Partial ovoids and partial spreads in hermitian polar spaces

Abstract

We present improved lower bounds on the sizes of small maximal partial ovoids in the classical hermitian polar spaces, and improved upper bounds on the sizes of large maximal partial spreads in the classical hermitian polar spaces. Of particular importance is the presented upper bound on the size of a maximal partial spread of H(3, q 2 ). For q = 2, 3, the presented upper bound is sharp. For q = 3, our results confirm via theoretical arguments properties, deduced by computer searches performed by Ebert and Hirschfeld, for the largest partial spreads of H(3, 9). An overview of the status regarding these results is given in two summarizing tables. The similar results for the classical symplectic and orthogonal polar spaces are presented in De Beule et al. [8].

Characterization results on small blocking sets of the polar spaces Q +(2n + 1, 2) and Q +(2n + 1, 3)

“…This ovoid of Q + (7, 3) is an ovoid of a parabolic quadric Q(6, 3) contained in Q + (7,3). Regarding this ovoid of Q(6, 3), we will use the following properties, found by computer [10] .…”

“…By the previous lemma, the 2-secants on P form an ovoid in the hyperbolic quadric Q + (7, q) seen in the quotient geometry of P . As mentioned in the introduction, Q + (7, 3) has a unique ovoid, which is in fact an ovoid lying in a parabolic quadric Q(6, 3) contained in Q + (7,3).…”

“…The first type of ovoid is the 3-dimensional elliptic quadric, corresponding under the Klein correspondence to a regular spread and hence to the Desarguesian projective plane P G(2, 9). The second type of ovoid is equal to a set (Q − (3, 3)\C)∪C ⊥ , where Q − (3, 3) is a 3-dimensional elliptic quadric contained in Q + (5, 3) and where C is a conic contained in Q − (3,3). Here ⊥ is the polarity related to Q + (5, 3).…”

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

“…Paper [8] is also an excellent source for bounds on the size of a maximal partial ovoid of H (n, q 2 ) for n ≥ 3.…”

Small maximal partial spreads in classical finite polar spaces

“…This research on the size of smallest maximal partial spreads in classical finite polar spaces is part of a detailed study on small and large maximal partial ovoids and spreads in classical finite polar spaces, performed in [2,3]. …”