Ovoid

Ovoids in 3-dimensional projective space.

Let $\mathrm{PG}(3,q)$ denote the $3$-dimensional projective space over the Galois field of order $q$. An ovoid in $\mathrm{PG}(3,q)$ is a set of $q^2+1$ points, no three of which are collinear. (Note that in [a2] and some other older publications, an ovoid is called an ovaloid.) If $q>2$, then an ovoid is a maximum-sized set of points, no three collinear, but in the case $q=2$ the complement of a plane is a set of $8$ points, no three collinear. More information about ovoids can be found in [a2] and [a5]. The survey paper [a4] gives further details, especially regarding more recent work, with the exception of the recent (1998) result in [a1].

The only known ovoids in $\mathrm{PG}(3,q)$ (as of 1998) are the elliptic quadrics (cf. also Quadric), which exist for all $q$, and the Tits ovoids, which exist for $q=2^h$ where $h\ge 3$ is odd. There is a single orbit of elliptic quadrics under the homography group $\mathrm{PGL}(4,q)$, and one can take as a representative the set of points

$$ \left\{ (t^2+st+as^2,1,s,t): s,t \in \mathrm{GF}(q) \right\} \cup \{(1,0,0,0)\}, $$

where $x^2+x+a$ is irreducible over $\mathrm{GF}(q)$. The stabilizer in $\mathrm{PGL}(4,q)$ of an elliptic quadric is $\mathrm{PO}^-(4,q)$, acting $3$-transitively on its points. There is a single orbit of Tits ovoids under $P\Gamma L(4,q)$, a representative of which is given by

$$ \left\{ (t^\sigma+st+s^{\sigma+2},1,s,t): s,t \in \mathrm{GF}(q) \right\} \cup \{(1,0,0,0)\}, $$

where $\sigma \in \operatorname{Aut} \mathrm{GF}(q)$ is such that $\sigma^2 \equiv 2 \pmod{q-1}$. The stabilizer in $\mathrm{PGL}(4,q)$ of a Tits ovoid is the Suzuki group $\mathrm{Sz}(q)$ acting $2$-transitively on its points. A plane in $\mathrm{PG}(3,q)$ meets an ovoid in either a single point or in the $q+1$ points of an oval. It is worth noting that in the case of an elliptic quadric, this oval is a conic while in the case of a Tits ovoid it is a translation oval with associated automorphism $\sigma$.

The classification of ovoids in $\mathrm{PG}(3,q)$ is of great interest, particularly in view of the number of related structures, such as Möbius planes, symplectic polarities, linear complexes, generalized quadrangles, unitals, maximal arcs, translation planes, and ovals.

If $q$ is odd, then every ovoid is an elliptic quadric, but the classification problem for $q$ even has been resolved only (as of 1998) for $q \le 32$, with the case $q=32$ involving some computer work. These classifications rely on the classification of ovals in the projective planes over fields of order up to $32$. On the other hand, there are several characterization theorems known, normally in terms of an assumption on the nature of the plane sections. One of the strongest results in this direction states that an ovoid with at least one conic among its plane sections must be an elliptic quadric. Similarly, it is known that an ovoid which admits a pencil of $q$ translation ovals is either an elliptic quadric or a Tits ovoid.

Ovoids in generalized polygons.

An ovoid $\mathcal{O}$ in a generalized $n$-gon $\Gamma$ (cf. also Polygon) is a set of mutually opposite points (hence $n=2m$ is even) such that every element $v$ of $\Gamma$ is at distance at most $m$ from at least one element of $\mathcal{O}$. One connection between ovoids in $\mathrm{PG}(3,q)$ and ovoids in generalized $n$-gons is that just as polarities of projective spaces sometimes give rise to ovoids, polarities of generalized $n$-gons produce ovoids. Further, every ovoid of the classical symplectic generalized quadrangle ($4$-gon; cf. also Quadrangle) usually denoted by $W(q)$, $q$ even, is an ovoid of $\mathrm{PG}(3,q)$ and conversely. It is worth noting in this context that a Tits ovoid in $\mathrm{PG}(3,q)$ is the set of all absolute points of a polarity of $W(q)$, $q=2^h$ and $h \ge 3$ odd. There are many ovoids known (1998) for most classes of generalized quadrangles, with useful characterization theorems. For the details and a survey of existence and characterization results for ovoids in generalized $n$-gons, see [a6] and especially [a7].

Ovoids in finite classical polar spaces.

An ovoid $\mathcal{O}$ in a finite classical polar space $\mathcal{S}$ of rank $r\geq 2$ is a set of points of $\mathcal{S}$ which has exactly one point in common with every generator of $\mathcal{S}$. Again, there are connections between ovoids in polar spaces and ovoids in generalized $n$-gons. For example, if $H(q)$ denotes the classical generalized hexagon ($6$-gon) of order $q$ embedded on the non-singular quadric $Q(6,q)$, then a set of points $\mathcal{O}$ is an ovoid of $H(q)$ if and only if it is an ovoid of the classical polar space $Q(6,Q)$. The existence problem for ovoids has been settled for most finite classical polar spaces, see [a3] or [a5] for a survey which includes a list of the open cases (as of 1998).

References

[a1]	M.R. Brown, "Ovoids of $PG(3,q)$, $q$ even, with a conic section" J. London Math. Soc. (to appear)
[a2]	J.W.P. Hirschfeld, "Finite projective spaces of three dimensions" , Oxford Univ. Press (1985)
[a3]	J.W.P. Hirschfeld, J.A. Thas, "General Galois geometries" , Oxford Univ. Press (1991)
[a4]	C.M. O'Keefe, "Ovoids in $PG(3,q)$ : a survey" Discrete Math. , 151 (1996) pp. 175–188
[a5]	J.A. Thas, "Projective geometry over a finite field" F. Buekenhout (ed.) , Handbook of Incidence Geometry, Buildings and Foundations , Elsevier (1995) pp. Chap. 7; 295–348
[a6]	J.A. Thas, "Generalized Polygons" F. Buekenhout (ed.) , Handbook of Incidence Geometry, Buildings and Foundations , Elsevier (1995) pp. Chap. 9; 295–348
[a7]	H. Van Maldeghem, "Generalized polygons" , Birkhäuser (1998)