Geodesic
Divisible Semiplanes, Arcs, and Relative Difference Sets
Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 1001-1024

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following: (i) S is the empty set. (ii) S consists of a line L with all its points and a point p with all the lines through it. (iii) S is a Baer subplane of Π. We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs). 
Jungnickel, Dieter. Divisible Semiplanes, Arcs, and Relative Difference Sets. Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 1001-1024. doi: 10.4153/CJM-1987-051-1
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