Piecewise regular spreads

Abstract

A spread $\cal S$ of the real projective 3-space PG(3,ℝ) is called piecewise regular, if, roughly speaking, the Klein image of $\cal S$ is composed of two elliptic caps and z elliptic zones (z ∈ {0,1,2,…}); we say that $\cal S$ is of segment number z + 2. We use piecewise regular spreads in order to give explicit examples of rigid and hyperrigid spreads. A spread $\cal T$ of a projective 3-space II is called rigid, if the only collineation of II leaving $\cal T$ invariant is the identity. A rigid spread ℌ is said to be hyperrigid, if there is no duality of II leaving ℌ invariant. We exhibit a 3-parameter family S″ of rigid piecewise regular spreads of segment number 4 and show that S″ contains spreads which represent non-isomorphic rigid 4-dimensional translation planes. Finally, we construct a 7-parameter family H of explicit examples of hyperrigid piecewise regular spreads of segment number 5. In H there are at least four spreads which represent mutually non-isomorphic rigid 4-dimensional translation planes.