Partitioning quadrics, symmetric group divisible designs and caps (Q676705)

Using oval cones in PG\((3,q)\) and hypercones (with ovoid base) in PG\((4,q)\) the authors construct certain block designs. In the 4-dimensional case, let \(\Omega\) be an ovoid in a hyperplane \(W\), \(V\) a point of the space not in \(W\) and \(C\) the hypercone with vertex \(V\) and base \(\Omega\). The elements of the design are the points of \(S = C \setminus \{ \Omega, V \}\), while the blocks are the sets \(H \cap S\), where \(H\) is any hyperplane other than \(W\) containing a tangent plane to \(\Omega\). The design is a symmetric group divisible design on \((q-1)(q^2 + 1)\) elements with block size \(q^2\). Each group (corresponding to the points of \(S\) on a generator of \(C\)) has \(q-1\) elements and every pair of elements from different groups are on \(q + 1\) blocks. If \(\Omega\) is an elliptic quadric this design is self-dual. In the 3-dimensional case, the analogous construction with \(q\) odd gives a pair of disjoint isomorphic semi-biplanes on \((q^2 -1)/2\) elements with block size \(q\), while a slight variation of the construction for even \(q\) (\(S = C\) and \(W\) is included as an \(H\)) results in a Desarguesian projective plane of order \(q\). In the 4-dimensional case with \(q = 3\), the points of \(S\) form a maximal cap, a set of points in the space with no three collinear, of size 20 in AG\((4,3)\). The authors prove that any cap of AG\((4,3)\) has at most 20 points with equality if and only if the cap is a set \(S\) coming from their construction. The title of the article is a slight misnomer as there are no partitions of quadrics to be found in this paper.