On hyperbolic sets of maxes in dual polar spaces

Suppose ∆ is a fully embeddable thick dual polar space of rank n ≥ 3. It is known that a hyperplane H of ∆ is classical if all its nontrivial intersections with quads are classical. In order to conclude that a hyperplane H is classical, it is perhaps not necessary to require in advance that all these intersections are classical. In fact, in this paper we show that for dual polar spaces admitting hyperbolic sets of maxes, the existence of certain classical quad-hyperplane intersections implies that other quad-hyperplane intersections need to be classical as well. We will also derive necessary and sufficient conditions for two disjoint maxes to be contained in a (necessarily unique) hyperbolic set of maxes. Dual polar spaces admitting hyperbolic sets of maxes include all members of a class of embeddable dual polar spaces related to quadratic alternative division algebras.