Projective spaces and substructures are well-studied objects in incidence geometry. In finite geometry, the projective space over the finite field of order q (GF(q)) plays a central role. Typically in this field, relations between different types of geometries and their substructures are very important. For instance, a spread of the finite three-dimensional projective space PG(3,q) over the field GF(q) is equivalent to a so-called translation plane (a projective plane satisfying certain symmetry conditions), and so results on spreads of PG(3,q) contribute to the study of translation planes.
A spread of PG(3,q) is a set of q2+1 lines partitioning the pointset of PG(3,q). A partial spread is a set of mutually skewed lines of PG(3,q). Sometimes partial spreads can be extended to spreads. A partial tube &Tgr; in PG(3,q) is a pair L,&Lgr;, where &Lgr; is a partial spread and L is a line meeting no line of &Lgr;, such that the intersection with the lines of &Lgr; and any plane &pgr; on L is an arc, that is, a set of points of PG(2,q) with no three collinear. Nonextendable arcs of PG(2,q) are called complete. When q is even, a complete arc is a hyperoval and contains q+2 points. When q is odd, a complete arc is an oval and contains q+1 points. A partial tube is called a tube when q is even and the induced arcs are hyperovals. A partial tube is called an oval tube when q is odd and the induced arcs are ovals. Using nontrivial geometric and algebraic arguments, the authors prove that every tube, respectively, oval tube, can be embedded in a regular spread of PG(3,q), q even, respectively, q odd. As a corollary, the authors prove a geometric characterization of a regulus of PG(3,q), q odd, in terms of oval tubes. This paper contributes to the study of spreads of PG(3,q) and related objects, and describes a nice alternative way to study these objects.
