Abstract
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 {\cal S}_{2,2} of \PG(8,q) into caps of size q^2+q+1 which are Veronese surfaces.
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