Blockades and Baer subspaces in finite projective spaces (Q806037)

A set \({\mathcal L}\) of lines in a projective space of dimension \(2t+1\) is a blockade if (i) every \(t+1\) dimensional subspace contains at least one line of \({\mathcal L}\). (ii) If x is on two lines of \({\mathcal L}\) then every \(t+1\) dimensional subspace which contains x contains at least one line of \({\mathcal L}\) through x. The points x are called blocking points. The following line sets in projective 3-space \((t=1)\) are examples of blockades (a) The set of all lines in a plane \(\pi\). (b) Let \({\mathcal L}_ 1\) be the blockade of (a); let \({\mathcal L}_ 2\) be a set of lines not in \(\pi\) such that two lines of \({\mathcal L}_ 2\) must either be skew or meet in a point of \(\pi\). Then \({\mathcal L}_ 1\cup {\mathcal L}_ 2\) is a blockade. (c) Let I be a set of points in \(\pi\) such that every line of \(\pi\) meets I. (Note that I need not be a blocking set of \(\pi\)). Take \({\mathcal L}_ 1\) as before and take \({\mathcal L}_ 2\) to be the set of all lines meeting I. (d) Let B be a blocking set (i.e. every line contains a point of B and point not in B.) Then the set of all lines containing at least two points of B is a blockade. A spread in \(PG(2t+1,q)\) is a set of \({\mathcal L}\) of mutually skew lines which cover the space. (Note the difference from the ``spreads'' used to define translation planes.) The spread is ``geometric'' if it induces a spread on each 3-space generated by two lines of \({\mathcal L}.\) Let us say (reviewer's definition) that a blockade \({\mathcal L}\) in \(PG(2t+1,q)\) is strong if there is a t-space T such that each point of T is on at most one line of \({\mathcal L}.\) The author characterizes strong blockades. They include the geometric spreads, Baer subspaces and some sets called t-pencils and t-double pencils.