The structure of the spin-embeddings of dual polar spaces and related geometries

In [B. De Bruyn, A. Pasini, Minimal scattered sets and polarized embeddings of dual polar spaces, European J. Combin. 28 (2007) 1890–1909], it was shown that every full polarized embedding of a dual polar space of rank n≥2n≥2 has vector dimension at least 2n2n. In the present paper, we will give alternative proofs of that result which hold for more general classes of dense near polygons. These alternative proofs allow us to characterize full polarized embeddings of minimal vector dimension 2n2n. Using this characterization result, we can prove a decomposition theorem for the embedding space. We will use this decomposition theorem to get information on the structure of the spin-embedding of the dual polar space DQ(2n,K).