In this post I want to continue discussing some of the constructions for nonclassical generalised quadrangles. First I need to introduce some new geometrical notions.

Ovals, hyperovals and ovoids

An oval in the projective plane ${\mathrm{PG}(2,q)}$ is a set of ${q+1}$ points such that no three are collinear. The typical example is a conic, that is, the set of zeros of some nondegenerate quadratic form. Up to equivalence we can take this quadratic form to be ${Q(x)=x_1x_2+x_3^2}$, and so any conic can be mapped to the conic defined by ${Q}$ by a collineation of ${\mathrm{PG}(2,q)}$. Lines of the projective plane are either

1. nondegenerate with respect to the quadratic form, and so contain two totally singular points, or
2. degenerate, and contain precisely one totally singular point ${v}$ and this point is perpendicular to all the remaining points on the line, that is, ${B(v,w)=0}$ for all remaining points on the line, where ${B}$ is the bilinear form associated with ${Q}$.

In fact, it is a theorem of Segre that for ${q}$ odd the only ovals are these conics.

For ${q}$ even, any oval can be extended to a set of ${q+2}$ points such that no three are collinear. Such a set is called a hyperoval. This can be seen in the case of a conic where there is a unique point of ${\mathrm{PG}(2,q)}$ which is perpendicular to all points of the conic. This point is referred to as the nucleus. For the quadratic form ${Q(x)}$ given above this is the point ${\langle (0,0,1)\rangle}$. This provides a construction for further ovals in the ${q}$ even case, for any ${q+1}$ points of a hyperoval is an oval. For ${q>4}$ the setwise stabiliser of this hyperoval fixes the nucleus and so these extra ovals are not conics. In general, given an oval ${O}$, for each point ${P}$ on the oval there is a unique line of ${\mathrm{PG}(2,q)}$ meeting ${O}$ at ${P}$ and this line is called a tangent line. The tangent lines meet in a common point outside the conic, and this is the nucleus.

In fact, there are many hyperovals which do not come from adding the nucleus to a conic, and hence lots of ovals. Of note are the Subiaco ovals (named as they were discovered here in Perth and Subiaco is a local suburb, and a pun on the fact that the main football ground in Perth is Subiaco Oval) and the Adelaide ovals (cricket fans will know of the Adelaide oval, the main cricket ground in South Australia). There is also the Lunelli-Sce hyperoval in ${\mathrm{PG}(2,16)}$, which along with the hyperovals formed by adding the nucleus to a conic when ${q=2}$ or ${4}$, are the only hyperovals for which the setwise stabiliser of the hyperoval is transitive on the set of points of the hyperoval. This was proved by Korchmáros in 1978.

Further details on ovals and hyperovals can be found in this survey and at Bill Cherowitzo’s hyperoval page.