Partitioning Quadrics, Symmetric Group Divisible Designs and Caps

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 inAG(4, 3) and its structure. Semi-biplanes make their appearance as part of our construction in the three dimensional case.