Commuting polarities and maximal partial ovoids of H(4,q²)

Abstract

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 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 xed point of O and X varies in O\{P}. q

On large partial ovoids of symplectic and Hermitian polar spaces

“…A straightforward check shows that a non-degenerate plane section of H(2n, q 2 ) is an example of maximal partial ovoid of H(2n, q 2 ) of size q 3 + 1. Other examples of maximal partial ovoids of H(4, q 2 ) of size 2q 3 + q 2 + 1 have been constructed in [4] and of size q 3 + 1 in [5,11].…”