A characterization of the SDPS-hyperplanes of dual polar spaces

In [B. De Bruyn, P. Vandecasteele, Valuations and hyperplanes of dual polar spaces, J. Combin. Theory Ser. A 112 (2005) 194–211], we introduced the class of the SDPS-valuations of dual polar spaces. We showed that these valuations and all their extensions give rise to hyperplanes of dual polar spaces. We call these hyperplanes SDPS-hyperplanes. In the present paper, we show that a hyperplane HH of a thick dual polar space is an SDPS-hyperplane if and only if every hex AA not contained in HH intersects HH in either a singular hyperplane or the extension of an ovoid.