Regularity conditions on the transversals of lines in a partial geometry
Let \(S=(P,B,I)\) be a partial geometry with parameters \(s, t, \alpha\) \((\alpha \ne t+1)\); that is, each line contains \(s+1\) points, through each point there are \(t+1\) lines and there are precisely \(\alpha\) lines joining any non-incident point and line. If \(S\) also satisfies the further property that, for any pair of skew lines \(\ell\) and \(m\) and any point \(P\) on neither \(\ell\) nor \(m\), there is a unique transversal of \(\ell\) and \m\) through \(P\), then \(S\) is one of three types: (1) a grid of order \(s+1\) with \(\alpha +1;\) (2) the points and lines of \(\mathrm{PG}(3,q)\) with \(\alpha =s+1;\) (3) the dual of a certain net of order 4 and degree 3 with \(\alpha =2\), \(s=2\), \(t=3\).
scientific article; zbMATH DE number 4101982