Mixed partitions of projective geometries
Starting from a linear collineation of PG(2n-1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n-1,q) consisting of two (n-1)-subspaces and caps, all having size (q(n)-1)/(q-1) or (q(n)-1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety S-2,S-2 of PG(8,q) into caps of size q(2)+q+1 which are Veronese surfaces.
mixed partition singer cycle cap veronese surface kronecker product