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 isot ropic subspaces and caps of equal size are also obtained. We finally consid er the possibility of partitioning the Segre variety S-2,S-2 of PG(8,q) int o caps of size q(2)+q+1 which are Veronese surfaces.