Configurations of planes in PG(5,2) Original Research Article

We discuss various configurations of planes in PG(5,2). The supporting point-sets of most of the configurations, or else the complements of these point-sets, have cubic equations and are of elliptic or hyperbolic type. In our discussion of conclaves of eight planes in PG(5,2) we introduce a second addition upon the space V(6,2)=X⊕X∗, where X∗ is the dual of X. We show that the 8 planes of a conclave lying on a hyperbolic quadric for one addition mutate into the 8 Conwell heptads lying off a hyperbolic quadric for the other addition. In our discussion of double-seven of planes in PG(5,2) we also consider their relation with the double-seven of planes in PG(8,2) which is supported by the Segre variety S2,2 over GF(2).