On a Family of Planes of a Polar Space

A Y -family S is a set of q + 3 planes ofPG (5, q) satisfying some conditions on their intersections. If every member of S is isotropic for a form f on PG(5, q), the affine expansion of S is covered by a geometry insidePG (7, q) (Proposition ?? ). Such families are classified if f is unitary (Proposition ?? ). Existence of such families with f symplectic is equivalent to that of two o -polynomials with certain properties (Lemma ?? ). A nonexistence result follows from this criterion (Proposition ?? ).