)-matrix representing h . Notethat ( v n ) and ( v n ) are suitable columns with entries respectively in the rings K [ u , . . . , u n ] and K [ u n , . . . , u n ].Suppose u n = 0. Since the entries of ¯ M and ¯ M are respectively linearfunctionals in u , . . . , u n and u n , . . . , u n if all entries u i of u with i ≥ n are null,then ¯ M is a zero matrix and rank M u ≤ n − 2. Likewise if all entries u i of u with i ≤ n are zero, then ¯ M is the zero matrix and rank M u ≤ n − 2. Assume nowthat u n = 0 and that there exist i, j with i < n and j > n such that u i = 0 = u j .Then, by Proposition 2.1, the first ( n − 1) columns of ¯ M and the last n − n columns of ¯ M are a linearly dependent set; in particular, rank M u ≤ n − 2. So wehave proved that the hyperplane Π n : u n = 0 is contained in the variety of poles P ( H ).By construction, rank ¯ M ≤ n − f ( u , . . . , u n ) = 0 andrank ¯ M ≤ n − n − f ( u n , . . . , u n ) = 0. Let ∆ be the variety ofequation f ( u , . . . , u n ) f ( u n , . . . , u n ) = 0. Observe that ∆ \ Π n ⊆ P ( H ) \ Π n ,as [ u ] ∈ ∆ \ Π n implies that either rank ¯ M ( n )1 ≤ rank ¯ M ≤ n − M ( n )2 ≤ rank ¯ M ≤ n − n − u n = 0), the n -th column of M is a linear combinations of the columns of ¯ M ( n )1 as well as a linear combination ofthe columns of ¯ M ( n )2 (so it does not contribute to the rank); so rank M u ≤ n − u ] ∈ P ( H ) \ Π n . Conversely, suppose [ u ] ∈ P ( H ) with u n = 0. Then, byTheorem 2.4,0 = det M ( n ) u = det M · det M = det ¯ M ( n )1 · det ¯ M ( n )2 . If [ u ] ∈ PG( V ) \ Π n , again by Theorem 2.4 applied to V and V we havedet ¯ M ( n )1 = 0 if and only if f ( u , . . . , u n ) = 0 and det ¯ M ( n )2 = 0 if and onlyif f ( u n , . . . , u n ) = 0. So P ( H ) \ Π n ⊆ ∆ \ Π n , whence P ( H ) \ Π n = ∆ \ Π n Since Π n ⊆ P ( H ), we have P ( H ) = ( P ( H ) \ Π n ) ∪ Π n = (∆ \ Π n ) ∪ Π n and, consequently f ( u , . . . , u n ) = u n · f ( u , . . . , u n ) · f ( u n , . . . , u n ) is an equa-tion for P ( H ). Corollary 2.6. Let V be a vector space of odd dimension n ≥ over a field K .Then there exists a hyperplane H of G ( V ) whose set of H -poles is the union of ( n − / distinct hyperplanes of PG( V ) . roof. We proceed by induction on n odd. If n = 5, consider the hyperplane of G ( V ) defined by the trilinear form h := 123 + 345 . It is easy to verify that itspoles are all the points of the hyperplane of equation u = 0 . By induction hypothesis, suppose that the thesis holds for vector spaces of odddimension n . We shall prove it also holds for vector spaces of (odd) dimension n + 2 . Let V with dim( V ) = n + 2 and let ( e i ) i =1 ,...,n +2 be a given basis of V. Put V := h e i i ≤ i ≤ n and V := h e n , e n +1 , e n +2 i . Clearly V = V + V . As dim( V ) = n we can apply the induction hypothesis. So, there exists a trilinear form h on V (defining a hyperplane H of G ( V )) such that its set of poles is the union of( n − / V ). Equivalently, we can assume withoutloss of generality that any H -pole satisfies the equation g ( u , . . . , u n ) = 0 where g ( u , . . . , u n ) := Q ( n − / i =1 u i +1 . Let h be the trilinear form on V defined by h = ( n )( n + 1)( n + 2). Clearly h has no pole in PG( V ) . By (the proof of) Theorem 2.5, we can consider the hyperplane H of G ( V )defined by the sum of the extensions h and h to V of h and h (see the beginningof the proof of Theorem 2.5 for the definition of h and h ). Then, the set of H -poles is a variety of PG( V ) with equation g ( u , . . . , u n +2 ) = 0 where g ( u , . . . , u n +2 ) := g ( u , . . . , u n ) · u n = ( n − / Y i =1 u i +1 · u n = ( n − / Y i =1 u i +1 . In this section we will explain some general constructions yielding large families ofhyperplanes of k -Grassmannians. More precisely, in Sections 3.1 and 3.2 we shallbriefly recall (without proofs) two constructions already introduced in [3] while inSection 3.3 we will present a new one.We first need to give the following definition which extends the definition of S p ( H ) given in Section 1.1. For a ( k − X of V , let ( X ) G k be theset of k -subspaces of V containing X . This is a subspace of G k ( V ). Let ( X ) G k be the geometry induced by G k ( V ) on ( X ) G k and put ( X ) H := ( X ) G k ∩ H .Then ( X ) G k ∼ = G ( V /X ) and either ( X ) H = ( X ) G k or ( X ) H is a hyperplane of( X ) G k . In either case, the point-line geometry S X ( H ) = (( X ) G k − , ( X ) H ) is apolar space of symplectic type (possibly a trivial one, when ( X ) H = ( X ) G k )). Let R X ( H ) := Rad ( S X ( H )) be the radical of S X ( H ).14 .1 Extensions and trivial extensions Let V = V ⊕ V be a decomposition of V as the direct sum of two non-trivialsubspaces V and V . Put n := dim( V ) and assume that n ≥ k ( ≥ χ : V × · · · × V → K be a non-trivial k -linear alternating form on V . The form χ can naturally be extended to a k -linear alternating form χ of V by setting χ ( x , . . . , x k ) = 0 if x i ∈ V for some 1 ≤ i ≤ k,χ ( x , . . . , x k ) = χ ( x , . . . , x k ) if x i ∈ V for all 1 ≤ i ≤ k, (cid:27) (4)and then extending by (multi)linearity. Let H χ be the hyperplane of G k ( V ) definedby χ . Then, the following properties hold: Theorem 3.1 ([3]) . Let χ be a k -alternating linear form on V and n = dim V ;define χ as in (4) . For n = k , put H = ∅ ; otherwise, let H be the hyperplaneof G k ( V ) defined by χ . Let also π : V → V be the projection of V onto V along V . Then,(1) H χ = { X ∈ G k ( V ) : either X ∩ V = 0 or π ( X ) ∈ H } .