Barwick, Susan G.; Butler, David K. A characterisation of the lines external to a quadric cone of PG\((3,q)\), \(q\) odd. (English) Zbl 1201.51009 Innov. Incidence Geom. 8, 39-48 (2008). 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. Reviewer: Steven T. Dougherty (Scranton) Cited in 2 Documents MSC: 51E20 Combinatorial structures in finite projective spaces Keywords:projective space; quadric cone; lines; characterisation PDFBibTeX XMLCite \textit{S. G. Barwick} and \textit{D. K. Butler}, Innov. Incidence Geom. 8, 39--48 (2008; Zbl 1201.51009)