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

Abstract

In this article, the lines not meeting an oval cone in PG(3, q) (q even) are characterised by their intersection properties with points and planes.

A characterization of the family of external lines to a quadratic cone of PG(3, q), q

“…In this paper we obtain a similar result for the family F of external lines to an oval cone of P G (3, q), q odd, case that has not been studied in [1]. (See [2] and [3] for analogous results regarding hyperbolic quadrics and ovoids of P G (3, q)).…”

“…In a recent paper [1] the authors, characterize the family of external lines to a hyperoval cone. The result contained in [1] is the following.…”

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

“…(i) Every point lies on 0 or 1 2 q(q − 1) lines of L ; (ii) Every plane contains 0 or 1 2 q(q − 1) lines of L ; (iii) In every plane there are 0, 1 2 (q − 1) or 1 2 (q + 1) lines of L through any point. Then the set of points on no lines of L forms either one line, two skew lines or a hyperbolic quadric.…”

“…(i) Every point lies on 0 or 1 2 q(q + 1) lines of L ; (ii) Every plane contains q 2 or 1 2 q(q − 1) lines of L . Then L is the set of external lines to an ovoid of PG(3, q).…”

A characterization of the external lines of a hyperoval cone in PG(3, q), q even

“…An important question is whether we may use these properties to characterize them. The following result [1] enters into this scheme of things. Barwick and Butler).…”