On Arcs in a Finite Projective Plane

Abstract

The aim of this paper is to generalize and unify results of B. Qvist, B. Segre, M. Sce, and others concerning arcs in a finite projective plane. The method consists of applying completely elementary combinatorial arguments.To the usual axioms for a projective plane we add the condition that the number of points be finite. Thus there exists an integer n ⩾ 2, called the order of the plane, such that the number of points and the number of lines equal n2 + n + 1 and the number of points on a line and the number of lines through a point equal n + 1. In the following, n will always denote the order of a finite plane. Desarguesian planes of order n, formed by the analytic geometry with coefficients from the Galois field of order n, are examples of finite projective planes. We shall not assume that our planes are Desarguesian, however.

Ovals in a plane coordinatised by a regular nearfield of dimension 2 over its centre

“…It turns out that ∩ 0,1,1,0 ≥ 1 2 (n − 1) + 2. But then = 0,1,1,0 by a result of Qvist [6], see also [3,Theorem 3.2.25] and [1,5]. Therefore, is a Rosati oval.…”

Necessary conditions for additive loops of finite projective planes

“…Count (1) O(j2)l = n-2 > (n/2)+ 1. This is impossible as, by a simple counting argument, different hyperovals cannot have more than (n/2) + 1 points in common [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

Finite Laguerre Near-planes of Odd Order Admitting Desarguesian Derivations

“…Examples of complete 9-arcs in projective planes of order 9 can be found in [3] and [6]. For finite Desarguesian projective planes, however, we have the following, cf.…”