A characterization of the external lines of a hyperoval in PG\((3,q),q\) even (Q409475)

Let \(PG(3,q)\), \(q\) even, be the projective space of dimension \(3\) and order \(q=2^h\). Let \(\pi\) be a plane of \(PG(3,q)\) and consider a hyperoval \(H\) in it, i.e., \(q+2\) points no three of which are collinear. A \textit{hyperoval cone} of \(PG(3,q)\) is the set of the points joining the hyperoval \(H\) to a vertex \(V\notin \pi\).NEWLINENEWLINE A characterization of the external lines of a hyperoval cone in \(PG(3,q)\), \(q\) even, has been given by M. Zanetti. In this work, the author gives a new proof, without using a particular hypothesis, called \textit{polynomial hypothesis} by the author.NEWLINENEWLINE In particular, in the \textit{old} proof the author does not consider that a certain quantity could be different from zero. Without assuming this hypothesis, F. Zuanni is able to prove the same characterization for the external lines of a hyperoval cone.NEWLINENEWLINE A set of lines \(L\) is of type \((m_1,\dots, m_r)\) with respect to the planes (resp. to the stars of lines) if each plane (resp. star) contains exactly either \(m_{1}\) or \dots or \(m_r\) lines of \(L\). The characterization of the external lines of a hyperoval cone is the following. If \(L\) is a set of \(q^3(q-1)/2\) lines having \((q+1)(q+2)/2\) external planes, \(q^4(q+1)(q-1)^2(q-2)/8\) pairs of skew lines, of type \((m,n)\) with respect to the stars of lines and of type \((0,a,b)\) with respect to the planes, then \(L\) is the set of the external lines to a hyperoval cone of \(PG(3,q)\), \(q\) even.