A characterization of the family of secant or external lines of an ovoid of PG$(3,q)$

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

“…Let P G(3, q) denote the three-dimensional Desarguesian projective space defined over a finite field of order q. Characterizations of the family of external or secant lines to an ovoid/quadric in P G (3, q) with respect to certain combinatorial properties have been given by several authors. 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.…”

“…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.…”

“…Proof. We have |H| = (q + 1) 2 by (6). It is enough to show that the points of H together with the black lines form a projective generalized quadrangle of order (q, 1).…”

A characterisation of the lines external to a quadric cone of PG(3,q), q odd

“…Recently, Durante and Olanda [4] and Di Gennaro, Durante and Olanda [3] have characterised the lines external to the non-singular quadrics in PG(3, q) using their combinatorial properties. These results are listed below.…”