Tubes in PG(3, q)

A tube (resp. an oval tube) in PG(3, q) is a pair T = {L, L}, where {L} ∪ L is a collection of mutually disjoint lines of PG(3, q) such that for each plane π of PG(3, q) containing L, the intersection of π with the lines of L is a hyperoval (resp. an oval). The line L is called the axis of T. We show that every tube for q even and every oval tube for q odd can be naturally embedded into a regular spread and hence admits a group of automorphisms which fixes every element of T and acts regularly on each of them. For q odd we obtain a classification of oval tubes up to projective equivalence. Furthermore, we characterize the reguli in PG(3, q), q odd, as oval tubes which admit more than one axis.