Ovoidal fibrations in $${\pmb {PG}}\varvec{(3,q)}, {\pmb {q}}$$ PG
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Ovoidal fibrations in PG(3, q), q even N S NARASIMHA SASTRY1 and R P SHUKLA2,∗ 1 Indian Institute of Technology, Walmi Campus, Belur Industrial Area, Near High Court, P.B. Road, Dharawad 580 011, India 2 Department of Mathematics, University of Allahabad, Prayagraj 211 002, India ∗ Corresponding author. E-mail: [email protected]; [email protected]
MS received 28 June 2019; revised 16 May 2020; accepted 18 May 2020 Abstract. Given a partition of a projective 3-space of odd cardinality by a set of ovoids, a line secant to one of the ovoids of the partition, and its polar relative to the symplectic polarity on the projective 3-space defined by this ovoid, are tangent to distinct ovoids of the partition (Theorem 2). The proof uses the fact that the radical of the linear code generated by the duals of the hyperbolic quadrics in a symplectic generalized quadrangle is of codimension one (Theorem 4). Keywords.
Dual grid; elliptic quadric; Frobenius reciprocity; Tits ovoid.
Mathematics Subject Classification.
05B25, 51E22, 94B05.
1. Introduction and statement of the result An ovoid in P G(3, q), q > 2, is a set of q 2 + 1 points, no three collinear. Each plane of P(3, q) meets an ovoid in 1 or q + 1 points. All ovoids in P G(3, q), q odd, are elliptic quadrics, as shown by Barlotti [4] and Panella [19] (see [18], Theorem 2.1, p. 178). From now on, let q = 2n , n > 1. Elliptic quadrics which exist for all q and the Suzuki–Tits ovoids ([25], see [15], 16.4) which exist, if and only if, n is odd, are the only known ovoids in P G(3, q); and these are the only ovoids if n ≤ 6 [18]. Classification of ovoids in P G(3, 2n ) and their ‘distribution’ in P G(3, 2n ) are fundamental problems in Incidence Geometry. Our object in this note is to make a small contribution (Theorem 2) to the later problem. Let h¯ be a non-degenerate symplectic bilinear form on V (q) = Fq4 , the vector space of dimension four over Fq , τ denotes the symplectic polarity on P G(3, q) defined by h¯ and G P Sp(4, q) be the symplectic group defined by h¯ . Then the incidence system W (q) = Wτ whose point-set is the set P of points of P G(3, q); the line-set L = L τ is the set of all absolute lines of P G(3, q) with respect to the symplectic polarity τ (equivalently, the lines on which h¯ restricts to the zero map); and symmetrized inclusion as the incidence is a regular generalized quadrangle of order q ([20], p. 37). The group G acts transitively on the sets P and L, preserving the incidence relation. In view of the uniqueness of a non-degenerate symplectic bilinear form on V (q) up to G L(V (q))-equivalence (see [17], Corollary 9.2, p. 516), W (q) is unique up to P G L(4, q)© Indian Academy of Sciences 0123456789().: V,-vol
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equivalence. A general linear complex in P G(3, q) is the set of all absolute lines with respect to a symplectic polarity of P G(3, q). An ovoid of W (q), q > 2 and even, is a set of q 2 +1 points, no two on a line of W (q). We note that each ovoid of P G(3, q)
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