Tubes in

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.