Blocking sets of type \((1,k)\) in a finite projective plane
In a finite projective plane \(\Pi\) of order \(n\), a set \(\mathcal B\) is called of type \((1,k)\), if any line intersects \(\mathcal B\) in 1 or \(k\) points. \textit{G. Tallini} [J. Geom. 29, 191-199 (1987; Zbl 0618.51004)] gave strong conditions on the values of \(n\) and \(|\mathcal B|\) in dependence of \(k\). The authors give several numerical results on Tallini's parameters. In particular, they give restrictions on the case \(|{\mathcal B}|=y^m\), \(m\geq 2\). Furthermore, they consider sets of type \((1,k)\) which are conserved by a ``large'' collineation group \(G\) of \(\Pi\). A classification is given in the case where \(G\) is transitive on the \(k\)-secants of \(\mathcal B\) and contains a nontrivial perspectivity, and in the case where \(G\) acts flag-transitively on \(\mathcal B\).
scientific article; zbMATH DE number 1436008