A characterisation of the lines external to a quadric cone of PG\((3,q)\), \(q\) odd
The authors characterise the lines not meeting a quadric cone in \(PG(3,q)\), \(q\) odd. Specifically, they prove the following. Let \({\mathcal L}\) be a non-empty set of lines in \(PG(3,q)\), \(q\) odd, such that: (1) Every point lies on \(0\), \({{1}\over{2}}q(q+1)\) or \({{1}\over{2}}q (q-1)\) lines of \({ \mathcal L}\); (2) Every plane contains \(0\), \(q^2\) or \({{1}\over{2}}q (q-1)\) lines of \({\mathcal L}\); (3) For any point \(P\), if \(P\) is on two planes which contain the same number of lines of \({\mathcal L}\), then \(P\) is on the same number of lines of \({\mathcal L}\) in both planes. Then \({\mathcal L}\) is the set of external lines to a quadric cone.
scientific article; zbMATH DE number 5788608