Tubes of even order and flat π. C₂ geometries

A tube of even order q=2^d is a set T={L, {Mathematical expression}} of q+3 pairwise skew lines in PG(3, q) such that every plane on L meets the lines of {Mathematical expression} in a hyperoval. The quadric tube is obtained as follows. Take a hyperbolic quadric Q=Q₃⁺(q) in PG(3, q); let L be an exterior line, and let {Mathematical expression} consist of the polar line of L together with a regulus on Q. In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PG{cyrillic}L(4, q) fixing a tube {L, {Mathematical expression}} cannot act transitively on {Mathematical expression}. As pointed out by a construction due to Pasini, this implies new results for the existence of flat π. C₂ geometries whose C₂-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3, q) for small q. Further, we define tubes for odd q (replacing 'hyperoval' by 'conic' in the definition), and consider briefly a related extremal problem.