Projective embedding of \([0,m]\)-spaces
A linear incidence space \((M, {\mathfrak M})\) is called a \([0, m]\)-space (with \(m \in \mathbb{N}\)) if for any pointline pair \((x,G) \in M \times {\mathfrak M}\) there are at most \(m\) lines \(X\) through \(x\) which are complanar with \(G\) and \(X \cap G = \emptyset\). For \(G, H \in {\mathfrak M}\) let ``\(G |H : \Leftrightarrow G = H\) or \(G \cap H = \emptyset\) and \(G\), \(H\) are complanar'', if \(x \notin G\), let \(\pi (x,G) := |\{H \in {\mathfrak M} \mid x \in H \wedge H |G\}|\) and let \(\pi (G) := \max \{\pi (x,G) \mid x \in M \setminus G\}\). Clearly \(0 \leq \pi(G) \leq m\). The author calls \(\text{ord } G := |G|+ \pi(G) - 1\) the order of the line \(G\), and proves that in a \([0,m]\)-space \(\text{ord }G\) is constant for all \(G \in {\mathfrak M}\). Therefore \(\text{ord }M = \text{ord } G\) is called the order of the space \((M, {\mathfrak M})\). The main result is read: Each \([0,m]\)-space \((M, {\mathfrak M})\) with \(\dim (M, {\mathfrak M}) \geq 3\) and \(\text{ord }M \geq 3m + 2\) can be embedded in a projective space \((P, {\mathfrak L})\) with \(\dim P = \dim M\) and \(\text{ord }P = \text{ord }M\).
scientific article; zbMATH DE number 740288
scientific article