Tubes in PG(3,q)

A tube (resp. an oval tube) in PG(3,q) is a pair T={L,L}T={L,L}, where {L}∪L{L}∪L is a collection of mutually disjoint lines of PG(3,q) such that for each plane ππ of PG(3,q) containing LL, the intersection of ππ with the lines of LL is a hyperoval (resp. an oval). The line LL is called the axis of TT. We show that every tube for qq even and every oval tube for qq odd can be naturally embedded into a regular spread and hence admits a group of automorphisms which fixes every element of TT and acts regularly on each of them. For qq 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.