A unital in the projective plane $PG(2, q^2)$ is a set $U$ of $q^3+1$ points of $PG(2,q^2)$such that every line meets $U$ in 1 or q+1 points. These lines are known as tangent and secant lines respectively. If we take the points of the unital together with the secant lines, we obtain a $3-(q^3+1, q+1, 1)$ design.

### An example of a unital; the Hermitian curve.

Consider the set of points $(x_0,x_1,x_2)$ of $PG(2,q^2)$ which satisfy

$x_0^{q+1}+x_1^{q+1}+x_2^{q+1}=0.$

Then these points form a unital known as a Hermitian curve or classical unital. A classical unital is defined to be a set of points which is the image of this example under the group $P\Gamma L(3,q^2)$.

### Another example: Buekenhout-Metz unitals.

In 1976, Francis Buekenhout gave a construction of a family of non-classical unitals of $PG(2,q^2)$ (and in fact, translation planes of order $q^2$ with kernel containing $GF(q)$) for $q>2$ even and not a square. This construction was extended by Rudolf Metz (1979) to all $q>2$. It is a complicated construction for the uninitiated, but we will try to explain it below. Moreover, we will say that the output are Buekenhout-Metz unitals, even though they include the classical unitals.

Consider the affine space one gets by having distinguished a hyperplane $\Sigma_\infty$ in $PG(4,q)$. Inside our 3-dimensional projective space $\Sigma_\infty$, suppose we have a partition $S$ of the points into $q^2+1$ lines; this is known as a spread. Furthermore, we will actually want to have a nice spread, known as the Desarguesian spread, which is constructed as follows. The points of the projective line $PG(1,q^2)$ are the 1-dimensional subspaces of $GF(q^2)^2$. By taking a basis for $GF(q^2)$ over $GF(q)$, we can take the $GF(q)$-subspaces arising from our old $GF(q^2)$-subspaces, which blows things up. We see that a 1-dimensional subspace blows up to a 2-dimensional subspace of $GF(q)^4$ when we do this. This is the Desarguesian spread.

Now define the following incidence structure:

• POINTS: (i) the points of $PG(4,q)$ not contained in $\Sigma_\infty$ (the affine points), together with (ii) the elements of the spread $S$; so we have $q^4+q^2+1$ points in total.
• LINES: (a) the planes of $PG(4,q)$ which intersect $\Sigma_\infty$ in an element of $S$, together with (b) the set $S$ (just one more line); so we have $q^4+q^2+1$ of these too.

The incidence relation is either containment of objects or natural incidence in the projective space. Believe it or not, this incidence structure is equivalent to the Desarguesian projective plane $PG(2,q^2)$. This is the André/Bruck-Bose representation of $PG(2,q^2)$. Here is an artist’s impression of it here:

Now take a line $\ell$ of the spread $S$, and two points $U$ and $V$ on $\ell$. Here comes the tricky part. Suppose we have an ovoid (see this previous post) $\mathcal{O}$ of $q^2+1$ points inside some other hyperplane of $PG(4,q)$ such that $\mathcal{O}$ meets $\Sigma_\infty$ in the point $U$, and $V\notin \mathcal{O}$. We then build a cone $V\mathcal{O}$ out of the ovoid $\mathcal{O}$ and the point $V$, and the line $\ell$ is one of the lines of this cone. Let $\mathcal{U}$ be the set of affine points which lie on the lines of this cone. Let’s count these. Each line of the cone has $q$ affine points, except for $\ell$ which has none. So there are $q^3$ affine points lying on the cone. If we add the POINT $\ell$ to our set $\mathcal{U}$, we end up with a unital of the corresponding $PG(2,q^2)$ incidence structure. These unitals are known as Buekenhout-Metz unitals, and the classical unital is an example of one of these. Here is a picture which illustrates what we have been discussing:

### The open problem

Do there exist unitals of $PG(2,q^2)$ which do not arise from the Buekenhout-Metz construction?

## 2 thoughts on “Open problems in Finite Geometry: Unitals of Desarguesian projective planes”

1. Another interesting open problem about unitals that I recently learned about is on existence of disjoint unitals in $\mathrm{PG}(2, q^2)$. The classical ones always intersect non-trivially (in fact, with number of points congruent to 1 mod p).