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Ovoids and ovoids

March 31, 2014

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.

 

 

 

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