Embedding theorems for locally projective three-dimensional linear spaces
Let \(L\) be a \(d\)-dimensional linear space such that no element of \(L\) is incident with all points. A line incident with only one point is called short, otherwise it is called long. For \(d=3\) the linear space \(L\) is called planar. If there is an incidence preserving injective map from \(L\) to some other linear space \(L'\), then \(L\) is said to be embedded into \(L'\). If \(d\geq 3\), then all elements of \(L\) incident with some point \(p\) form a \((d-1)\)-dimensional linear subspace, which is called the residue of \(p\) and is denoted by \(L/p\). If the residue at every point is a \((d-1)\)-dimensional projective space (of order \(q\)), then \(L\) is called locally projective (of order \(q\)). In the paper under review the author proves several embedding theorems for locally projective planar spaces \(L\) in projective spaces. Several such embedding theorems are known in the literature [for an overview see, e.g., \textit{A. Kreuzer}, `Zur Theorie der Bündelräume' (1993; Zbl 0798.51007)]. \textit{W. M. Kantor} [J. Comb. Theory, Ser. A 17, 173-195 (1974; Zbl 0302.06018)] proved that \(L\) can always be embedded into a projective space of the same dimension and the same order as \(L\), if \(d \geq 4\). If \(L\) can be embedded into a projective space, then it must satisfy the Bundle Theorem: If \(L_1,L_2,L_3,L_4\) are mutually non-intersecting lines, no three of them lying on a common plane, and if \(L_i\) and \(L_j\) are coplanar for \(1 \leq i < j \leq 4\) and \((i,j) \neq (3,4)\), then \(L_3\) and \(L_4\) are also coplanar. \textit{J. Kahn} [Math. Z. 175, 219-247 (1980; Zbl 0443.06010)] gave a sufficient condition for a locally projective planar space of order \(q\) satisfying the bundle theorem to be embeddable. In this pleasant paper the author gives interesting new sufficient conditions which are based on the number of points in \(L\) rather than the number of short lines through some point (which essentially was Kahn's condition). Some of these conditions are: (1) if \(L\) has more than \({1\over2}q(q^2-1)\) points, (2) if \(q \geq 2\) and \(L\) has more than \(q^3-q^2+q+2\) points, (3) if \(q \geq 2\), and every residue of \(L\) is a Desarguesian projective plane, every line of \(L\) is long, and \(L\) has at least \(0.763(q+1)^3+(q+1)^2\) points. If any of these conditions are satisfied, then \(L\) can be embedded into \(\text{PG}(3,q)\).
scientific article; zbMATH DE number 1110051