Tubes of even order and flat \(\pi\). \(C_ 2\) geometries (Q1900041)

The following combinatorial-geometric problem is considered in this paper: for a given line \(L\) in \(PG(3,q)\), find a set \({\mathcal L}\) of \(q + 2\) skew lines (all skew to \(L\) too), such that the intersection of any plane through \(L\) with these \(q + 2\) lines is a hyperoval. The pair \((L, {\mathcal L})\) is called a tube. By unpublished work of A. Pasini, a tube gives rise to a flat \(\pi \cdot C_2\) geometry. Pasini noticed only one example, namely, let \({\mathcal L}'\) be the set of lines of one regulus of a ruled quadric. Let \(L\) be any exterior line. Then \((L, {\mathcal L})\), with \({\mathcal L} = {\mathcal L}' \cup \{L'\}\), where \(L'\) is the polar line of \(L\) w.r.t. the quadric, is a tube. In the paper under review, the situation is translated into an algebraic problem of 2 by 2 matrices and new examples are discovered. These may be constructed as follows: consider a regular line spread of \(PG(3,q)\), let \(L\) be a line of that spread. Let \({\mathcal O}\) be any hyperoval in any plane through \(L\) such that \(L\) is an exterior line w.r.t. \({\mathcal O}\). Then the lines of the spread meeting \({\mathcal O}\) form a set \({\mathcal L}\) such that \((L, {\mathcal L})\) is a tube. Some related problems are touched, such as an analogue in the odd characteristic case.