Characterizing the line set of a Baer subspace (Q1087831)

This is a 3-dimensional generalization of a theorem of \textit{A. Bruen} on blocking sets in finite projective planes [SIAM J. Appl. Math. 21, 380- 392 (1971; Zbl 0252.05014)]. Let \({\mathcal L}\) be a set of lines in \({\mathfrak P}=PG(3,q)\). A blocking set of a projective plane is a set \({\mathcal B}\) of points such that every line contains a point of \({\mathcal B}\) and a point off \({\mathcal B}\). A plane \(\pi\) is said to be a blocking plane if the set of lines in \(\pi\) forms a blocking set of \(\pi\). Let \({\mathcal L}\) now satisfy the properties: (0) There exists at least one blocking plane in \({\mathfrak P}\), (1) Every point of \({\mathfrak P}\) is on at least one line of \({\mathcal L}\), (2) For any two intersecting lines \(\ell\), \(\ell '\in {\mathcal L}\), their span \(<\ell,\ell '>\) is a blocking plane. Then the theorem states that \(| {\mathcal L}| \geq (q+1)(q+\sqrt{q}+1),\) with equality if and only if \({\mathcal L}\) is the line set of a Baer subspace of \({\mathfrak P}\), that is, a subspace of order \(\sqrt{q}\). Thus the examples of minimal cardinality are exactly the line sets of Baer subspaces of \({\mathfrak P}\).