Oval (projective Plane) - Even q

Even q

When q is even, the situation is completely different.

In this case, sets of q + 2 points, no three of which collinear, may exist in a finite projective plane of order q and they are called hyperovals; these are maximal arcs of degree 2.

Given an oval there is a unique tangent through each point, and if q is even Qvist (1952) showed that all these tangents are concurrent in a point p outside the oval. Adding this point (called the nucleus of the oval or sometimes the knot) to the oval gives a hyperoval. Conversely, removing any one point from a hyperoval immediately gives an oval.

As all ovals in the even order case are contained in hyperovals, a description of the (known) hyperovals implicitly gives all (known) ovals. The ovals obtained by removing a point from a hyperoval are projectively equivalent if and only if the removed points are in the same orbit of the automorphism group of the hyperoval. There are only three small examples (in the Desarguesian planes) where the automorphism group of the hyperoval is transitive on its points (see (Korchmáros 1978)) so in general there are different types of ovals contained in a single hyperoval.