On Sets of Lines Corresponding to Affine Spaces

1. Given a 3-dimensional Pappian projective space, it is well known that the Grassmannian which is representing its set of lines is a quadric (called the Pltcker quadric) in a 5-dimensional projective space. The link between the set of lines and the Pltcker quadric is the bijective Klein map g. Under g pencils of lines become lines of the Pltcker quadric. Cf. e.g. [5,287], [15,28], [16,13], [20,176], [21,327]. Now, taking any 3-dimensional projective space, we may ask for an injective map g from the set of lines into the set of points of (another) projective space, such that every pencil of lines is mapped onto a line. If we have a non-Pappian space, then such a map does not exist [9,172]. For a Pappian space any such map is the product of the Klein map with a suitable collineation [9,174], [22,377]. If j denotes a stereographic projection of the Pltcker quadric g through a point At=ta (for some line a), then the restriction of gj to the set of lines which are skew to a is a bijection l, say, onto a 4-dimensional affine space. The line a has no image under gj and all lines which meet a in one point are mapped (in a non-injective way) onto a ruled quadric in the hyperplane at infinity of this affine 4-space. Cf. [10,109], [14] for these results and their generalization to higher dimensions as well as [18]. Any subspace of this affine 4-space may be interpreted as the stereographic image of those points of the Pltcker quadric which belong to a subspace passing through A. The corresponding sets of lines are well known (e.g. linear complexes of lines). However, as has been shown in [13] there is an alternative construction of this bijection l which will work for Pappian as well as non-Pappian spaces. Thus this l may serve as a substitute for the Klein map in the non-Pappian case. Throughout this article the inverse of such a bijection l will be labeled b. The present paper is concerned with those sets of lines which correspond under b to the subspaces of such an affine 4-space. Clearly, only non-Pappian spaces are of interest.