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On Absolute Points of Correlations in PG $(2, Q^ N) $

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On Absolute Points of Correlations in PG $(2, Q^ N) $ On absolute points of correlations in PG(2,qn) Jozefien D’haeseleer ∗ Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University Gent, Flanders, Belgium [email protected] Nicola Durante Dipartimento di Matematica e Applicazioni ”Caccioppoli” Universit`adi Napoli ”Federico II”, Complesso di Monte S. Angelo Napoli, Italy [email protected] Submitted: Aug 1, 2019; Accepted: May 8, 2020; Published: TBD c The authors. Released under the CC BY-ND license (International 4.0). Abstract Let V bea(d+1)-dimensional vector space over a field F. Sesquilinear forms over V have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the d-dimensional projective space PG(V ). Everything is known in this case for both degenerate and non-degenerate reflexive forms if F is either R, C or a finite field Fq. In this paper we consider degenerate, non- F3 reflexive sesquilinear forms of V = qn . We will see that these forms give rise to degenerate correlations of PG(2,qn) whose set of absolute points are, besides cones, m the (possibly degenerate) CF -sets studied in [8]. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in PG(2,qn) induced by a non-degenerate, F3 non-reflexive sesquilinear form of V = qn . Mathematics Subject Classifications: 51E21,05B25 arXiv:2005.05698v1 [math.CO] 12 May 2020 1 Introduction and definitions Let V and W be two vector spaces over the same field F. A map f : V −→ W is called σ-semilinear if there exists an automorphism σ of F such that f(v + v′)= f(v)+ f(v′) and f(av)= aσf(v) ∗Supported by FWO grant. the electronic journal of combinatorics 27 (2020), #P00 https://doi.org/10.37236/abcd for all vectors v ∈ V and all scalars a ∈ F. If σ is the identity map, then f is a linear map. Let V be an F-vector space with dimension d + 1 and let σ be an automorphism of F. A map h , i :(v, v′) ∈ V × V −→ hv, v′i ∈ F is a σ-sesquilinear form or a σ-semibilinear form on V if it is a linear map on the first argument and it is a σ-semilinear map on the second argument, that is: hv + v′, v′′i = hv, v′′i + hv′, v′′i hv, v′ + v′′i = hv, v′i + hv, v′′i hav, v′i = ahv, v′i, hv, av′i = aσhv, v′i for all v, v′, v′′ ∈ V , a ∈ F. If σ is the identity map, then h , i is a bilinear form. If σ B =(e1, e2,...,ed+1) is an ordered basis of V , then for x, y ∈ V we have hx, yi = XtAY . Here A = (aij)=(hei, eji) is the associated matrix to the σ-sesquilinear form in the ordered basis B, and X, Y are the columns of coordinates of x, y w.r.t. B. In what follows we will denote by At the transpose of a matrix A. The term sesqui comes from the Latin and it means one and a half. For every subset S of V put S⊥ := {x ∈ V : hx, yi =0 ∀y ∈ S} S ⊥ := {y ∈ V : hx, yi =0 ∀x ∈ S} ⊥ Both S⊥ and S ⊥ are subspaces of V . The subspaces V ⊥ and V are called the left and the right radical of h, i, respectively. Proposition 1. The right and left radical of a sesquilinear form of a vector space V have the same dimension. Let d+1 be the dimension of V . Then dimV ⊥ = d+1−rk(A) and dimV ⊥ = d+1−rk(B) σ−1 where B =(aij ). The assertion follows from rk(B) = rk(A). A sesquilinear form h, i is called non-degenerate if V ⊥ = V ⊥ = {0}. It is called reflexive if hx, yi = 0 implies hy, xi = 0 for all x, y ∈ V . A duality or correlation of PG(d, F) is a bijective map preserving incidence between points and hyperplanes of PG(d, F). It can be seen as a collineation of PG(d, F) into its dual space PG(d, F)∗. We will call degenerate duality or degenerate correlation of PG(d, F) a non-bijective map preserving incidence between points and hyperplanes of PG(d, F). If V is equipped with a sesquilinear form h , i, then there are two induced (possibly degenerate) dualities of PG(d, F): σ σ Y 7→ XtAY = 0 and Y 7→ YtAX =0. The following holds. the electronic journal of combinatorics 27 (2020), #P00 2 Theorem 2. Any duality of PG(d, F),d > 2, is induced by a non-degenerate sesquilinear form on