(2) R ↓ ( H χ ) = h R ↓ ( H ) ∪ [ V ] i where the span in taken in PG( V ) .(3) R ↑ ( H χ ) = { X ∈ G k − ( V ) : either X ∩ V = 0 or π ( X ) ∈ R ↑ ( H ) } . (4) When k = 3 , the points [ p ] [ V ] have degree δ ( p ) = δ ( π ( p )) + n − n , where δ ( π ( p )) is the degree of [ π ( p )] with respect to H . The points p ∈ [ V ] havedegree n − . We call H χ the trivial extension of H centered at V (also extension of H by V , for short) and we denote it by the symbol Ext V ( H ) . When k = n we have H = ∅ ; we shall call Ext V ( ∅ ) the trivial hyperplane centered at V . In this caseTheorem 3.1 can also be rephrased as follows, with no direct mention of H . Proposition 3.2 ([3]) . Let H = Ext V ( ∅ ) be a trivial hyperplane of G k ( V ) . Then H = { X ∈ G k ( V ) : X ∩ V = 0 } . Moreover, R ↓ ( H ) = [ V ] , R ↑ ( H ) = { X ∈ G k − ( V ) : X ∩ V = 0 } and, for X ∈G k − ( V ) , if X ∩ V = 0 then R X ( H ) = [ V /X ] , otherwise R X ( H ) = [( V + X ) /X ] . By construction, the lower radical of a trivial extension is never empty. Thefollowing theorem shows that the converse is also true, namely if R ↓ ( H ) = ∅ then H is a trivial extension, possibly a trivial hyperplane.If S is a subspace of V with dim( S ) > k , denote by H ( S ) := G k ( S ) ∩ H thehyperplane of G k ( S ) induced by H. heorem 3.3 ([3]) . Suppose R ↓ ( H ) = ∅ and let S, S ′ be complements in V of thesubspace R < V such that [ R ] = R ↓ ( H ) . Then(1) H = Ext R ( H ( S )) ;(2) H ( S ) ∼ = H ( S ′ ) ;(3) R ↓ ( H ( S )) = ∅ . Each hyperplane H of G k ( V ) defined by a k -linear alternating form h withrank ( h ) < dim V is clearly a trivial extension of a hyperplane H ′ of G k ( V ′ ) withdim V ′ = rank ( h ), since V = Rad ( h ) ⊕ V ′ . Let V be a hyperplane of V and H a given hyperplane of G k − ( V ). Assume k ≥ 3; hence V has dimension n ≥ 4. PutExp( H ) := { X ∈ G k ( V ) : either X ⊂ V or X ∩ V ∈ H } . Theorem 3.4 ([3]) . The set Exp( H ) is a hyperplane of G k ( V ) . Moreover,(1) R ↓ (Exp( H )) = R ↓ ( H ) .(2) R ↑ (Exp( H )) = H ∪ { X ∈ G k − ( V ) \ G k − ( V ) : X ∩ V ∈ R ↑ ( H ) } .(3) For X ∈ G k − ( V ) , if X ⊆ V with X ∈ R ↑ ( H ) then R X (Exp( H )) = S X (Exp( H )) = ( X ) G k − (the latter being computed in V ). If X ⊆ V but X R ↑ ( H ) then R X (Exp( H )) = ( X ) H (a subspace of PG( V /X ) ). Fi-nally, if X V , then R X (Exp( H )) = {h x, Y i : Y ∈ R X ∩ V ( H ) } for agiven x ∈ X \ V , no matter which. We call Exp( H ) the expansion of H . A form h : V k V → K associated toExp( H ) can be constructed as follows. Suppose h : V k − V → K is the ( k − H . Suppose V = h e , . . . , e n − i where E = ( e i ) ni =1 is the given basis of V . Recall that { e i ∧ · · · ∧ e i k : 1 ≤ i < . . . < i k ≤ n } is abasis of V k V . Put h ( e i ∧ · · · ∧ e i k ) = (cid:26) i k < n,h ( e i ∧ · · · ∧ e i k − ) if i k = n. (5)and extend it by linearity. It is easy to check that the form h defines Exp( H ).We now recall some properties linking expansions and trivial extensions whichmight be of use in investigating the geometries involved.16 heorem 3.5 ([3]) . Let H be a hyperplane of G k ( V ) ; then1. R ↓ (Exp( H )) = ∅ if and only if R ↓ ( H ) = ∅ ;2. denote by S a complement of R ≤ V such that [ R ] = R ↓ ( H ) ; then, Exp( H ) = Ext R (Exp( H ( S ))) where H ( S ) is the hyperplane induced on S by H ;3. if H is trivial, then Exp( H ) is also trivial with center R ↓ ( H ) . Assume now k = 3 and take H to be a hyperplane of G ( V ) (hence defined bya bilinear alternating form of V ). The point-line geometry S ( H ) = ( G ( V ) , H )having as points all points of [ V ] and as lines all elements in H , is a polar spaceof symplectic type. The upper and lower radical of H are mutually equal andcoincide with the radical R ( H ) of S ( H ).First suppose that S ( H ) is non-degenerate. Then n − n ≥ 5. Claims (1) and (2) of Theorem 3.4 imply that R ↓ (Exp( H )) = ∅ and R ↑ (Exp( H )) = H ; thus the geometry of poles P (Exp( H )) of Exp( H ) coin-cides precisely with the symplectic polar space S ( H ) . In particular, the points of[ V ] \ [ V ] are smooth while those of [ V ] are poles of degree 1. Motivated by theabove remark we call Exp( H ) a symplectic hyperplane whenever R ( H ) = ∅ . Assume now that S ( H ) is degenerate, i.e. Rad ( S ( H )) = 0. In this case, R ↓ (Exp( H )) = 0 since R ↓ (Exp( H )) = Rad ( S ( H )); so Exp( H ) is either atrivial extension of a symplectic hyperplane by Rad ( S ( H )) (this happens whendim(Rad ( S ( H )) < n − 3) or a trivial hyperplane centered at Rad ( S ( H )) (thishappens when dim(Rad ( S ( H )) = n − The construction of block decomposable hyperplanes can be done for general k ≥ k = 3 . Suppose V = V ⊕ V . Any vector x ∈ V can