On partial ovoids of Hermitian surfaces

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

“…Recently, attention has turned to the problem of the cardinality of the smallest maximal partial ovoids and the smallest maximal partial spreads in finite classical generalized quadrangles and polar spaces. We mention in particular [1,2,9,12,15,25,27]. In particular, [24,27] addressed these problems for the classical generalized quadrangles.…”

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

“…We now present a general construction of such smaller maximal partial spreads in H(3, q 2 ), q = 2 2h+1 , h ≥ 1. We will make the construction on the dual generalized quadrangle Q − (5, q) to H(3, q 2 ), by constructing a maximal partial ovoid of size 3q 2…”

“…In Table 1, the results for Q(4, q) and W(q) are from [9], for Q − (5, q) and H(3, q 2 ) from [2,12,18], and for H(4, q 2 ) from Theorems 2.1 and 2.2.…”

“…Partial ovoids and partial spreads of generalized quadrangles have received a lot of attention. Research was concentrated on the existence problem of ovoids and spreads [15,25,26], on the extendability of partial ovoids and partial spreads with small deficiency δ to ovoids and spreads [13,16,25], and on the size of the smallest maximal partial ovoids and spreads in generalized quadrangles [1,2,9,12].…”

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

“…Hirschfeld and Korchmáros [10], proved that a maximal partial ovoid of H (n, q 2 ) has at least q 2 +1 points. They showed that this lower bound is sharp for n = 3 and even q. Aguglia, Ebert and Luyckx [1] dealt with small partial ovoids in H (3, q 2 ). They also give a proof that the previous lower bound holds for n = 3 and showed that this bound is reached if and only if q is even.…”