Uniform hyperplanes of finite dual polar spaces of rank 3

Let Delta be a finite thick dual polar space of rank 3. We say that a hyperplane H of Delta is locally singular (respectively, quadrangular or ovoidal) if H boolean AND Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Delta. 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 Delta or Delta is the dual of L(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 L(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 Delta H induced by Delta on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with Delta H flag-transitive are classified. (C) 2001 Academic Press.