Geometric hyperplanes of the half-spin geometries arise from embeddings (Q1330777)

The paper investigates embeddings of half-spin geometries of type \(D_ n\) in projective spaces. Suppose that \(\Gamma = ({\mathcal P},{\mathcal L})\) is a half-spin geometry of type \(D_ n\) and that \(e : \Gamma \to {\mathbb{P}}(V)\) is an embedding in the projective space \({\mathbb{P}} (V)\), where \(\dim V \geq 2^{n - 1}\). It is shown that the dimension of \(V\) equals \(2^{n - 1}\) and that every geometric hyperplane of \(\Gamma\) arises from the embedding \(e\), i.e., every geometric hyperplane \(H\) can be obtained as \(H = e^{- 1} (e({\mathcal P}) \cap {\mathbb{H}})\), where \(\mathbb{H}\) is a hyperplane of \(\mathbb{P} (V)\). The embedding \(e\) is universal, and as a corollary, using the existence of Veldkamp lines, the author obtains that the Veldkamp space of any half-spin geometry \(D_ n\), \(n \geq 4\), is a projective space. The paper introduces all relevant notation and prerequisites in sufficient detail. The proof of the main theorem works by induction. For \(n = 4\) the result is readily deduced from results of \textit{K. J. Dienst} [Arch. Math. 35, 177-186 (1980; Zbl 0437.51005)] and \textit{F. Buekenhout} and \textit{C. Lefèvre} [Arch. Math. 25, 540-552 (1974; Zbl 0294.50022)]. For \(n > 4\) one takes two subspaces \(D_ 1\) and \(D_ 2\) which are themselves half-spin geometries of type \(D_{n - 1}\) and obtains \(\langle e(\Gamma) \rangle = \langle e(D_ 1) \rangle \oplus \langle e(D_ 2) \rangle\). This implies that \(V\) has the required dimension. Moreover, if \(H\) is a geometry hyperplane of \(\Gamma\), then \(H_ 1 := H \cap D_ i\) is a hyperplane of \(D_ i\) or \(H_ i = D_ i\), \((i = 1,2)\). The case \(H_ 1 = D_ 1\) and \(H_ 2 = D_ 2\) is impossible. If precisely one of the \(H_ i\) is a hyperplane, then one obtains rather easily that \(H\) arises from a hyperplane of \(V\). The case where \(H_ 1\) and \(H_ 2\) are hyperplanes is the difficult one. It requires a considerable amount of work. Proceeding step by step, exploiting basic results on half-spin geometries provided earlier, the author arrives at the desired result, namely that also in this case \(H\) arises from a hyperplane of \(V\).