then be uniquely written as x = x + x with x ∈ V and x ∈ V . For i = 0 , h i : V V i → K bea linear functional defining the hyperplane H i of G ( V i ). Consider the extension h i : V V → K of ¯ h i to V given by h i ( x ∧ y ∧ z ) = ¯ h i ( x i ∧ y i ∧ z i )where x = x + x , y = y + y , z = z + z ∈ V and x i , y i , z i ∈ V i . h := h + h be the trilinear form of V defined by the sum of h and h . So, h (( x + x ) ∧ ( y + y ) ∧ ( z + z )) = ¯ h ( x ∧ y ∧ z ) + ¯ h ( x ∧ y ∧ z ) . Then the hyperplane of G ( V ) defined by h is called a block decomposablehyperplane arising from H and H and it will be denoted by Dec( H , H ) . Theorem 3.6. Let H := Dec( H , H ) be a block decomposable hyperplane of G ( V ) . Then the following hold:1. The poles of H are all the points of PG( V ⊕ V ) ;2. R ↑ ( H ) = { ℓ ∈ G ( V ) : ( π ( ℓ ) ∈ R ↑ ( H ) or dim( π ( ℓ )) < and ( π ( ℓ ) ∈ R ↑ ( H ) or dim( π ( ℓ )) < } where π i : V → V i is the projection of V onto V i along V j ( j = i , i = 0 , ). Denote by ε : G ( V ) → PG( V V ) the Pl¨uckerembedding of the -Grassmannian G . We have ε ( R ↑ ( H )) = [ ε ( R ↑ ( H )) + ε ( R ↑ ( H )) + V ∧ V ] ∩ ε ( G ) where V ∧ V := h v ∧ v : v ∈ V , v ∈ V i . Proof. Put n = dim V and n = dim V . Let u ∈ V where u = u + u ∈ V , u ∈ V and u ∈ V . Denote by M u the matrix of the bilinear form χ u ( x, y ) := h ( u ∧ x ∧ y ). Then, M u is a block matrix of the form M u = (cid:18) M u M u (cid:19) where M u i is the matrix representing the form χ iu ( x, y ) := ¯ h i ( u ∧ x ∧ y ) associatedto the hyperplane H i of G ( V i ) . For any x, y, u ∈ V with x = x + x , y = y + y , u = u + u and x i , y i , u i ∈ V i ,we have by definition of decomposable hyperplane, χ u ( x, y ) = x T M u y = x T M u y + x T M u y . By Corollary 2.2, rank ( M u i ) ≤ n i − 1. So, rank M u ≤ ( n − 1) + ( n − 1) = n − u ] = [ u + u ] is a pole.A line ℓ = h x, y i is in the upper radical R ↑ ( H ) if, and only if, for any choice of u ∈ V we have χ u ( x, y ) = 0. Since χ u ( x, y ) = χ u ( x , y ) + χ u ( x , y ) , where x i , y i ∈ V i and x = x + x , y = y + y , we have ℓ ∈ R ↑ ( H ) if andonly if for all u i ∈ V i and i = 0 , χ u i ( x i , y i ) = 0 . This holds if and only if18 π ( ℓ ) ∈ R ↑ ( H ) or dim( π ( ℓ )) < 2) and ( π ( ℓ ) ∈ R ↑ ( H ) or dim( π ( ℓ )) < . The first part of claim 2 is proved.Consider the following subspace of PG( V V ) L := [ ε ( R ↑ ( H )) + ε ( R ↑ ( H )) + V ∧ V ]where V ∧ V := h v ∧ v : v ∈ V , v ∈ V i . We claim that any point in ε ( R ↑ ( H )) + ε ( R ↑ ( H )) + V ∧ V is in ε ( R ↑ ( H )) . Since R ↑ ( H i ) ⊆ R ↑ ( H ) (for i = 0 , 1) and ε − (( V ∧ V ) ∩ ε ( G ( V ))) ⊆ R ↑ ( H ), bythe first part of claim 2, the inclusion L ⊆ ε ( R ↑ ( H )) is immediate.Suppose now h ( x + x ) ∧ ( y + y ) i ∈ ε ( R ↑ ( H )). We can write (by the firstpart of claim 2)( x + x ) ∧ ( y + y ) = ( x ∧ y ) | {z } ∈ ε ( R ↑ ( H )) + ( x ∧ y ) + ( x ∧ y ) | {z } ∈ V ∧ V + ( x ∧ y ) | {z } ∈ ε ( R ↑ ( H )) ∈ L . The thesis follows.We remark that, with some slight abuse of notation, the extension Ext V ( H ) ofan hyperplane H can always be regarded as a special case of a block decomposablehyperplane, where the form ¯ h defined over V is identically null.The definition given for block decomposable hyperplane arising from two hy-perplanes H and H can be extended by induction to the definition of blockdecomposable hyperplane Dec( H , · · · , H n − ) arising from n hyperplanes H i (0 ≤ i ≤ n − H i is a hyperplane of G ( V i ) and V = ⊕ i V i . In general, given two linear subspaces V , V of V such that V = V ⊕ V ,and given two hyperplanes H and H of G ( V ) and G ( V ), there exist severalpossible hyperplanes H of G ( V ⊕ V ) which are block decomposable and arisefrom H and H , namely all of those induced by the forms h α,β := αh + βh with α, β ∈ K \ { } . Even if all these hyperplanes are in general neither equivalent nornearly equivalent, they turn out to be always geometrically equivalent and theirgeometry of poles depends only on the geometries of H and H . In this section we will prove our Theorems 1, 2 and 3. As in the previous sections,let E := ( e i ) ni =1 be a given basis of V. Let H be a given hyperplane of G ( V ) and P ( H ) = ( P ( H ) , R ↑ ( H )) be the geometry of poles of H. In Section 2.1 we haveexplained how to algebraically describe the pointset P ( H ) and the lineset R ↑ ( H )of the geometry of poles associated to H. The main steps to describe P ( H ) arethe following: 19 Consider the bilinear form χ u associated to the trilinear form defining H andwrite the matrix M u representing χ u . This is done in Table 2 for forms ofrank up to 6 and in Table 3 for forms of rank 7 . Recall that for dim( V ) ≤ K perfect with cohomological dimension at most 1, all trilinear formsare classified: they are listed in Table 1. • By Theorem 2.4, we know that the set of poles is either the pointset of PG( V )or an algebraic variety. If dim( V ) = 6 all points of PG( V ) are poles, for anyhyperplane H of G ( V ) . If dim( V ) = 7, to get the equations describing thevariety P ( H ) we rely on Corollary 2.3, which gives algebraic