Arcs and ovals in derivable planes

Abstract

A k-arc S in a projective plane rc (see [8], p. 265) is a set of k points in with no 3 collinear. An (n+ 1)-arc in a finite plane of order n is called an oval in ~. In this note we construct examples of (2q+2)-arcs in a class of planes of order q2 (with q odd) which are known as derivable planes. As is pointed out in Ostrom ([5] and [6]) many of the known planes are of this form. In particular the 4 known planes of order 9 are derivable. Our construction then yields examples of 8-arcs in these 4planes: sometimes these arcs are embeddable in ovals. It turns out that, for the non-Desarguesian translation plane of order 9, the oval we construct is the same as that discussed in Rodriquez [7] and Nizette [3, 4]. In Section 4 we also show the existence of a complete 8-arc in the Desarguesian projective plane of order 9 (see [8], p. 283).