Embedding Polar Spaces in Absolutes
In Theorems 8.3.16 and 8.4.25, we saw that the most common nondegenerate polar spaces (cf. Definition 7.4.1) of rank at least three are embeddable in a projective space. So, in the continued study of a nondegenerate polar space Z of rank at least three, i
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Embedding Polar Spaces in Absolutes
In Theorems 8.3.16 and 8.4.25, we saw that the most common nondegenerate polar spaces (cf. Definition 7.4.1) of rank at least three are embeddable in a projective space. So, in the continued study of a nondegenerate polar space Z of rank at least three, it is a mild restriction to assume that Z is embedded in a projective space P. Still, the methods used in this chapter only require that the rank of Z be at least two. Our main goal is to show that such a space Z is a subspace of the polar space Pπ related to a quasi-polarity (cf. Definition 7.1.9) π of P. This goal is achieved in Theorem 9.5.8. Before we reach it, we will construct a subspace π(p) of P for each point p of P that will play the role of the image of p under a quasi-polarity of P whose absolute (cf. Definition 7.1.9) contains Z. It is easy to see that π(p) is a hyperplane or coincides with P (see Lemma 9.2.3). It is harder to show that π is a quasi-polarity. In Sect. 9.1, we discuss the notion of a projective embedding and some properties of spaces embedded in a projective space. The construction of π takes place in Sect. 9.2. We define a set, called the defect of Z in P, that will turn out to be the kernel of π (cf. Definition 7.1.9) once this map has been established to be a quasipolarity. In Sect. 9.3, we prove that the map π is a quasi-polarity if the defect of Z in P is empty and P does not belong to the image of π . We use this fact to provide a relatively easy proof that, if Z is nondegenerate of rank at least two and spans P and if dim(P) ≥ 4, then π is a quasi-polarity and Z is a subspace of the absolute Pπ with respect to π . In Theorem 9.3.3 of this section, we prove that the point set of Z is invariant under special perspectivities of P. These perspectivities will play a role in the case where dim(P) = 3 (Sect. 9.5) and in Chap. 10, where we classify proper subspaces of absolutes. In Sect. 9.4, we deploy some algebraic machinery underlying polarities. This is used in Sect. 9.5 to prove that π is a quasi-polarity and Z embeds in Pπ even if dim(P) = 3. Throughout this chapter, P will be a (possibly non-Desarguesian) projective space of dimension n ≥ 1, possibly infinite. F. Buekenhout, A.M. Cohen, Diagram Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 57, DOI 10.1007/978-3-642-34453-4_9, © Springer-Verlag Berlin Heidelberg 2013
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Embedding Polar Spaces in Absolutes
9.1 Embedded Spaces We consider (line) spaces (cf. Definition 2.5.8) occurring in the projective space P. Later, the choice of spaces will be narrowed down to polar spaces. Definition 9.1.1 Let Z = (P , L) be a line space. It is said to be embedded in P if (1) P is a set of points of P; (2) each member of L (i.e., line of Z) is a line of P. This means that the identity map P → P determines an embedding of the line space Z in P (cf. Definition 2.5.8). We will say that Z is ruled in P if it is embedded in P and (3) P is the union of all members of L; (4) P generates P.
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