On 1-systems of \(Q(6,q)\), \(q\) even (Q1404327)

This paper is a continuation of previous work [Dev. Math. 3, 257-275 (2001; Zbl 1016.51008)] of the authors, where they described a method for investigating locally Hermitian 1-systems of the quadric \(Q(6,q)\), \(q\) odd, by associating some kind of flocks in \(\text{PG}(4,q)\) to them. In the paper under review this is extended to the case where \(q\) is even. As it turns out this is a highly nontrivial generalization, because for \(q\) odd the method relies on utilizing the polarity that comes with the quadric \(Q(6,q)\). Such a polarity, however, does not exist for \(q\) even. For \(q\) even, the following equivalence is used instead: each locally Hermitian 1-system of \(Q(6,q)\) gives rise to some flock of a dual cone in \(\text{PG}(4,q)\) with a line vertex and, conversely, from every such flock a locally Hermitian 1-system of \(Q(6,q)\) can be constructed. The authors then construct a new class of 1-systems of \(Q(6,q)\), \(q\) even, by starting with a dual conic cone \(K\) with vertex line \(L\) in \(\text{PG}(4,q)\). If \(\pi_{0}\) is a plane skew to \(L\), if \(C^*\) is the intersection of \(\pi_{0}\) with \(K\), and if \(C\subseteq \pi_{0}\) is a conic, disjoint to \(C^*\), then any rational normal cubic scroll \(R^3\) with \(C \subseteq R^3\) and \(L\) as directrix line defines a 1-systems of \(Q(6,q)\). In the last section of the manuscript all locally Hermitian, semi-classical 1-systems \(Q(6,q)\) are classified for \(q>2\) even.