A class of topological spreads with unisymplecticly complemented regulization

Abstract

We give an application of the extended Thas-Walker construction. The major tool is the concept of Thas-Walker line sets which can be seen as link between flockoids of a Lie quadric on the one hand and spreads with symplecticly complemented regulization on the other hand. We construct a class c of Thas-Walker line sets with respect to a Lie quadric of the real projective 4-space. In c we find 4-spatial, solid, and planar Thas-Walker line sets such that the corresponding classes D and C of flockoids resp. spreads contain also 3 types each. The class C consists of topological spreads ${\cal A}_{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$ with unisymplecticly complemented regulization. If ε₁ε₂ε₃ ≠ 0, then the spread ${\cal A}_{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$ is algebraic and rigid, i.e., the group Aut ${\cal A}_{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$ of all collineations leaving ${\cal A}_{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$ invariant contains only the identity. If ε₁ε₂ ≠ 0 and ε₃ = 0, then $\#({\rm Aut}\ {\cal A}_{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}})=2$.