conditions on( u i ) ni =1 for [( u i ) ni =1 ] to be a H -pole. In the second column of Table 5 we havewritten down those equations, according to the type of H. • To describe the lines of P ( H ) we rely on the last part of Section 2.1. Inparticular, ℓ := [ x, y ] ∈ R ↑ ( H ) if and only if the functional ˜ h xy described inEquation (2) is the null functional. This immediately reads as some linearequations in the Pl¨ucker coordinates | x, y | ij of the line ℓ . The results of these(straightforward) computations are reported in the third column of Tables 4for forms of rank at most 6 and Table 5 for forms of rank 7.So, Tables 1, 2, 3, 4, 5 provide an algebraic description of points and lines of P ( H ) , for any hyperplane H of G ( V ) . In the remainder of this section we shallalso provide a geometrical description of P ( H ) . T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 3equivalent to h = 123, see the first row of Table 1.Suppose dim( V ) ≥ e i ) ni =1 be a given basis of V. Then Rad ( h ) = h e i i i ≥ . According to Section 3.1, H is a trivial hyperplane Ext Rad ( h ) ( ∅ ) centeredat Rad ( h ) . By Proposition 3.2, the set of Ext Rad ( h ) ( ∅ )-poles is the whole pointsetof PG( V ) and the lines of the geometry of poles, i.e. the elements in the upperradical R ↑ (Ext Rad ( h ) ( ∅ )), are those lines of PG( V ) meeting Rad ( h ) non-trivially.When n = dim( V ) ≤ 6, this proves part 1 of Theorem 1. T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 5equivalent to h = 123 + 145, see the second row of Table 1.Suppose dim( V ) > . Then dim(Rad ( h )) ≥ . Let V = Rad ( h ) ⊕ V ′ . BySection 3.1, H is a trivial extension Ext Rad ( h ) ( H ′ ) of a hyperplane H ′ of G ( V ′ ) . 20y Theorem 3.3, we can assume without loss of generality V ′ = h e i i i =1 . Put V := h e i i i =2 and consider the hyperplane H of G ( V ) defined by the functional 23 + 45 . By Section 3.2, H ′ is the expansion Exp( H ) of H . By Subsection 3.2.1, since thegeometry S ( H ) = ( G ( V ) , H ) is a non-degenerate symplectic polar space, thegeometry of poles of Exp( H ) coincides precisely with S ( H ). Hence, if dim( V ) > H is a trivial extension Ext Rad ( h ) (Exp( H )) of a symplectic hyperplane Exp( H ) . The H -poles are all the points of PG( V ) and the lines of the geometry of polesare all the lines ℓ of PG( V ) meeting Rad ( h ) non-trivially or such that π ( ℓ ) ∈ H where π is the projection onto V along Rad ( h ) . If dim( V ) = 5 then H is the expansion Exp( H ) of a non-degenerate symplecticpolar space S ( H ) defined by the functional 23 + 45 in V = h e i i i =2 . By Subsec-tion 3.2.1, the geometry of poles of H coincides with the symplectic polar space S ( H ) . Part 2 of Theorem 1 is proved. T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 6equivalent to h = 123 + 456, see the third row of Table 1.Suppose dim( V ) = 6. Let ( e i ) i =1 be a given basis of V . Put V = h e , e , e i and V = h e , e , e i . Clearly, V = V ⊕ V . For i = 0 , 1, denote by ¯ h i :=(3 i + 1)(3 i + 2)(3 i + 3) the trilinear form induced by the restriction of h to V V i . By Section 3.3, H is a decomposable hyperplane Dec( H , H ) of G ( V ) arising fromthe hyperplanes H i , i = 0 , 1, of G ( V i ) defined by the forms ¯ h i . Since R ↑ ( H i ) = ∅ ,by Theorem 3.6, all points of PG( V ) are elements of the geometry of poles P ( H )and the lines of P ( H ) are exactly those lines of PG( V ) intersecting both PG( V )and PG( V ) . This proves part 3 of Theorem 1.Suppose dim( V ) > 6. Let ( e i ) i ≥ be a given basis of V . Then Rad ( h ) = h e i i i ≥ . By last part of Section 3.1, H is a trivial extension Ext Rad ( h ) (Dec( H , H )) of adecomposable hyperplane Dec( H , H ) of G ( V ′ ) where V = Rad ( h ) ⊕ V ′ and H i , i = 0 , 1, are the hyperplanes of G ( V i ), V i = h e i +1 , e i +2 , e i +3 i , V ′ = V ⊕ V ,defined by (3 i + 1)(3 i + 2)(3 i + 3). T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 6equivalent to h = 162 + 243 + 135, see the fourth row of Table 1.Suppose dim( V ) = 6 . For any u ∈ V , rank ( M u ) ≤ M u ). If ( e i ) i =1 is a basis of V and V = V ⊕ V with V = h e , e , e i and V = h e , e , e i , by a direct computation we have that theelements of V are poles of degree 3 while all remaining poles have degree 1.21et ℓ = [ u, v ] be a line of PG( V ). By the forth row of Table 4, ℓ ∈ R ↑ ( H ) ifand only if its Pl¨ucker coordinates satisfy 6 linear equations. More explicitly, wehave that ℓ = [ u, v ] ∈ R ↑ ( H ) if and only if u = ¯ u + ¯ u and v = ω (¯ u ) ∈ V with¯ u ∈ V , ¯ u ∈ V and ω : V → V , ω ( u , u , u , u , u , u ) = ( u , u , u , u , u , u ) . Note that ω interchanges V and V . Indeed, if u = ¯ u + ¯ u and v = ¯ v + ¯ v with¯ u , ¯ v ∈ V and ¯ u , ¯ v ∈ V , the Pl¨ucker coordinates of the line ℓ = [ u, v ], satisfythe equations | u, v | = 0 , | u, v | = 0 , | u, v | = 0 if and only if ¯ v = λ ¯ u with λ ∈ K . Hence ℓ = [ u, v ] = [ u, v − λu ] = [¯ u + ¯ u , ¯ v ′ ] with ¯ v ′ = ¯ v − λ ¯ u ∈ V . The remaining three equations | x, y | −| x, y | = 0 , | x, y | −| x, y | = 0 , | x, y | −| x, y | = 0 are