A generalization of the quadratic cone of $$\mathop {\mathrm{PG}}(3,q^n)$$ PG
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A generalization of the quadratic cone of PG(3, q n ) and its relation with the affine set of the Lüneburg spread Giorgio Donati1 · Nicola Durante1
Received: 10 April 2017 / Accepted: 2 April 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Using a variation of Seydewitz’s method of projective generation of quadratic cones, we define an algebraic surface of PG(3, q n ), called σ -cone, whose Fq n -rational points are the union of a line with a set A of q 2n points. If q n = 22h+1 , h ≥ h 1, and σ is the automorphism of Fq n given by x → x 2 , then the set A is the affine set of the Lüneburg spread of PG(3, q n ). If n = 2 and σ is the involutory automorphism of Fq 2 , then a σ -cone is a subset of a Hermitian cone and the set A is the union of q non-degenerate Hermitian curves. Keywords Luneburg spread · Quadratic cone · Hermitian curve
1 Introduction Let A and B be two distinct points of a three-dimensional projective space. Let S A be the star of lines through A, let S B∗ be the star of planes through B, and let be a projectivity between S A and S B∗ . In 1848 F. Seydewitz proved that quadrics may be generated as the set of points of intersection of corresponding elements under (see e.g., [10]). If the line AB is mapped under into a plane, say π AB , containing AB and the lines through A of the plane π AB are mapped into the planes through the line AB, then the set of points of intersection of corresponding elements under is a quadratic cone. In this
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Nicola Durante [email protected] Giorgio Donati [email protected]
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Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II” Complesso di Monte S. Angelo, Edificio T via Cintia, 80126 Naples, Italy
123
J Algebr Comb
paper we define an algebraic surface of PG(3, q n ) by using a variation of Seydewitz’s projective generation of quadratic cones by means of a suitable collineation instead of a projectivity. Let A and B be two distinct points of PG(3, q n ), and let σ be an automorphism of Fq n such that Fi x(σ ) = Fq . Let be a σ -collineation between the star of lines with center A and the star of planes with center B. Suppose that the line AB is mapped under into a plane containing AB and that the lines through A of the plane π AB are mapped into the planes through the line AB, then the set of points of intersection of corresponding elements under is a σ -cone of PG(3, q n ) with vertices A and B. We will prove the following results: Theorem 1.1 Every σ -cone K of PG(3, q n ) is projectively equivalent to the set of Fq n -rational points of the algebraic surface with equation x1 σ +1 + x2 x3σ + x3 x2σ − x4 x2 σ = 0. It has size q 2n + q n + 1, it is of type (0, 1, 2, q + 1, q n + 1)1 , every (q + 1)-secant line meets K in a subline over Fq . Moreover AB is the unique line contained in K and π AB is the unique plane that meets K exactly in AB. Theorem 1.2 A σ -cone and a σ -cone of PG(3, q n ) are PL-equivalent if, and only if, either σ = σ or σ = σ −1 . Theorem 1.3 Let K be
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