A class of topological spreads with unisymplecticly complemented regulization
The author constructs a \(3\)-parameter family of Thas-Walker line sets \(c_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) with respect to a particular Lie quadric \(L_4\) in \(\text{PG}(4,\mathbb R)\) (i.e. \(c_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) is a set of lines such that the intersections of \(L_4\) and the images of these lines under the polarity associated with \(L_4\) form a so-called flockoid of \(L_4\)). By the extended Thas-Walker construction [the author, Bull. Belg. Math. Soc. 6, No. 2, 237-247 (1999; Zbl 0940.51006)], each set \(c_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) defines a spread \({\mathcal A}_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) in \(\text{PG}(3,\mathbb R)\). It turns out that all \({\mathcal A}_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) are topological spreads with unisymplecticly complemented regulizations. In particular, the corresponding translation planes are \(4\)-dimensional compact projective planes. Moreover, if \(\varepsilon_1\varepsilon_2\varepsilon_3 \not= 0\), then \({\mathcal A}_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) is algebraic and rigid (its automorphims group contains only the identity), and hence every collineation of the corresponding translation plane fixes the translation axis pointwise. For \(\varepsilon_1\varepsilon_2 \not= 0\), the automorphism group of \({\mathcal A}_{\varepsilon_1,\varepsilon_2,0}\) consists of precisely two elements.
scientific article; zbMATH DE number 1563154
scientific article