satisfied by the Pl¨ucker coordinates of ℓ = [ u, v ] = [¯ u + ¯ u , ¯ v ′ ]if and only if either ¯ u is the null vector and in this case ℓ = [¯ u , ¯ v ′ ] ⊂ V or[¯ v ′ ] = [(0 , , , u , u , u )] where [ u ] = [( u , u , u , u , u , u )] . This proves part 4 of Theorem 1.Suppose dim( V ) > 6. Let ( e i ) i ≥ be a given basis of V . Then Rad ( h ) = h e i i i ≥ . By last part of Section 3.1, H is a trivial extension Ext Rad ( h ) ( H ′ ) of a hyperplane H ′ of G ( V ′ ) where V = Rad ( h ) ⊕ V ′ , V ′ = h e i i i =1 and H ′ is defined by a trilinearform equivalent to 162 + 243 + 135 . A description of the geometry of poles of H follows from Theorem 3.1 and the already done case of hyperplanes of type T of G ( V ) for dim( V ) = 6 . T ( i )10 ,λ A hyperplane H of G ( V ) of type T ( i )10 ,λ is defined by a trilinear form of rank 6as written in row 10 or 11 of Table 1, according as char ( K ) is odd or even. Weremark that forms of type T (1)10 ,λ make sense also in even characteristic, providedthat the field K is not perfect.Let dim( V ) = 6 . For any u ∈ V , the matrix M u representing the bilinear alter-nating form χ u associated to H is written in Table 2. We have that rank ( M u ) ≤ u ∈ V , i.e. every point of PG( V ) is a pole. Actually, we will prove thatrank ( M u ) = 4 for any u ∈ V , i.e. every point of PG( V ) is a pole of degree 1,equivalently, R ↑ ( H ) is a line spread of PG( V ) . Consider first a hyperplane of type T (1)10 ,λ . Suppose by way of contradiction thatrank ( M u ) < , i.e. all minors of order 4 vanish. With u = ( u , . . . , u ), take thethree 4 × M u given byCase i = 1Principal submatrix of M u corresponding to rows/columns Value of the minor1 , , , λ ( λu − u ) , , , λ ( λu − u ) , , , λ ( λu − u ) . 22y the third column of Table 1 corresponding to T (1)10 ,λ , we have that p λ ( t ) = t − λ is an irreducible polynomial in K [ t ] . Hence the minors mentioned are null if andonly if u i = 0 for all i = 1 , . . . , . We argue in a similar way for case T (2)10 ,λ with char ( K ) = 2, by choosing theminors of M u given by Case i = 2Principal submatrix of M u corresponding to rows/columns Value of the minor1 , , , u + λu u + u ) , , , u + λu u + u ) , , , u + λu u + u ) . By the third column of Table 1 corresponding to T (2)10 ,λ , we have that p λ ( t ) := t + λt + 1 is irreducible in K [ t ]; hence rank M u = 4 unless u = (0 , . . . , R ↑ ( H ) is a line-spread of PG( V ). Lemma 4.1. Let H be a hyperplane of G ( V ) with dim V = 6 whose upper radical R ↑ ( H ) is a line-spread of PG( V ) . Then R ↑ ( H ) is a Desarguesian line-spread of PG( V ) .Proof. For simplicity of notation, denote by S the line-spread of PG( V ) inducedby H. Then, by duality, H induces also a line spread S ∗ in the dual space PG( V ∗ ),where V ∗ is the dual of V . In particular, a 4-dimensional vector space Σ is in S ∗ if and only if all planes contained in Σ are elements of H . We will prove that S isa normal spread , i.e. given any two distinct elements ℓ , ℓ ∈ S and Σ = ℓ + ℓ ,the set S Σ := { ℓ ∈ S : ℓ ⊆ Σ } is a line-spread of Σ.Let π be a plane contained in Σ. Then, π ∩ ℓ = ∅ 6 = π ∩ ℓ and we can write π = [ p , p , q + q ] with p , q ∈ ℓ , p , q ∈ ℓ suitably chosen. Denote by h a formdefining the hyperplane H . Then h ( p ∧ p ∧ ( q + q )) = h ( p ∧ p ∧ q ) + h ( p ∧ p ∧ q ) = 0 as h is identically zero on all planes through either ℓ or ℓ . Hence π ∈ H and, consequently, Σ ∈ S ∗ .So the 3-dimensional projective space spanned by any two elements of S is in S ∗ ; by duality, the intersection of any two elements of S ∗ is in S . Take now Σ ∈ S ∗ and p ∈ Σ; denote by ℓ p the unique line of S with p ∈ ℓ p . Let ℓ ′ ∈ S such that ℓ ′ is not contained in Σ. Then, Σ ′ = ℓ p + ℓ ′ ∈ S ∗ and, by the argument above,Σ ′ ∩ Σ ∈ S . Since p ∈ Σ ′ ∩ Σ, it follows that Σ ′ ∩ Σ = ℓ p . This for all points p ∈ Σ;so S Σ is a spread of Σ. Hence S is a normal spread and by [1, Theorem 2] S is aDesarguesian line spread of PG( V ) . Part 5 of Theorem 1 is now proved. 23 roof of Theorem 2. By Lemma 4.1, line-spreads of PG(6 , V ) induced byhyperplanes of G ( V ) are Desarguesian. As Desarguesian line-spreads are coordi-natized over division rings, there must exist a division ring D having dimension2 over K (see [2]) with either D commutative or K being the center of D . Weshow that D is commutative. Take a ∈ D \ K . Then the algebraic extension K ( a ) is a proper field extension of K contained in D ; so 2 = [ D : K ] ≥ [ K ( a ) : K ] ≥ 2. It follows that D = K ( a ) and D is a field which is an algebraic exten-sion of degree 2 of K . So, for D to exist, K must not be quadratically closed. (cid:3) Remark . The commutativity of D in the proof of Theorem 2 is also a consequenceof the Artin-Wedderburn theorem. We have provided a short argument.Suppose dim( V ) > 6. Let ( e i ) i ≥ be a given basis of V . Then Rad ( h ) = h e i i i ≥ . By last part of Section 3.1, H is a trivial extension Ext Rad ( h ) ( H ′ ) of a hyperplane H ′ of G ( V ′ ) where V = Rad ( h ) ⊕ V ′ , V ′ = h e i i i =1 and H ′ is defined by a trilinear formof type T ( i )10 ,λ . A description of the geometry of poles of H follows from Theorem 3.1and the already done case of hyperplanes of type T ( i )10 ,λ of G ( V ) with dim( V ) = 6 . Throughout this section let [ u ] = [( u i ) i =1 ]. T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 7equivalent to h = 123 + 456 + 147, see the fifth row of Table 1.Suppose dim( V ) = 7 . Straightforward computations shows that rank ( M u ) ≤ M u ) if and only if u = 0 or u = 0and rank ( M u ) = 2 if and only if u = u = u = u = 0 or u = u = u = u = 0 . Let S and S the hyperplanes of PG( V ) with equations respectively u = 0and u = 0. Denoted by P ( H ) the set of poles of H , we then have P ( H ) = S ∪ S . A point [ u ] has degree 4 if and only if [ u ] ∈ A ∪ A where A is the plane of PG( V )of equation u = u = u = u = 0 and A is the plane of PG( V ) of equation u = u = u = u = 0 . Take [ u ] = [(0 , u , . . . , u )] ∈ S . In this case, by the equations of Table 5, aline ℓ := [ u, v ] of PG( V ) through [ u ] is in R ↑ ( H ) if and only if ℓ ⊆ S , intersects A non-trivially and it is totally isotropic for the non-degenerate bilinear alternatingform of S defined by β ( u, v ) = u v − u v + u v − u v + u v − u v . A similarargument shows that if [ u ] is taken in S , then any line of R ↑ ( H ) through it meets A and it is totally isotropic for the non-degenerate bilinear alternating form of S defined by β ( u, v ) = u v − u v + u v − u v + u v − u v . T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 7equivalent to h = 152 + 174 + 163 + 243, see the sixth row of Table 1. Supposedim V = 7; straightforward computations show that rank ( M u ) ≤ M u ) if and only if u = 0. Let S be the hyperplane ofPG( V ) of equation u = 0. Then the set of the H -poles is precisely the point-setof S . Also, a point [ u ] has degree 4 if and only if [ u ] ∈ A where A is the plane ofequation u = u = u = u = 0. Take u = [(0 , u , . . . , u )] ∈ S . By the equations | u, v | = | u, v | = | u, v | , of Table 5, each line of the upper radical must intersectthe plane A . By the equation | u, v | + | u, v | + | u, v | = 0, we have that a line ℓ = [ u, v ] is in R ↑ ( H ) if and only if ℓ ⊆ S , ℓ ∩ A = 0 and ℓ is totally isotropicfor the non-degenerate alternating form β ( u, v ) := | u, v | + | u, v | + | u, v | = 0.This proves part 2 of Theorem 3. T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 7equivalent to h = 146 + 157 + 245 + 367 , see the seventh row of Table 1. Supposedim V = 7; straightforward computations show that rank ( M u ) ≤ M u ) if and only if u u + u u = 0. Let A bethe plane of PG( V ) of equations u = u = u = u = 0 and denote by Q thehyperbolic quadric of equation u u + u u = 0 embedded in the subspace W ofequations u = u = u = 0. The set of H -poles is the point-set of the quadraticcone of PG( V ) with vertex A and basis Q . Also, a point [ u ] has degree 4 if andonly if [ u ] ∈ C where C is the conic of A with equation u − u u = 0. Using theequations of Table 5, it is possible to associate to any [ x ] = [( st, t , s , , , , ∈ C a unique line ℓ x = [(0 , , , s, , , t ) , (0 , , , , s, − t, Q .With [ x ] ∈ C , denote by Res h A,ℓ x i ( x ) the projective geometry whose points areall the 2-dimensional vector spaces through x in the 5-dimensional vector space h A, ℓ x i and whose lines are the 3-dimensional vector spaces of h A, ℓ x i through x .Note that Res h A,ℓ x i ( x ) ∼ = PG(3 , K ). Proposition 4.2. For any [ x ] ∈ C there exists a line spread S x of Res h A,ℓ x i ( x ) such that ℓ ∈ R ↑ ( H ) if and only if ℓ ⊆ h x, s i for some x ∈ C and s ∈ S x .Proof. Suppose x, x ′ ∈ C with x = x ′ . Then, h A, ℓ x i ∩ h A, ℓ x ′ i = A . We shall nowdefine S x as follows S x := { π p : p ∈ h A, ℓ x i \ A } ∪ { A } π p is the unique plane of PG( V ) spanned by all the lines through p in R ↑ ( H ).Note that δ ( p ) = 2 and x ∈ π p ⊆ h A, ℓ x i .Let now [ q ] be a point in π p with π p ∈ S x . If [ q ] [ x, p ], then [ x, q ] and [ p, q ] areboth lines in the upper radical of H through [ q ]; since δ ( q ) = 2 we have π q = π p .If [ q ] ∈ [ x, p ] we consider a point [ r ] [ x, p ] and apply the same argument to showthat π q = π r = π p . Hence all lines of PG( V ) in any plane π p ∈ S x are in theupper radical and for any pole p A , the lines of the upper radical through p arecontained in π p .Suppose π p and π q are two lines of S x with non-trivial intersection, i.e. π p ∩ π q is a line through x not contained in A . Any point on this line would have degreeat least 3 — a contradiction.Finally all lines contained in A are elements of the upper radical and A readsas one line of S x .This proves part 3 of Theorem 3. T A hyperplane H of G ( V ) of type T is defined by a trilinear form of rank 7equivalent to h = 123+145+167, see the eighth row of Table 1. Suppose dim V = 7.Put V = h e i i i =2 and consider the hyperplane H of G ( V ) define by the functional23 + 45 + 67. By Subsection 3.2, H is the expansion Exp( H ) of H and since thegeometry S ( H ) := ( G ( V ) , H ) is a non-degenerate symplectic polar space, thegeometry of poles of Exp( H ) coincides