Innovations in Incidence Geometry

Embedded polar spaces revisited

Antonio Pasini

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Pseudo-quadratic forms have been introduced by Tits in his Buildings of spherical type and finite BN-pairs (1974), in view of the classification of polar spaces. A slightly different notion is proposed by Tits and Weiss. In this paper we propose a generalization. With its help we will be able to clarify a few points in the classification of embedded polar spaces. We recall that, according to Tits’ book, given a division ring K and an admissible pair (σ,ε) in it, the codomain of a (σ,ε)-quadratic form is the group K¯:=KKσ,ε, where Kσ,ε:={ttσε}tK. Our generalization amounts to replace K¯ with a quotient K¯R¯ for a subgroup R¯ of K¯ such that λσR¯λ=R¯ for any λK. We call generalized pseudo-quadratic forms (also generalized (σ,ε)-quadratic forms) the forms defined in this more general way, keeping the words pseudo-quadratic form and (σ,ε)-quadratic form for those defined as in Tits’ book. Generalized pseudo-quadratic forms behave just like pseudo-quadratic forms. In particular, every non-trivial generalized pseudo-quadratic form admits a unique sesquilinearization, characterized by the same property as the sesquilinearization of a pseudo-quadratic form. Moreover, if q:VK¯R¯ is a non-trivial generalized pseudo-quadratic form and f:V×VK is its sesquilinearization, the points and the lines of PG(V) where q vanishes form a subspace Sq of the polar space Sf associated to f. In this paper, after a discussion of quotients and covers of generalized pseudo-quadratic forms, we shall prove the following, which sharpens a celebretated theorem of Buekenhout and Lefèvre. Let e:S PG(V) be a projective embedding of a non-degenerate polar space S of rank at least 2; then e(S) is either the polar space Sq associated to a generalized pseudo-quadratic form q or the polar space Sf associated to an alternating form f. By exploiting this theorem we also obtain an elementary proof of the following well known fact: an embedding e as above is dominant if and only if either e(S)=Sq for a pseudo-quadratic form q or char(K)2 and e(S)=Sf for an alternating form f.

Article information

Innov. Incidence Geom., Volume 15, Number 1 (2017), 31-72.

Received: 24 November 2014
Accepted: 8 August 2015
First available in Project Euclid: 28 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51A45: Incidence structures imbeddable into projective geometries 51A50: Polar geometry, symplectic spaces, orthogonal spaces 51E12: Generalized quadrangles, generalized polygons 51E24: Buildings and the geometry of diagrams

polar spaces embeddings


Pasini, Antonio. Embedded polar spaces revisited. Innov. Incidence Geom. 15 (2017), no. 1, 31--72. doi:10.2140/iig.2017.15.31.

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