On Ovoids of Parabolic Quadrics

Abstract

It is known that every ovoid of the parabolic quadric Q(4, q), q = p h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p = 2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points.We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q > 3.We conclude with a 1 mod p result for ovoids of Q(6, q), q = p h , p prime.

Blocking sets in PG(2, qn) from cones of PG(2n, q)

“…Very deep results about ovoids of Q(2n, q) have been recently obtained by S. Ball in [2] and by S. Ball, P. Govaerts and L. Storme in [3]. In particular, these authors prove that an ovoid O of Q(2n, q), n = 2, 3, meets every elliptic quadric Q − (2n − 1, q) on Q(2n, q) in 1 mod p points, p the characteristic of G F(q) (see [2] for n = 2, [3] for n = 3). So, since every hyperplane of PG(2n, q) intersecting Q(2n, q) not in a Q − (2n − 1, q) has some points on O, the following useful result on blocking set can be stated.…”

“…Moreover S n is not contained in K . Actually, under this assumption, (3) implies that the n−dimensional subspace S n , ∩ is contained inB; a contradiction by (ii). In conclusion, the cone K blocks any ndimensional subspace S n of defining a line in not through Y and no such S n is contained in it.…”

Blocking All Generators of Q+(2n + 1,3), n ≥ 4

“…Presently, ovoids of Q(6, q) are only known when q ≡ 0 mod 3. Furthermore, all ovoids of Q(4, p), p prime, are elliptic quadrics Q − (3, p) [1], which is a sufficient condition for the non-existence of ovoids of Q(6, p), p > 3 prime [12]. Finally we mention that Q(6, 3) has, up to collineations, a unique ovoid [9].…”

“…Using the classification of ovoids of Q(4, p), p prime [1], similar results for small minimal blocking sets of Q(2n, p), p > 3 prime, n ≥ 3, were obtained in [4], while results on small minimal blocking sets of Q(2n, 3) were obtained in [6,7]. General results on small minimal blocking sets of Q(6, q), q 32, q even, were obtained in [5].…”

Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings

“…As remarked above such quads are isomorphic to Q(4, q). Now, all ovoids of Q(4, q) are classical if q is prime ( [1]), q = 4 ([2], [24]) or q = 16 ([22], [23]). Non-classical ovoids of Q(4, q) are known to exist for every q = p h where p is an odd prime and h ≥ 2 ( [21], [27], [32]) and for every q = 2 2n+1 where n ≥ 1 ( [33] …”