Uniform Hyperplanes of Finite Dual Polar Spaces of Rank 3

Abstract

Let Δ be a finite thick dual polar space of rank 3. We say that a hyperplane H of Δ is locally singular (respectively, quadrangular or ovoidal) if H∩Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Δ. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally singular, then either H is the set of points at non-maximal distance from a given point of Δ or Δ is the dual of Q(6, q) and H arises from the generalized hexagon H(q). In this paper we prove that only two examples exist for the locally quadrangular case, arising in Q(6, 2) and H(5, 4), respectively. We fail to rule out the locally ovoidal case, but we obtain some partial results on it, which imply that, in this case, the geometry Δ\\H induced by Δ on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with Δ\\H flag-transitive are classified.