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Generalized line stars and topological parallelisms of the real projective 3-space. (English) Zbl 1168.51003

A simple covering \(\mathfrak S\) of the non-interior points of an elliptic quadric \(Q\) in real projective three-space is called a generalized line star. It is continuous, if the mapping that assigns a line of \(\mathfrak S\) to a (non-interior) point is continuous. A family \(\mathbf P\) of spreads such that every line is contained in exactly one spread is called a parallelism. It is called topological if the mapping that assigns to a point \(p\) and a line \(L\) a line incident with \(p\) and in the same spread as \(L\) is continuous. In this case, \(\mathfrak S\) is also called topological.
From a given generalized line star \(\mathfrak S\) a parallelism \({\mathbf P}({\mathfrak S})\) can be constructed [D. Betten and R. Riesinger, Result. Math. 47, No. 3–4, 226–241 (2005; Zbl 1088.51005)]). In Section 2 the authors prove that continuity of \(\mathfrak S\) implies that \({\mathbf P}({\mathfrak S})\) is topological.
Section 3 is dedicated to a generalization of a construction of a topological generalized line star \(\mathfrak A\) that was originally presented in [D. Betten and R. Riesinger, Adv. Geom. 8, No. 1, 11–32 (2008; Zbl 1149.51002)]. This line star is axial: any line of \(\mathfrak A\) intersects a fixed line \(A\). In Section 4 the authors construct the apparently first example of a generalized line star \(\mathfrak S\) that is topological but not axial. The elaborate construction requires a quadratic cone \(\Gamma\) with vertex \(o\) in the interior of \(Q\), a quadric \(\Psi\) tangent to \(\Gamma\) along a conic \(T\) contained in the interior of \(Q\) and a covering of parts of \(\Psi\) by a continuous one-parametric family of conics \(C(t)\). The lines of \(\mathfrak S\) are obtained as tangents of \(Q\) that meet the poles \(C(t)\) with respect to \(\Psi\) plus certain lines incident with \(o\). Limiting cases and generalizations of this construction are considered as well.

MSC:

51H10 Topological linear incidence structures
51A40 Translation planes and spreads in linear incidence geometry
51M30 Line geometries and their generalizations
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