Tangency sets in PG(3,q)

A tangency set of PG(d, q) is a set Q of points with the property that every point P of Q lies on a hyperplane that meets Q only in P. It is known that a tangency set of PG(3, q) has at most q(2) + 1 points with equality only if it is an ovoid. We show that a tangency set of PG(3, q) with q(2) - 1, q >= 19, or q(2) points is contained in an ovoid. This implies the non-existence of minimal blocking sets of size q(2) - 1, q >= 19, and of q(2) with respect to planes in PG(3, q), and implies the extendability of partial 1-systems of size q(2)-1, q >= 19, or q(2) to 1-systems on the hyperbolic quadric Q(+)(5, q)