Bilinear flocks of quadratic cones
The authors investigate the translation planes, isomorphisms, the derived flocks and the generalized quadrangles of a class of bilinear flocks of a quadratic cone over ordered fields, whose construction is modelled on the construction of a flock of the sphere by considering all planes through a line \(L\) (with \(L\) lying in the plane of the equator) meeting the sphere in the north half and all planes through a line \(M\neq L\) (also \(M\) lies in the plane of the equator) meeting the sphere in the south half (since the field is ordered, ``north'' and ``south'' make sense -- as ``positive'' and ``negative'', respectively). So ``bilinear'' means that all planes of the flock contain one of two fixed lines, and these lines are contained in at least two planes of the flock. It turns out that the associated projective planes admit \(SL_2\) as a collineation group, and the authors also investigate, in a more general setting, this situation more closely. There is also an interesting characterization of bilinear flock planes as translation planes over ordered fields (in which all positive elements are squares) whose associated spread is doubly covered by reguli.
scientific article; zbMATH DE number 1292215
scientific article