A* unital *in the projective plane *tangent* and *secant* lines respectively. If we take the points of the unital together with the secant lines, we obtain a

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

Consider the set of points

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

### Another example: Buekenhout-Metz unitals.

In 1976, Francis Buekenhout gave a construction of a family of non-classical unitals of

Consider the affine space one gets by having distinguished a hyperplane *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

Now define the following incidence structure:

**POINTS:**(i) the points ofnot contained in (the *affine*points), together with (ii) the elements of the spread; so we have points in total. **LINES:**(a) the planes ofwhich intersect in an element of , together with (b) the set (just one more line); so we have 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

Now take a line *ovoid* (see this previous post) *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

which do not arise from the Buekenhout-Metz construction?