On Bruen chains (Q1903719)

A Bruen chain in the projective 3-space \(PG (3,q)\), \(q\) odd, is a set of \({q + 3 \over 2}\) reguli of a regular spread such that any two reguli have exactly two lines in common and no three reguli share a line. Such a Bruen chain is replaceable if one can partition the point set covered by the chain by using only lines of the complementary reguli of the Bruen chain. Using the connection of regular spreads with inverse Miquelian planes, the author gives, in the quite impressive paper under review, an equivalent description of Bruen chains in terms of sets of conics in the affine plane \(AG (2,q)\), and these conics are called circles; hence sets of circles which correspond with Bruen chains are called Bruen chains in \(AG (2,q)\). The author characterizes algebraically the Bruen chains in \(AG (2,q)\) using certain (algebraic) invariants of the circles. This way, he is able to prove that all Bruen chains are replaceable and also to give a universal construction of Bruen chains. Applied to small values of \(q\), i.e., \(5 \leq q \leq 23\) and \(q\) odd, he uses the computer to enumerate all Bruen chains for these values of \(q\).