Projective Embeddings of Polar Spaces
This is the second chapter devoted to polar spaces. The motivation for studying these spaces is their occurrence as partial subspaces—in fact, as absolutes of polarities—in projective spaces. This chapter shows that the following converse holds: a polar s
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Projective Embeddings of Polar Spaces
This is the second chapter devoted to polar spaces. The motivation for studying these spaces is their occurrence as partial subspaces—in fact, as absolutes of polarities— in projective spaces, whose lines are projective lines. This chapter shows that the following converse holds: a polar space with a thick polar geometry is isomorphic to a partial subspace of a projective space if it is nondegenerate of rank at least three and all of its singular subspaces are Desarguesian; this is the content of Corollary 8.4.26. The conditions mentioned in this statement are, in a sense, best possible. For, if a space is a partial subspace of a projective space, its linear subspaces must be projective as well. Thus, if Z is a degenerate polar space whose radical is not a generalized projective space, it will not be embeddable. There are numerous examples of thick generalized quadrangles that cannot be embedded in a projective space; for instance there are finite examples whose point orders are not prime powers; see Example 8.3.17. Finally, there are examples of nondegenerate polar spaces of rank three whose planes are Moufang but not Desarguesian. These are most easily defined in terms of algebraic groups and will not be discussed here. The construction of embeddings of polar spaces of rank at least three comes in three main steps. The first step, taken in Sect. 8.1, is the study of geometric hyperplanes of polar spaces. For every point x of a polar space, the set x ⊥ of all points collinear with x is a geometric hyperplane. The notion of a geometric hyperplane was introduced in Definition 5.2.7 (see also Exercise 5.7.12). The second step is the embedding of the polar space in a space whose points are geometric hyperplanes by means of the map x → x ⊥ , which takes place in Sect. 8.2. The final and hardest step, namely the proof that the space of geometric hyperplanes is actually a projective space is carried out in Sects. 8.3 and 8.4. The proof of the embedding result for rank three, given in Sect. 8.4, is long. If the rank is four or more, there is a more elementary proof, given in Sect. 8.3. By Corollary 6.3.2, the singular subspaces of these polar spaces are Desarguesian. The chapter ends with Sect. 8.5 in which automorphisms of polar spaces of rank at least three are found. 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_8, © Springer-Verlag Berlin Heidelberg 2013
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Projective Embeddings of Polar Spaces
8.1 Geometric Hyperplanes and Ample Connectedness Recall from Definition 5.2.7 that a geometric hyperplane of a line space Z is a proper subspace of Z with the property that every line of Z meets it in at least a point. The study of geometric hyperplanes will be of use both for the embedding of polar spaces in linear spaces, the main goal of this chapter, and for the construction of automorphisms of polar spaces, to be discussed at the end of
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