Commuting polarities and maximal partial ovoids of H(4,q 2 )

In PG(4, q 2 ), q odd, let Q (4, q 2 ) be a non‐singular quadric commuting with a non‐singular Hermitian variety H (4, q 2 ). Then these varieties intersect in the set of points covered by the extended generators of a non‐singular quadric Q 0 in a Baer subgeometry Σ 0 of PG(4, q 2 ). It is proved that any maximal partial ovoid of H(4, q 2 ) intersecting Q 0 in an ovoid has size at least 2( q 2 +1). Further, given an ovoid O of Q 0 , we construct maximal partial ovoids of H(4, q 2 ) of size q 3 +1 whose set of points lies on the hyperbolic lines 〈 P,X 〉 where P is a fixed point of O and X varies in O \{ P }. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 307–313, 2009