with S ( H ).This proves part 4 of Theorem 3. T and T ,µ Hyperplanes H of G ( V ) of types either T or T ,µ are defined by trilinear formsof rank 7 nearly equivalent to h = 123 + 456 + 147 + 257 + 367, see rows 9and 14 of Table 1. Suppose dim V = 7; straightforward computations show thatrank ( M u ) ≤ M u ) if and only if[ u ] is a point of a non-degenerate parabolic quadric Q of PG( V ). More precisely,each point of Q has degree 2.The equations appearing in Table 5 for cases T (and T ,µ ) are the same asthose in [15, § H ( K ) in PG(6 , K ); see also [14]. Hence, the geometry of the poles of H is preciselya Split-Cayley hexagon. For these reasons, hyperplanes of this type are called hexagonal . This proves part 5 of Theorem 3.26 .2.6 Hyperplanes of type T ( i )11 ,λ A hyperplane H of G ( V ) of type T ( i )11 ,λ ( i = 1 , 2) is defined by a trilinear form ofrank 7 as written in row 12 or 13 of Table 1, according as char ( K ) is odd or even.As in the case T (1)10 ,λ , we remark that forms of type T (1)11 ,λ may also be considered ineven characteristic, provided that the field is not perfect. The geometries of polesarising in both cases i = 1 , V = 7. By the proof of Theorem 2.4, straightforward computa-tions show that rank ( M u ) ≤ M (7) u ) /u = 0 where by M (7) u isthe submatrix of M u (see Table 5 for the description of M u ) obtained by deletingits last row and column.First consider the case T (1)11 ,λ . Then [ u ] is a pole if and only if its coordinatessatisfy the equation λu − u = 0. Since the polynomial p λ ( x ) = x − λ is ir-reducible in K , the points satisfying the above equation have coordinates with u = u = 0. Hence, the set of poles is S := S ∩ S where S is the hyperplane ofPG( V ) of equation u = 0 and S is the hyperplane of PG( V ) of equation u = 0.Considering the 4 × M u given byCase i = 1Principal submatrix of M u corresponding to rows/columns Value of the minor1 , , , λ ( λu − u ) , , , λ ( λu − u ) , we have that the only point of degree 4 is [ e ].In the case T (2)11 ,λ , then [ u ] is a pole if and only if its coordinates satisfy theequation u + λu u + u = 0. Since the polynomial p λ ( x ) = x + λx + 1 isirreducible in K , the points satisfying the above equation have coordinates with u = u = 0. Hence, the set of poles is S := S ∩ S where S is the hyperplane ofPG( V ) of equation u = 0 and S is the hyperplane of PG( V ) of equation u = 0.Considering the 4 × M u given byCase i = 2Principal submatrix of M u corresponding to rows/columns Value of the minor1 , , , u + λu u + u ) , , , u + λu u + u ) so we have that the only point of degree 4 is [ e ] as above.We now provide a geometric description of R ↑ ( H ) holding for both i = 1 and i = 2. Denote by Res S ( e ) the projective geometry whose points are all the lines27f S through [ e ] and whose lines are the planes of S through [ e ]. It is well-knownthat Res S ( e ) ∼ = PG(3 , K ). Consider the following set F := { π p : p ∈ S \ [ e ] } where π p is the plane of PG( V ) spanned by the lines in R ↑ ( H ). Proposition 4.3. The set F is a line-spread of Res S ( e ) and R ↑ ( H ) = { ℓ ⊆ π p : π p ∈ F } . Proof. Take [ p ] ∈ S \ [ e ]. Since [ p ] a pole of degree 2, then there exists a plane π p with [ p ] ∈ π p spanned by lines of R ↑ ( H ). Note that [ p, e ] ∈ R ↑ ( H ); so [ e ] ∈ π p .Any line ℓ ⊆ π p is in the upper radical of H . Indeed, let now [ q ] be a point in π p with π p ∈ F . If [ q ] [ x, p ], then [ x, q ] and [ p, q ] are both lines in the upper radicalof H through [ q ]; since δ ( q ) = 2 we have π q = π p . If [ q ] ∈ [ x, p ] we consider a point[ r ] [ x, p ] and apply the same argument to show that π q = π r = π p . Hence alllines of PG( V ) in any plane π p ∈ F are in the upper radical of H .Suppose now π p , π q ∈ F and π p ∩ π q = r where r is a line through [ e ]. Thenany point on r has degree at least 3, a contradiction. We have proved that the set F is a line-spread of Res S ( e ). The characterization of the upper radical is nowstraightforward.This proves part 6 of Theorem 3. 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Van Maldeghem, Generalized polygons , Monographs in Mathematics, vol. 93, Birkh¨auserVerlag, Basel, 1998. Authors’ addresses:Ilaria CardinaliDepartment of Information Engineer-ing and MathematicsUniversity of SienaVia Roma 56, I-53100, Siena, [email protected] Luca GiuzziD.I.C.A.T.A.M.Section of MathematicsUniversity of BresciaVia Branze 43, I-25123, Brescia,[email protected] ppendix A Tables Table 1: Types for linear functionals on V V . Type Description Rank Special conditions, if any T 123 3 T 123 + 145 5 T 123 + 456 6 T 162 + 243 + 135 6 T 123 + 456 + 147 7 T 152 + 174 + 163 + 243 7 T 146 + 157 + 245 + 367 7 T 123 + 145 + 167 7 T 123 + 456 + 147 + 257 + 367 7 T (1)10 ,λ 123 + λ (156 + 345 + 426) 6 p λ ( t ) := t − λ irreducible in K [ t ] .T (2)10 ,λ 126 + 153 + 234+( λ + 1)456 + λ (156 + 345 + 426) 6 char ( K ) = 2 and p λ ( t ) := t + λt + 1 irreducible in K [ t ] .T (1)11 ,λ [the same as at T (1)10 ,λ ] + 147 7 same conditions as for T (1)10 ,λ T (2)11 ,λ [the same as at T (2)10 ,λ ] + 147 7 