On finite \(\{\) 1,2,3\(\}\)-semiaffine planes (Q1092404)

A (proper) \{1,2,3\(\}\)-semiaffine plane is a linear space (P,L) such that for every anti-flag (p,\(\ell)\) there are exactly 1, 2 or 3 lines through p which are disjoint from \(\ell\) (and each of the three possibilities does occur). Variations of this concept have been investigated by several people, cp. \textit{P. Dembowski} [Arch. Math. 13, 120-131 (1962; Zbl 0135.393)], \textit{A. Beutelspacher} and \textit{K. Metsch} [Ann. Discrete Math. 30, 39-56 (1986; Zbl 0586.51012)]. The authors prove the following result. Let (P,L) be a finite proper \{1,2,3\(\}\)-semiaffine plane of order \(n>12\). If (P,L) has a point of degree n, then (P,L) is obtained from a projective plane of order n by deleting all but one of the points on the sides of a triangle. In the remaining cases every point has degree \(n+1,\) and if \(n>51\), then (P,L) is obtained from a projective plane of order n by deleting either two lines \(\ell_ 1\), \(\ell_ 2\) and one or two points collinear with \(\ell_ 1\cap \ell_ 2\), or a k-arc and one of its exterior lines, where \(2\leq k\leq n+1.\)