On the type of partial t- spreads in finite projective spaces

Abstract

A partial t- spread in a projective space P is a set of mutually skew t- dimensional subspaces of P . In this paper, we deal with the question, how many elements of a partial spread L can be contained in a given d- dimensional subspace of P . Our main results run as follows. If any d- dimensional subspace of P contains at least one element of L , then the dimension of P has the upper bound d−1+( d/ t). The same conclusion holds, if no d- dimensional subspace contains precisely one element of L . If any d- dimensional subspace has the same number m>0 of elements of L , then L is necessarily a total t- spread. Finally, the ‘type’ of the so-called geometric t- spreads is determined explicitely.