A characterization of the family of lines external to a hyperbolic quadric of PG (3, q)

Abstract

In this paper we characterize the family of external lines to a hyperbolic quadric of PG(3, q). (2000): 51E20. Mathematics Subject Classification

A Characterisation of the Lines External to an Oval Cone in PG(3, q), q Even

“…Recently, Di Gennaro, Durante and Olanda [1], and Durante and Olanda [2] have characterised the lines external to each of the two non-singular quadrics in PG(3, q). Durante and Olanda proved this result for the elliptic quadric: Theorem 1.1 ( [2]).…”

A characterization of the planes meeting a hyperbolic quadric of $\PG(3,q)$ in a conic

“…In a series of two recent papers [1,2], characterizations of elliptic hyperplanes and hyperbolic hyperplanes in PG(4, q) are given. A characterization of the family of external lines to a hyperbolic quadric in PG(3, q) was given in [4] for all q (also see [6] for a different characterization in terms of a point-subset of the Klein quadric in PG (5, q)). In a recent paper [8], a characterization of secant lines to a hyperbolic quadric is given for all odd q, q ≥ 7.…”

A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd

“…A characterization of the family of secant lines to an ovoid in P G(3, q) was obtained in [7] for q odd and in [4] for q > 2 even, which was further improved in [6] for all q > 2. A characterization of the family of external lines to a hyperbolic quadric in P G(3, q) was given in [5] for all q (also see [9] for a different characterization in terms of a point-subset of the Klein quadric in P G(5, q)) and to an ovoid in P G(3, q) was obtained in [6] for all q > 2. One can refer to [1,2,11,12] for characterizations of external lines in P G (3, q) with respect to quadric cone, oval cone and hyperoval cone.…”