Abstract
We obtain a classification of the nonclassical hyperplanes of all finite thick dual polar spaces of rank at least 3 under the assumption that there are no ovoidal and semi-singular hex intersections. In view of the absence of known examples of ovoids and semi-singular hyperplanes in finite thick dual polar spaces of rank 3, the condition on the nonexistence of these hex intersections can be regarded as not very restrictive. As a corollary, we also obtain a classification of the nonclassical hyperplanes of $DW(2n-1,q)$, $q$ even. In particular, we obtain a complete classification of all nonclassical hyperplanes of $DW(2n-1,q)$ if $q \in \{ 8,32 \}$.
Highlights
Suppose Π is a finite thick polar space of rank n ≥ 2 which is fully embeddable in a projective space Σ
If we look at the projective space Res(x), we see that Q ∩ M ⊆ H is either Q or a line, in contradiction with the fact that Q ∩ H is an ovoid of Q
If F is a hex through Q, by Lemma 3.6 F ∩ H is the extension of an ovoid of a quad of F and there exists a unique line through x contained in F ∩ H
Summary
Suppose Π is a finite thick polar space of rank n ≥ 2 which is fully embeddable in a projective space Σ. Proposition 2.4 Let ∆ be a thick dual polar space, F a convex subspace of even diameter of ∆, X an SDPS-set in F and HF the SDPS-hyperplane of F associated with X. If H is a hexagonal hyperplane of DQ−(7, q), there are no points x for which x⊥ ⊆ H and every quad is singular or ovoidal with respect to H.
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