V = Fd+1. A duality applied twice gives a collineation of PG(d, F). If h , i is given by hx, yi = σ XtAY , then the associated collineation is the following map: −1 σ σ2 f −1 σ 2 : X 7→ A A X (see [14]). At A ,σ t The most important and studied types of dualities are those whose square is the identity, −1 σ 2 namely the polarities. Note that in this case At A = ρI and σ = 1. Proposition 3. [1, 2] A duality is a polarity if and only if the sesquilinear form defining it is reflexive. Non-degenerate, reflexive forms have been classified long ago (not only on finite fields but also on R or C) . A reflexive σ-sesquilinear form h, i is: • σ-Hermitian if hy, xi = hx, yiσ, for all x, y ∈ V. If σ = 1 such a form is called symmetric. • Alternating if hy, xi = −hx, yi. Theorem 4. A non-degenerate, reflexive σ-sequilinear form is either alternating, or a scalar multiple of a σ-Hermitian form. In the latter case, if σ =1(i.e. the form is symmetric), then the scalar can be taken to be 1. For a proof of the previous theorem we refer to [1] or [2]. Note that in case of reflexive σ σ-sesquilinear forms, left- and right-radicals coincide. The set Γ : XtAX = 0 of absolute points of a polarity is one of the well known objects in the projective space PG(d, F): subspaces, quadrics and Hermitian varieties. We recall that the classification theorem for reflexive σ-sesquilinear forms also holds if the form is degenerate. Remark 5. Let Γ be the set of absolute points of a degenerate polarity in the projec- tive space PG(d, F), then Γ is one of the following: subspaces, degenerate quadrics and degenerate Hermitian varieties. All of these sets are cones with vertex a subspace V of dimension h (corresponding to V ⊥) and base a non-degenerate object of the same type in a subspace of dimension d − 1 − h skew with V. So everything is known for reflexive sesquilinear forms. Up to our knowledge (at least on finite fields) almost nothing is know on non-reflexive Fd+1 sesquilinear forms of V = qn . Given a sesquilinear form of V we may consider the induced correlation in PG(d, qn) and the set Γ of its absolute points, that is the points X ⊥ ⊥ σ σ such that X ∈ X (or equivalently X ∈ X ). If hx, yi = XtAY , then Γ : XtAX = 0. We will determine in the next sections the set Γ in PG(1, qn) and in PG(2, qn). Fd+1 σ Let V = qn , d ∈{1, 2} and let Γ : XtAX = 0, σ =6 1, be the set of absolute points of a (possibly degenerate) correlation of V . We start with the following definition. the electronic journal of combinatorics 27 (2020), #P00 3 Fd+1 Definition. Let α be a σ-sesquilinear form of qn . A σ-quadric (σ-conic if d = 2) of PG(d, qn), d ∈ {1, 2} is the set of absolute points of the induced correlation of α in PG(d, qn). See [10] for the definition and the properties of σ-quadrics of PG(d, qn) for every integer d > 1. It is an easy exercise (that we include in this paper) to determine the set Γ in PG(1, qn). For d = 2 the set Γ in PG(2, qn) has been studied by B.C. Kestenband in case h , i is non-degenerate and the form is non-reflexive. The results are contained in 10 different papers from 2000 to 2014. We will summarize some of his results in the last section. Contrary to the reflexive case, we will see that in general the knowledge of the set Γ of absolute points of a correlation induced by a non-degenerate, non-reflexive sesquilinear form will not help to determine the set Γ in the degenerate case. 2 Absolute points of correlations of PG(1,qn) In the sequel, we will determine the set of absolute points in PG(2, qn) for a degenerate correlation induced by a degenerate σ-sesquilinear form with associated automorphism qm F3 σ : x 7→ x , (m, n)=1, of V = qn . F2 First assume that V = qn and consider the set of absolute points of a correlation induced n σ by a σ-sesquilinear form, that is the set of points in PG(1, q ) given by Γ : XtAX = 0, where a b A = c d qm is a 2×2-matrix over Fqn . Here again, let σ be the automorphism σ : x 7→ x ,(m, n) = 1, F2 of V = qn . The points of Γ satisfy σ σ (ax1 + cx2)x1 +(bx1 + dx2)x2 =0.
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