On the hyperplanes of the half-spin geometries and the dual polar spaces DQ(2n, K)

摘要

We describe relationships between locally singular hyperplanes of the dual polar space DQ (2n K), n >= 2, and hyperplanes of the half-spin geometries HS(2n - 1, K) and HS(2n + 1, K) for the respective hyperbolic quadrics Q(+)(2n - 1, K) and Q+(2n + 1, K). We use these relationships to classify all hyperplanes of HS(9, K) and to provide a inethod for constructing locally singular hyperplanes of DQ(2n + 2, K) from locally singular hyperplanes of DQ(2n, K). Along our way, we also obtain a new proof for the fact that all hyperplanes of the half-spin geometries arise from embeddings. (c) 2006 Elsevier Inc. All rights reserved.