A unified construction for \(c^*\cdot c\)- and \(L\cdot L^*\)-geometries in projective spaces (Q1976896)

The author presents two new constructions of \(c^{\ast }\cdot c\)-geometries, where \(c\) denotes the class of circular spaces and \(c^{\ast }\) the class of dual circular spaces. One of these constructions is realized in \(PG(3,q)\) (for \(q\) even) and the other one in \(PG(4,q)\). For \(q\) even, both of them provide models of the same geometry, related to the Cameron-Fisher geometries. The paper contains a description of these geometries, as well as of those constructed by Meixner and Pasini, related to the second construction presented here, for \(q\) odd. Finally, a new family of \(L\cdot L^{\ast }\)-geometries embedded in \(PG(3,q)\) is obtained, using some complementary models for \(c^{\ast }\) and \(L\) in a projective plane.