A Remark on Non-Uniform Hyperplanes of Finite Thick Dual Polar Spaces

Let Δ be a finite thick dual polar space, and let H be a hyperplane ofΔ . Calling the elements of Δ of type 2 quads, we call a quadα ≠ ⊂ H singular (respectively subquadrangular or ovoidal) ifH meets α in the perp of a point (respectively in a full subquadrangle or in an ovoid). A hyperplane is said to be uniform if all quads not contained in it are either singular or subquadrangular or ovoidal. We prove that each non-uniform hyperplane H of a finite thick dual polar space meets some quad in the perp of a point.