On the hyperplanes of the half-spin geometries and the dual polar spaces

We describe relationships between locally singular hyperplanes of the dual polar space DQ ( 2 n , K ) , n ⩾ 2 , and hyperplanes of the half-spin geometries HS ( 2 n − 1 , K ) and HS ( 2 n + 1 , K ) for the respective hyperbolic quadrics Q + ( 2 n − 1 , K ) and Q + ( 2 n + 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 ( 2 n + 2 , K ) from locally singular hyperplanes of DQ ( 2 n , K ) . Along our way, we also obtain a new proof for the fact that all hyperplanes of the half-spin geometries arise from embeddings.