Generalized semiaffine linear spaces (Q1683902)

In this paper, the authors introduce the notion of ``generalized $T$-semiaffine spaces''. More precisely, let $T$ be a set of natural numbers, and let $S$ be a linear space admitting a finite dimension $n$ (many linear spaces do not admit a dimension) in the sense of \textit{F. Buekenhout} [J. Comb. Theory, Ser. A 27, 121--151 (1979; Zbl 0419.51003)]. A subspace of dimension $n-1$ is called a hyperplane. Then $S$ is a generalized $T$-seminaffine space if for every point-hyperplane pair $(p,H)$, the number of lines through $p$ missing $H$ belongs to $T$. \par The authors show that generalized $\{0,1\}$-semiaffine spaces of dimension at least 3 with all lines of size at least 4, are projective spaces with at most one point deleted. They claim that this improves a result of the reviewer [Electron. J. Comb. 16, No. 1, Research Paper R18, 10 p. (2009; Zbl 1169.51005)], but they fail to see that the reviewer does not use any dimension hypothesis (since many linear spaces do not admit a dimension, far from a finite dimension, this is quite a difference) and that the spaces in the conclusion of the present paper are much more restricted (only two isomorphism types given a skew field and a dimension $n$, whereas the reviewer obtained $2^n$ many isomorphism types). Besides, the result of the reviewer they refer to what was already proved by \textit{A. Kreuzer} [J. Comb. Theory, Ser. A 64, No. 1, 63--78 (1993; Zbl 0806.51003)], so it should be accredited to Kreuzer. \par The second main result of the authors is that every generalized $\{0,1,2\}$-semiaffine space of dimension at least 3 whose lines have size at least 9 are projective spaces with at most two points deleted. Again, they claim that this improves a result of \textit{A. Kreuzer} [J. Comb. Theory, Ser. A 70, No. 1, 66--81 (1995; Zbl 0826.51005)], but again Kreuzer does not hypothesize a dimension, and his conclusion is that the hypothesized linear space (which is more generally a linear space with $[0,m]$-semiaffine planes) embeds in a projective space, leaving a lot more possibilities. \par Finally, the authors prove that there do not exist generalized $\{1,2\}$-semiaffine spaces. \par The techniques of the paper are obvious: using the planes, one reduces to ordinary $T$-semiaffine planes and then use well-known theorems of Teirlinck and Kreuzer.