Subplane covered nets and semipartial geometries
Let \({\mathbf N}\) be a net and assume the existence of a set \({\mathbf B}\) of affine subplanes of \({\mathbf N}\) satisfying the following requirements: (i) Each plane in \({\mathbf B}\) has the same set of parallel classes as \({\mathbf N}\); (ii) for each distinct pair of collinear points \(p\), \(p'\) of \({\mathbf N}\), there exists a unique plane in \({\mathbf B}\) containing \(p\) and \(p'\). Then \({\mathbf N}\) is called a subplane covered net. (In the special case, where each plane in \({\mathbf B}\) is a Baer subplane, \({\mathbf N}\) is a derivable net. Hence subplane-covered nets generalize the well-known derivable nets.) The authors completely characterize the finite subplane- covered nets as those nets which are determined by an \((n-1)\)-regulus in \(PG(2n-1,q)\). This extends a previous result by the second author [Abh. Math. Semin. Univ. Hamb. 58, 245-253 (1988; Zbl 0686.51002); Part II: Arch. Math. 55, No. 1, 94-104 (1990; Zbl 0721.51004)], where the finite derivable nets were characterized as those nets determined by a regulus in \(PG(3,q)\).
scientific article; zbMATH DE number 94517
scientific article