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

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 method 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.