A characterization of the partial geometry \(T^ *_ 2(K)\)
A partial geometry S has parameters s, t, and \(\alpha\), which are respectively the number of lines through a point, the number of points on a line, and the number of lines connecting a point and a line. Two lines are called parallel if either they are the same or they are skew and have a pair of transversals intersecting at a point on neither line. Let \(\ell\) and m be distinct lines which are either parallel or meet, and for any point P not on \(\ell\) or m let \(\beta\) (P) be the number of lines through P meeting both \(\ell\) and m. If S has the properties that \(1<\alpha <\min (s,t)\) and that \(\beta (P)=\alpha -| \ell \cap m|\) for all such pairs (\(\ell,m)\), then S is called net-inducible; in the particular case that \(\ell\) and m meet, they lie in a net N(\(\ell,m)\), called a generated net, which is also determined by any non- incident pair (P,\(\ell)\) with P on m. The partial geometry \(T^*_ 2(K)\) has as its points the complement of a plane H in PG(3,q) and as its lines the lines of PG(3,q) meeting a maximal arc K in H in a single point; its parameters are \(s=q-1,\) \(t=(d-1)(q+1),\alpha =d-1\) for some d. The first theorem is that if S is a net-inducible partial geometry, then (a) every pair of non-collinear points is contained in precisely \((t+1)/(\alpha +1)\) generated nets; (b) every line of S is contained in t/\(\alpha\) generated nets; (c) \(t\geq \alpha (s+2)\). The main result is that when equality holds in (c) and every triad of points is contained in precisely 0, 1 or \((t+1)/(\alpha +1)\) generated nets of order \(s+1\) and degree \(\alpha +1\), then S is isomorphic to \(T^*_ 2(K)\).
scientific article; zbMATH DE number 4009207
scientific article