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Open problems in “Finite Geometry”: Ovoids of projective spaces

March 11, 2010
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An ovoid of a finite 3-dimensional projective space PG(3, q) is a set of q^2 + 1 points such that no three are on a common line. Besides the case q=2, the bound q^2+1 is the maximum size of a set of points with no three collinear. Ovoids are central to finite geometry, and two examples of their significance are the constructions of inversive planes and generalised quadrangles from ovoids. Peter Dembowski (1963) proved that every inversive plane of even order arises from an ovoid of PG(3,q), q even, and Jacques Tits (see Dembowksi’s book) gave a construction of a generalised quadrangle of order (q,q^2) from an ovoid. Here is a simple example of an ovoid:

Example (Elliptic Quadric):

Consider an element n of GF(q) such that x^2+x+n is irreducible over GF(q). If we take the points of PG(3,q) of the form

(s,t, s^2+st+nt^2,1) : s,t\in GF(q)

together with the point (0,0,1,0), we obtain an ovoid known as an elliptic quadric. All non-singular quadrics (zeros of quadratics) of elliptic type are equivalent to the one we have just described in the sense that PGL(4,q) acts transitively on elliptic quadrics.

Barlotti and Panella independently proved in 1955, that every ovoid of PG(3,q), with q odd, is an elliptic quadric. Segre (1959) had discovered an ovoid of PG(3,8) which was not an elliptic quadric, and Tits showed that there exists such a non-classical ovoid of PG(3,q) for q an odd power of 2 (Fellagara later showed in 1962 that Tits’ family of examples included the Segre example). The stabilisers of these ovoids are the famous Suzuki groups, and so they are sometimes called Suzuki-Tits ovoids.

Example (Suzuki-Tits ovoids):

Let \sigma be the automorphism of GF(2^{2e+1}) defined by x^\sigma=x^{2^{e+1}}. If we take the points of PG(3,2^{2e+1}) of the form

(s,t, s^{\sigma+2}+st+t^\sigma,1) : s,t\in GF(2^{2e+1})

together with the point (0,0,1,0), we obtain an ovoid known as a Suzuki-Tits ovoid.

O’Keefe, Penttila and Royle (1994) classified the ovoids for PG(3,32), and until the time of writing, all the known examples are the two we have just described; which brings us to the conjecture…

Every ovoid of PG(3,q) is an elliptic quadric or a Suzuki-Tits ovoid.

The best result so far in this area is a theorem of Matthew Brown (2000):

Brown (2000): If any hyperplane section of an ovoid is a conic, then the ovoid is an elliptic quadric.

There is of course a lot lot more on this topic, and a good survey (up to the year 2000) can be found here.

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