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 $q^2-1, q\geq 19$, or $q^2$ points is contained in an ovoid. This implies the non‐existence of minimal blocking sets of size $q^2-1$, $q\geq 19$, and of $q^2$ with respect to planes in PG (3,q), and implies the extendability of partial 1‐systems of size $q^2-1$, $q\geq 19$, or $q^2$ to 1‐systems on the hyperbolic quadric $Q^+(5,q)$. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 462–476, 2008