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.

An ovoid is a set of ${q^2+1}$ points of ${\mathrm{PG}(3,q)}$ such that no three are collinear. John discussed ovoids in a previous post. As John discussed, it was proved by Barlotti and Panella that when ${q}$ is odd the only ovoids are the elliptic quadrics, while when ${q}$ is even it is conjectured that the only ovoids are the elliptic quadrics and the Suzuki-Tits ovoids.

${T_2(O)}$ and ${T_3(O)}$

Now I can give the constructions of the generalised quadrangles ${T_2(O)}$ and ${T_3(O)}$ due to Jacque Tits.

Let ${O}$ be an oval in ${H=\mathrm{PG}(2,q)}$ and embed ${H}$ as a hyperplane in ${A=\mathrm{PG}(3,q)}$. We construct an incidence geometry whose points are:

• (i) the points of ${A}$ not on ${H}$,
• (ii) the hyperplanes of ${A}$ which meet ${O}$ in a single point,
• (iii) an extra point denoted by ${(\infty)}$.

and whose lines are

• (a) the lines of ${A}$ which are not contained in ${H}$ and meet ${O}$ in a unique point,
• (b) the points of ${O}$.

Incidence is as follows:

• a point of type (i) is incident only with lines of type (a) and here incidence is that inherited from ${A}$.
• a point of type (ii) is incident with all lines of type (a) contained in it and with the unique point of ${O}$ that it meets.
• ${(\infty)}$ is incident with all lines of type (b) and none of type (a).

This incidence structure is a GQ of order ${(q,q)}$ denoted by ${T_2(O)}$.

As mentioned previously, the only ovals for ${q}$ odd are the conics and in this case ${T_2(O)}$ is isomorphic to ${\mathsf{Q}(4,q)}$. For ${q}$ even, ${T_2(O)}$ is isomorphic to ${\mathsf{W}(3,q)}$ if and only if ${O}$ is a conic. Thus for ${q}$ even, the ovals which are not conics give rise to new GQs of order ${(q,q)}$.

It was proved by Tim Pentilla and Christine O’Keefe in their paper `Subquadrangles of generalised quadrangles of order ${(q^2,q)}$, ${q}$ even’, that the full automorphism group of ${T_2(O)}$, for ${O}$ not a conic, is the stabiliser in ${\mathrm{P}\Gamma\mathrm{L}(4,q)}$ of ${O}$. (Thanks Tim for the reference.) Since there are several types of points and lines in ${T_2(O)}$, it follows that it is not possible for these GQs to be point- or line-transitive when ${O}$ is not a conic. It also follows that if ${T_2(O)}$ is isomorphic to ${T_2(O')}$ for ${O}$ and ${O'}$ ovals which are not conics, then there is a collineation of ${\mathrm{PG}(2,q)}$ which maps ${O}$ to ${O'}$.

Next let ${O}$ be an ovoid of ${H=\mathrm{PG}(3,q)}$. We can embed ${H}$ in ${A=\mathrm{PG}(4,q)}$ and we can define a new incidence structure in the same way as for ${T_2(O)}$. This gives us a generalised quadrangle of order ${(q,q^2)}$ denoted by ${T_3(O)}$.

The only GQs of order ${(q,q^2)}$ that we have seen already are ${Q(5,q)}$. It turns out that ${T_3(O)}$ is isomorphic to ${Q(5,q)}$ if and only if ${O}$ is an elliptic quadric. As mentioned earlier, for ${q}$ odd elliptic quadrics are the only ovoids in ${\mathrm{PG}(3,q)}$, while for ${q}$ even we can obtain new GQs by taking ${O}$ to be a Suzuki-Tits ovoid.

It was proved by Tim Pentilla and Christine O’Keefe in their paper mentioned earlier that the full automorphism group of ${T_3(O)}$, for ${O}$ not an elliptic quadric, is the stabiliser in ${\mathrm{P}\Gamma\mathrm{L}(5,q)}$ of ${O}$. This result was also previously proved by Kantor. Again, if ${O}$ is not an elliptic quadric it follows that ${T_3(O)}$ is neither point- nor line-transitive. Also, if ${T_3(O)}$ is isomorphic to ${T_3(O')}$ for ${O}$ and ${O'}$ ovals which are not elliptic quadrics, then there is a collineation of ${\mathrm{PG}(3,q)}$ which maps ${O}$ to ${O'}$.

${T_2^*(O)}$

The next construction is due to Ahrens and Szekeres and independently due to Marshall Hall.

Let ${O}$ be a hyperoval of ${H=\mathrm{PG}(2,q)}$ for ${q}$ even. Embed ${H}$ in ${P=\mathrm{PG}(3,q)}$. We define a new incidence structure with

• points the points of ${A}$ not in ${H}$, and
• whose lines are the lines of ${A}$ which are not contained in ${H}$ and meet ${O}$ in a unique point.

