Ruled sets of plane-type \((m,h)_2\) in \(\mathrm{PG}(3,q)\) with \(m\leq q\)
The author generalizes the result that a set \(K\) of points in \(\mathrm{PG}(d,q)\) intersected by any hyperplane in 1 or \(h\) points is either a line or \(d=3\) and \(K\) is an ovoid in two ways. The first is as follows. Let \(0<m<h\) and \(s \geq m+1\) be three integers. Let \(K\) be a set of \(k\) points of \(\mathrm{PG}(3,q)\) of plane-type \((m,h)_2\). If there exists a line \(\ell\) intersecting \(K\) in \(s\) points then either: (1) \(K\) is ruled, \(k=m(q+1)\), \(h-m=q\) and either \(K\) is a line or \(m \geq 2\) and \(K = \ell \cup K'\) is a set of points of \(\mathrm{PG}(3,q)\) of plane-type \((m-1,h-1)_2\) and of size \((m-1)(q+1)\) or (2) \(K\) is an ovoid. The second is as follows. Let \(K\) be a set of points of \(\mathrm{PG}(3,q)\) of plane type \((m,h)_2\) with \(m \geq 1\). If there exists a line intersecting \(K\) in at least \(m+1\) points then \(K\) contains at most \(m\) lines, and \(K\) contains exactly \(m\) lines if and only if \(K\) is the set of points on \(m\) pairwise skew lines.
scientific article; zbMATH DE number 6818624