On pairs of permutable Hermitian surfaces

Abstract

We investigate the intersection R of two permutable Hermitian surfaces of PG(3, q 2 ), q odd. We show that R is a determinantal variety. From the combinatorial point of view R comprises a complete (q 2 + 1)-span of the two corresponding Hermitian surfaces.

Partial ovoids and partial spreads in hermitian polar spaces

“…Every free generator π contains an (n − 2)-dimensional space π 1 lying in a pencil of q 2 + 1 hyperplanes P ⊥ ∩ π, P ∈ O, which cover all the points of π. 2 ) of size at most q 2 + q. The proof of the preceding theorem shows that every free generator π to O contains a point P lying on q 2 + 1 lines S ⊥ ∩ π, S ∈ O , covering all the points of π.…”

“…Recently, attention was also paid 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 [1,2,6,7,10,15,16]. In particular, [13,16] addressed these problems for the classical generalized quadrangles.…”

“…The finite classical generalized quadrangles are the non-singular parabolic quadric Q (4, q), the non-singular elliptic quadric Q − (5, q), the non-singular hyperbolic quadric Q + (3, q), the non-singular hermitian varieties H(3, q 2 ) and H(4, q 2 ), and the symplectic generalized quadrangle W (3, q) in PG (3, q). The generalized quadrangles Q (4, q) and W (3, q) are dual to each other.…”

On Some Maximal Subgroups of Unitary Groups

“…It turns out that the stabilizer of an elliptic congruence is actually the stabilizer of a complete (q 2 + 1)-span of H(3, q 2 ) as shown in [1], [2]. Notice that P SU 4 (q 2 ) has one or two classes of subgroups isomorphic to P Sp 4 (q) according as q is even or odd.…”

Shult sets and translation ovoids of the Hermitian surface

“…In particular, there is no Baer elliptic quadric Q − (3, q) contained in O 3 , as any such quadric embedded in H must be a complete partial ovoid for even q (see [1], for instance). However, we can find a Baer elliptic quadric that will determine O 3 in an analogous fashion to the previous two cases.…”