On regular parallelisms in PG(3,q) (Q797826)

A parallelism of \(S_ 3=PG(3,q)\) is a set \({\mathcal P}\) of \(q^ 2+q+1\) spreads such that no two distinct spreads of \({\mathcal P}\) have a line in common. A spread is said to be regular if for any 3 distinct lines of the spread the regulus (that is, one ruling of a nondegenerate hyperbolic quadric in \(S_ 3)\) determined by these 3 lines is contained in the spread. If all spreads of \({\mathcal P}\) are regular, then \({\mathcal P}\) is called a regular parallelism of \(S_ 3\). The author proves that for odd q there are no regular parallelisms of \(S_ 3\).