same conditions as for T (2)10 ,λ T ,µ µ [the same as at T ] 7 p µ ( t ) = t − µ irreducible in K [ t ] According to the clauses assumed on λ , types T ( r ) s,λ ( r ∈ { , } , s ∈ { , } )can be considered only when K is not quadratically closed. Moreover, when λ = λ ′ the types T ( r ) s,λ and T ( r ) s,λ ′ are different up to linear and near equivalence, even if theymight describe geometrically equivalent forms.It follows from Revoy [12] and Cohen and Helminck [4] that two functionals oftypes T i and T j with 1 ≤ i < j ≤ T i with i ≤ T ( r ) s,λ ; two functionalsof type T ( r ) s,λ and T ( r ′ ) s ′ ,λ ′ with ( r, s ) = ( r ′ , s ′ ) are never nearly equivalent while twofunctionals of type T ( r ) s,λ and T ( r ) s,λ ′ are nearly equivalent if and only if, denoted by µ and µ ′ respectively a root of p λ ( t ) and a root of p λ ′ ( t ) in the algebraic closureof K , we have K ( µ ) = K ( µ ′ ) (see the fourth column of Table 1 for the definitionof p λ ( t )). The forms T and T ,µ are not linearly equivalent; however they are, byconstruction, nearly equivalent. 30lso, the forms T ( i )10 ,λ and T ( i )10 ,λ ′ as well as T ( i )11 ,λ and T ( i )11 ,λ ′ are not in generalnearly-equivalent however they are geometrically equivalent. In particular both T (1)10 ,λ and T (2)10 ,λ ′ induce a Desarguesian spread on PG( V ). Note that the forms T (1)10 ,λ exist only if char ( K ) = 2 or if char ( K ) = 2 and K is not perfect. The forms oftype T (2)10 ,λ are equivalent to form of type T (1)10 ,λ if char ( K ) = 2; however, they arenot equivalent if char ( K ) = 2.Table 2: Matrices associated to forms of rank up to 6.Type Rk Matrix T (cid:16) u − u − u u u − u (cid:17) T u − u u − u − u u u − u − u u u − u ! T u − u − u u u − u u − u − u u u − u Type Rk Matrix T − u u − u u u − u u − u − u u − u u − u u u − u − u u T (1)10 ,λ u − u λu − λu − u u − λu λu u − u λu − λu λu − λu λu − λu − λu λu − λu λu λu − λu λu − λu Type Rk Matrix T (2)10 ,λ u − u λu + u − λu − u − u u − λu − u λu + u u − u λu + u − λu − u λu + u − λu − u λ + 1) u + λu ( − λ − u − λu − λu − u λu + u − ( λ + 1) u − λu λ + 1) u + λu λu + u − λu − u λ + 1) u + λu − ( λ + 1) u − λu T u − u u − u − u u u − u − u u − u u − u u 00 0 0 u − u u − u T − u − u − u u u u u − u u − u u u − u − u u − u u − u − u u − u u − u u T u u − u − u u − u u − u − u − u u u − u u − u u u − u − u u u u − u − u Type Matrix T u − u u − u u − u − u u u − u − u u u − u − u u u − u T u − u u − u − u u u − u u − u u − u − u u − u u − u − u u u − u u − u u u u u − u − u − u T (1)11 ,λ u − u u λu − λu − u − u u − λu λu u − u λu − λu − u λu − λu λu − λu u − λu λu − λu λu λu − λu λu − λu u − u Type Matrix T (2)11 ,λ u − u u λu + u − λu − u − u − u u − λu − u λu + u u − u λu + u − λu − u − u λu + u − λu − u λ + 1) u + λu ( − λ − u − λu u − λu − u λu + u − ( λ + 1) u − λu λ + 1) u + λu λu + u − λu − u λ + 1) u + λu − ( λ + 1) u − λu u − u T PG( V ) | x, y | = 0 , | x, y | = 0 , | x, y | = 0 .T PG( V ) | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | + | x, y | = 0 .T PG( V ) | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 .T PG( V ) | x, y | − | x, y | = 0 , | x, y | − | x, y | = 0 | x, y | − | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 .T (1)10 ,λ PG( V ) | x, y | + λ | x, y | = 0 , | x, y | − | x, y | = 0 | x, y | + λ | x, y | = 0 , | x, y | − | x, y | = 0 | x, y | + λ | x, y | = 0 , | x, y | − | x, y | = 0 T (2)10 ,λ PG( V ) | x, y | − | x, y | + λ | x, y | = 0 , −| x, y | + | x, y | − λ | x, y | = 0 , | x, y | − | x, y | + λ | x, y | = 0 , | x, y | + λ | x, y | − λ | x, y | + ( λ + 1) | x, y | = 0 , −| x, y | − λ | x, y | + λ | x, y | − (1 + λ ) | x, y | = 0 , | x, y | + λ | x, y | − λ | x, y | + (1 + λ ) | x, y | = 0 . T x x = 0 | x, y | + | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | − | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 T x = 0 | x, y | + | x, y | + | x, y | = 0 , | x, y | = 0 , | x, y | − | x, y | = 0 , | x, y | + | x, y | = 0 , | x, y | − | x, y | = 0 , | x, y | = 0 , | x, y | = 0 .T x x + x x = 0 | x, y | + | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | + | x, y | = 0 , | x, y | − | x, y | = 0 , | x, y | − | x, y | = 0 , | x, y | + | x, y | = 0 .T x = 0 | x, y | + | x, y | + | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 , | x, y | = 0 .T x − x x − x x − x x = 0 | x, y | + | x, y | = 0 , | x, y | − | x, y | = 0 | x, y | + | x, y | = 0 , | x, y | − | x, y | = 0 | x, y | + | x, y | = 0 , | x, y | − | x, y | = 0 | x, y | + | x, y | + | x, y | = 0 .T (1)11 ,λ λx − x = 0 | x, y | + λ | x, y | = 0 , | x, y | + λ | x, y | = 0 | x, y | + | x, y | + λ | x, y | = 0 , | x, y | = 0 , −| x, y | + λ ( | x, y | − | x, y | ) = 0 , | x, y | − | x, y | = 0 , | x, y | − | x, y | = 0 .T (2)11 ,λ x + λx x + x = 0 | x, y | − | x, y | + | x, y | + λ | x, y | = 0 , | x, y | − | x, y | + λ | x, y | = 0 , | x, y | = 0 , | x, y | − | x, y | + λ | x, y | = 0 , | x, y | − | x, y | − λ ( | x, y | − | x, y | ) −− ( λ + 1) | x, y | = 0 , | x, y | + λ ( | x, y | − | x, y | )++( λ + 1) | x, y | = 0 , | x, y | + λ | x, y | − λ | x, y | ++( λ + 1) | x, y | = 0 ..