Characterising ovoidal cones by their intersection numbers

In this paper, we characterise ovoidal cones by their intersection numbers. We first show that a set of points of PG(4,q) which intersects planes in $1$, $q+1$ or $2q+1$ points is either an ovoidal cone or a parabolic quadric, unless $q=3$, in which case also a sporadic example with automorphism group M₁₁ exists. We then show that a set of points of PG(4,q) which blocks all planes and intersects solids in $q+1$, q²+1 or q²+q+1 points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted.