On the family of linear flocks of an absolutely irreducible hyperbolic quadric in a finite projective space (Q1195009)

Let \(H\) be the absolutely irreducible hyperbolic quadric in \(PG(3,q)\). A flock of \(H\) is a set of \(q+1\) disjoint circles in \(H\). The set of all intersections of \(H\) with the planes through a fixed line in \(PG(3,q)\) is an example of a flock and these flocks are called linear flocks. There are \(q^ 2(q-1)^ 2/2\) linear flocks and each pair of disjoint circles of \(H\) determines a unique linear flock. The authors prove the following theorem: If \(F\) is a set of partitions of \(H\) in circles such that each pair of disjoint circles of \(H\) is contained in exactly one member of \(F\) then \(F\) is the set of all linear flocks of \(H\).