Incidence is inherited from ${A}$.

This provides us with a GQ of order ${(q-1,q+1)}$ and is denoted by ${T_2^*(O)}$. When ${q=4}$, we get the ${Q^-(5,2)}$ which we have seen previously.

It was proved by Bichara, Mazzocca and Somma that for ${q\geq 4}$, the automorphism group of ${T_2^*(O)}$ is the stabiliser in ${\mathrm{PGL}(4,q)}$ of ${O}$, that is, it is an elementary abelian ${2}$-group of order ${q^3}$ extended by the stabiliser in ${\Gamma L(3,q)}$ of the hyperoval ${O}$. The elementary abelian 2-group acts regularly on the points of the GQ. The automorphism group will be transitive on lines if and only if the stabiliser of the hyperoval is transitive on the points of the hyperoval. If it is line-transitive it will also be flag-transitive as the stabiliser of a point is the stabiliser of the hyperoval. Hence the only new line-transitive (and also flag-transitive) GQ we get that is the one of order ${(15,17)}$ obtained from the Lunelli-Sce hyperoval.

Payne-derivation

Now the GQ’s ${T_2(O)}$ have order ${(q,q)}$ and so are potential candidates for Payne-derivation (outlined in the previous post in this series). To find new GQ’s we only need to consider the case where ${q}$ is even and ${O}$ is not a conic.

The point ${(\infty)}$ is a regular point of ${T_2(O)}$. The proof goes as follows: Let ${x=(\infty)}$. Then the points of ${T_2(O)}$ collinear with ${x}$ are the hyperplanes of ${A=\mathrm{PG}(3,q)}$ other than ${H}$ which meet ${O}$ in a single point. Thus the points not collinear with ${(\infty)}$ are the points of ${A}$ not on ${H}$. Let ${y}$ be one of these points. The lines of ${T_2(O)}$ incident with ${y}$ are the lines of ${A}$ containing ${y}$ and so the points of ${T_2(O)}$ collinear with both ${x}$ and ${y}$ are the hyperplanes of ${P}$ not on ${H}$ which contain ${y}$. Now given one of these hyperplanes ${H'}$, the points of ${T_2(O)}$ incident with it are ${(\infty)}$, the points of the hyperplane which are not in ${H}$, the hyperplanes of ${A}$ which meet ${H'}$ in a line not on ${H}$ and meet ${O}$ in a unique point, and the hyperplanes of ${A}$ which meet ${O}$ in the same unique point as ${H'}$. Running over all ${H'\in\{x,y\}^{\perp}}$ it follows that ${\{x,y\}^{\perp\perp}}$ does not contain any hyperplanes. Moreover, since ${H'}$ meets ${O}$ in a point, it contains the nucleus of ${O}$. Labelling the nucleus by ${n}$, it follows that ${H'}$ contains the unique line ${\ell}$ through ${y}$ and ${n}$. Thus ${\{x,y\}^{\perp\perp}}$ is equal ${(\infty)}$ together with all the points of ${\ell}$ other than the nucleus ${n}$. Thus it has size ${q+1}$ and so ${(\infty)}$ is indeed regular. Moreover, we can think of the hyperbolic lines of ${T_2(O)}$ as the line of ${A}$ not on ${H}$ which meet ${n}$.

As mentioned before, the only points of ${T_2(O)}$ collinear with ${(\infty)}$ are the points of ${\mathrm{PG}(3,q)}$ not contained in the hyperplane ${H}$. The lines of ${T_2(O)}$ not collinear with ${(\infty)}$ are the lines of type (a), that is the lines of ${\mathrm{PG}(3,q)}$ not contained in ${H}$ and meet ${O}$ in a unique point. Moreover, as seen in the previous paragraph, the hyperbolic lines containing ${\infty}$ are the lines of ${\mathrm{PG}(3,q)}$ not contained in ${H}$, and which meet ${H}$ in the nucleus of ${O}$. Thus the Payne-derivative of ${T_2(O)}$ with respect to ${(\infty)}$ is ${T_2^*(O)}$.

For ${q\geq 8}$ even, there are examples of ovals ${O}$ for which ${T_2(O)}$ contains regular points other than ${(\infty)}$, and Payne showed that Payne derivation with respect to these points give GQs of order ${(q-1,q+1)}$ not isomorphic to ${T_2^*(O)}$.

More GQs

I still haven’t constructed all the GQs. Originally I had thought that the only ones left were the flock GQs (which I aim to discuss next) but John pointed out to me yesterday that there are in fact more so there will have to be at least one more post after that.