Hyper-reguli
Let \(\pi\) be an affine Desarguesian plane of order \(q^ 4\), \(V\) the vector space of dimension \(2d\) whose elements are points of \(\pi\) and \(\Theta\) the projective space of dimension \(2d-1\) whose points are the 1- spaces in \(V\). Let \(N=(q^ d-1) (q-1)^{-1}\). A hyper-regulus is a set \(A=\{A_ 1,\dots,A_ N\}\) of mutually skew projective subspaces of \(\Theta\) of dimension \(d-1\) such that there exists another set \(B=\{B_ 1,\dots,B_ N\}\) of mutually skew projective subspaces of dimension \(d- 1\) with \(| A_ i \cap B_ j |=1\), \(i,j=1,\dots,N\). When \(d=2\), \(A\) and \(B\) are reguli in projective 3-space. Obvious examples come from André planes. A non-André example is constructed. Partial hyper- reguli are considered and some results concerning their extension to larger partial hyper-reguli and ultimately to hyper-reguli are obtained.
scientific article; zbMATH DE number 493209