I’m writing this post so that I can direct my students to it as I often have to go over the evolution of the concepts of ovoids of projective spaces and polar spaces, and to explain that they are (i) different, and (ii) connected. It’s  a bit dry, but it will serve as a reference for future posts.

## Ovoids of projective spaces

I will start with a result of Jacques Tits in 1962, though ovoids arose earlier in the work of Bose and Qvist. An ovoid of a projective space $\mathcal{S}$ is a set of points $\mathcal{O}$ such that for any point $P$ of $\mathcal{O}$, the union of the lines incident with $P$ that are tangent to $\mathcal{O}$ forms a hyperplane of $\mathcal{S}$. Tits showed that an ovoid of $PG(n,q)$ exists if and only if $n\le 3$. For $n\le 3$, we should be careful to stipulate that $q>2$, since the size of an ovoid of $PG(3,2)$ is just 5 and things are a bit messy to state. By a simple counting argument, the number of points of an ovoid of $PG(3,q)$ is $q^2+1$, and for a projective plane of order $q$, an ovoid has size $q+1$. For example, the elliptic quadric of $PG(3,q)$ is an example of an ovoid, and a non-singular conic is an example of an ovoid of $PG(2,q)$. For more on the elliptic quadric example, see this post.

In the planar case, we now call these objects ovals, and reserve the name ‘ovoid’ just for the 3-dimensional case. Now an ovoid and oval have the property that no 3 points are collinear. We have the following way to think of an ovoid in three different ways:

Let $\mathcal{O}$ be a set of points of $PG(3,q)$. The following are equivalent:

1. No 3 points of $\mathcal{O}$ are collinear and $|\mathcal{O}| = q^2+1$;
2. For any point $P$ of $\mathcal{O}$, the union of the lines incident with $P$ that are tangent to $\mathcal{O}$ span a plane of $PG(3,q)$;
3. At every point $P$ of $\mathcal{O}$, there is a unique plane that meets $\mathcal{O}$ only in $P$ (i.e., a tangent plane).

## Ovoids of polar spaces

Jef Thas defined ovoids of polar spaces in his seminal paper “Ovoidal translation planes” in 1972. It is a set of points such that every maximal totally isotropic subspace meets it in precisely one point. Alternatively, we could define it as a set of points no two collinear, with size the number of points divided by the number of points in a maximal. We have:

Let $\mathcal{O}$ be a set of points of a finite polar space $\mathcal{S}$, and let $\mu$ be the number of points of $\mathcal{S}$ divided by the number of points lying in a maximal totally istropic subspace. The following three are equivalent:

1. No 2 points of $\mathcal{O}$ are collinear and $|\mathcal{O}| = \mu$;
2. Every maximal totally isotropic subspace has exactly one point of $\mathcal{O}$ incident with it.
3. At every point $P$ of $\mathcal{O}$, there exists a maximal totally isotropic subspace through $P$ tangent to $\mathcal{O}$.

So where did Thas’ definition come from? The most impressive result of his 1972 paper is the connection between ovoids of the rank 2 symplectic space and ovoids of the 3-dimensional projective spaces.

Theorem (Thas 1972): Let $q$ be an even prime power. Then an ovoid of $W(3,q)$ is also an ovoid of $PG(3,q)$. Conversely, if $\mathcal{O}$ is an ovoid of $PG(3,q)$, then we can define a null polarity $\rho$ defining a $W(3,q)$ such that $\mathcal{O}$ are absolute points for $\rho$.

This result is quite remarkable. It certainly is easier to study and enumerate ovoids of $W(3,q)$, than in $PG(3,q)$. The null polarity hinted at by the theorem is truly beautiful and was first known to Segre (1959): given a point $P$ of $\mathcal{O}$, we define $P^\rho$ to be the unique tangent plane at $P$; for each secant plane $\pi$ of $\mathcal{O}$ we define $\pi^\rho$ to be the nucleus of the oval carved out (i.e., the nucleus of $\mathcal{O}\cap\pi$).

The situation for $q$ odd was already classified independently by Barlotti and Panella (1965): the only ovoids in this case are quadrics. So the open problem on classifying ovoids of $PG(3,q)$ boils down to the even case.

Another reason for defining ovoids this way is that it encapsulates natural examples. A non-degenerate hyperplane section of the Hermitian polar space $H(3,q^2)$ is an ovoid, and a non-degenerate hyperplane section of minus type of the parabolic quadric $Q(4,q)$ is an ovoid. Plus, we can think of the non-singular conic as a rank 1 polar space $Q(2,q)$, and hence an ‘ovoid’ of $Q(2,q)$ coincides trivially with an oval of $PG(2,q)$.

In a future post, I’ll talk about the open problems on ovoids of polar spaces.