New results on covers and partial spreads of polar spaces

Abstract

We investigate blocking sets of projective spaces that are contained in cones over quadrics of rank two. As an application we obtain new results on partial ovoids, partial spreads, and blocking sets of polar spaces. One of the results is that a partial ovoid of $H\left(3,q^2\right)$ with more than $q^3-q+1$ points is contained in an ovoid. We also give a new proof of the result that a partial spread of $Q\left(4,q\right)$ with more than $q^2-q+1$ lines is contained in a spread; this is the first common proof for even and odd $q$. Finally, we improve the lower bound on the size of a smallest blocking set of the symplectic polar space $W\left(3,q\right)$, $q$ odd.