Partitioning Quadrics, Symmetric Group Divisible Designs and Caps *

Dedicated to Hanfried Lenz on the occasion of his 80th birthday. Abstract. Using partitionings of quadrics we give a geometric construction of certain symmetric group divisible designs. It is shown that some of them at least are self-dual. The designs that we construct here relate to interesting work — some of it very recent — by D. Jungnickel and by E. Moorhouse. In this paper we also give a short proof of an old result of G. Pellegrino concerning the maximum size of a cap in AG(4,3) and its structure. Semi-biplanes make their appearance as part of our construction in the three dimensional case. Keywords: Quadrics, Symmetric group divisible designs, Caps, Semi-biplanes, Ovoids For n ≥ 2, we denote by PG(n, q) the finite projective space of dimension n over F: = GF(q), the field of order q. Similarly AG(n, q) denotes the affine space of dimension n over F. A subset of PG(n, q) or AG(n, q) is a cap if no three of its points are collinear. For 1 ≤ k ≤ n, a subset K ⊆ PG(n, q) is a k-flat if K is isomorphic to PG(k, q). A line is a 1-flat, a plane is a 2-flat and a solid is a 3-flat. The complement in PG(n, q) of an (n−1)-flat is isomorphic to AG(n, q). We will denote by ℓ(x, y) the unique line containing both x and y. We begin by recalling the structure of maximal caps in low dimensions. Proofs of thes