Finite partially \(\{0,1\}\)-semiaffine linear spaces
A finite linear space is a finite set of objects called points and a family of proper subsets, called lines, each with at least two points, such that any two distinct points lie on a unique line. The degree of a point is the number of lines through it, and the length of a line is its cardinality. The order of a finite linear space is one less than the maximum degree among all the points. For any nonincident point-line pair \((p, \ell)\), we let \(\Pi (p, \ell)\) denote the number of lines through \(p\) parallel to \(\ell\). Let \(H\) be any set of nonnegative integers. A finite linear space is called partially \(H\)-semiaffine if the line set may be partioned into two sets \({\mathcal V}\) and \({\mathcal I}\), called the visible and invisible lines, such that i) a line \(\ell\) is visible if and only if for every point \(p\) not on \(\ell\), \(\Pi (p, \ell) \in H\), and ii) through any point there are at most two invisible lines. If \({\mathcal V} = \phi\), then the linear space is a near-pencil on three points, and this trivial case will thus be ignored. In the paper under review the authors restrict to \(H = \{ 0, 1 \}\). Then, if \({\mathcal I} = \phi\), the linear space must be a near-pencil, a finite projective plane (possibly punctured), or a finite affine plane (possibly plus a point at infinity) by early work of \textit{P. Dembowski} [Arch. Math. 13, 120-131 (1962; Zbl 0135.39304)]. Here, using a long and involved counting argument, the authors enumerate the possiblities when \({\mathcal I} \neq \phi\). Namely, it is shown that a partially \(\{ 0 \}\)-semiaffine linear space with invisible lines must be a punctured or doubly punctured projective plane. A partially \(\{ 1 \}\)-semiaffine linear space of order \(n\) with \({\mathcal I} \neq \phi\) must have greater than \(n^2 + n + 1\) lines or be one of eight explicitly described examples. A proper partially \(\{ 0, 1\}\)-semiaffine linear space with \({\mathcal I} \neq \phi\) and all points of degree \(n + 1\) must be one of seven explicitly given examples. Finally, a proper partially \(\{ 0, 1\}\)-semiaffine linear space with \({\mathcal I} \neq \phi\) and points of different degrees must be one of nine listed possibilities, two of which are hypothetical.
scientific article; zbMATH DE number 1347330