Projective plane

Definition

Let P be a set, which elements are called points, L is a collection of subsets of P, called lines and I is a relation between points and lines, called relation of incidence.

The triple $Pi$=(P,L,I) is called a projective plane, if 1. any pair of distinct points are incident with exactly one line; 2. any pair of distinct lines is incident exactly with one point; 3. there exists four points no three of which are incident with the same line.

Points which are incident with the same line are called collinear. \\ For any projective plane $\Pi$ there exists integer $q\geq 2$ such that 1. Any point (line) of projective plane $\Pi$ is incident exactly with $q+1$ lines (points). 2. A projective plane $\Pi$ has exactly q^{2+q+1$ points (lines).}

This number q is called the dimension of projective plane and $\Pi$ is denoted by PG(2,q).

	 Colliniation of projective plane is an authomorphism of projective plane which preserve incidentness. $P\Gamma L(3,q)$ is the group of all colliniations of a projective plane of order q. 

k- arc is a set of k point of a projective plane no three of which are collinear.

In a projective plane of order q the maximal size of k-arc is q+1, if q$is odd and q+2, if q is even. (q+1)-arc is called oval, (q+2)-arc - hyperoval. Hyperovals exist only in projective planes of even dimension.

A line is called tangent to an oval if it meets the oval in precisely one point.

There is a unique tangent line to each point of oval in projective plane of even dimension. All tangent lines of an oval has intersection at one point and all lines through this point are tangent lines to the oval. The intersection point of all tangent lines to an oval is called nucleus.

In projective planes of even dimension every oval is contained in a unique hyperoval. This hyperoval is obtained by adding to the points of oval the nucleus. On the other hand, if we start with a hyperoval H and remove a point N ∈H then we are left with an oval H\{N} which has nucleus N and completes to H. Thus, one hyperoval give rise to q+1 ovals.