## Innovations in Incidence Geometry

### Embedded polar spaces revisited

Antonio Pasini

#### Abstract

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 $K¯:=K∕Kσ,ε$, where $Kσ,ε:={t−tσε}t∈K$. Our generalization amounts to replace $K¯$ with a quotient $K¯∕R¯$ for a subgroup $R¯$ of $K¯$ such that $λσR¯λ=R¯$ for any $λ∈K$. 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 $q:V→K¯∕R¯$ is a non-trivial generalized pseudo-quadratic form and $f:V×V→K$ 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

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

Dates
Accepted: 8 August 2015
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323009

Digital Object Identifier
doi:10.2140/iig.2017.15.31

Mathematical Reviews number (MathSciNet)
MR3713356

Zentralblatt MATH identifier
1384.51003

Keywords
polar spaces embeddings

#### Citation

Pasini, Antonio. Embedded polar spaces revisited. Innov. Incidence Geom. 15 (2017), no. 1, 31--72. doi:10.2140/iig.2017.15.31. https://projecteuclid.org/euclid.iig/1551323009

#### References

• A. Bak, On modules with quadratic forms, Algebraic K-Theory and its Geometric Applications (Conf., Hull, 1969), pp. 55–66, Springer, Berlin.
• F. Buekenhout and A. M. Cohen, Diagram Geometries, Springer, Berlin, 2013.
• F. Buekenhout and C. Lefèvre, Generalized quadrangles in projective spaces, Arch. Math. 25 (1974), 540–552.
• F. Buekenhout and E. E. Shult, On the foundations of polar geometry, Geom. Dedicata 3 (1974), 155–170.
• B. De Bruyn and A. Pasini, On symplectic polar spaces over non-perfect fields of characteristic $2$, Linear Multilinear Algebra 47 (2009), 567–575.
• K. J. Dienst, Verallgemainerte Vierecke in projectiven Räumen, Arch. Math. 35 (1980), 177–186.
• A. Hahn and O. T. O'Meara, The classical groups and K-theory, with a foreword by J. Dieudonné, Grundlehren Math. Wiss. 291, Springer-Verlag, Berlin, 1989.
• A. Kasikova and E. E. Shult, Absolute embeddings of point-line geometries, J. Algebra 238 (2001), 100–117.
• M. A. Ronan, Embeddings and hyperplanes of discrete geometries, European J. Combin. 8 (1987), 179–185.
• J. Tits, Buildings of Shperical Type and Finite $BN$-pairs, Springer Lecture Notes 386 (1974), Springer, Berlin.
• J. Tits and R. M. Weiss, Moufang Polygons, Springer, Berlin, 2002.
• F. D. Veldkamp, Polar Geometry I–V, Indag. Math. 21 (1959), 512–551 and 22 (1959), 207–212.