Bulletin of the Belgian Mathematical Society - Simon Stevin

Dominant lax embeddings of polar spaces

Antonio Pasini

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Abstract

It is known that every lax projective embedding $e:\Gamma\rightarrow PG(V)$ of a point-line geometry $\Gamma$ admits a {\em hull}, namely a projective embedding $\tilde{e}:\Gamma\rightarrow PG(\tilde{V})$ uniquely determined up to isomorphisms by the following property: $V$ and $\tilde{V}$ are defined over the same skewfield, say $K$, there is morphism of embeddings $\tilde{f}:\tilde{e}\rightarrow e$ and, for every embedding $e':\Gamma\rightarrow PG(V')$ with $V'$ defined over $K$, if there is a morphism $g:e'\rightarrow e$ then a morphism $f:\tilde{e}\rightarrow e'$ also exists such that $\tilde{f} = gf$. If $e = \tilde{e}$ then we say that $e$ is {\em dominant}. Clearly, hulls are dominant. Let now $\Gamma$ be a non-degenerate polar space of rank $n\geq 3$. We shall prove the following: A lax embedding $e:\Gamma\rightarrow PG(V)$ is dominant if and only if, for every geometric hyperplane $H$ of $\Gamma$, $e(H)$ spans a hyperplane of $PG(V)$. We shall also give some applications of the above result.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 5 (2006), 871-882.

Dates
First available in Project Euclid: 10 January 2006

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1136902622

Digital Object Identifier
doi:10.36045/bbms/1136902622

Mathematical Reviews number (MathSciNet)
MR2241350

Zentralblatt MATH identifier
1133.51002

Subjects
Primary: 51A45: Incidence structures imbeddable into projective geometries 51A50: Polar geometry, symplectic spaces, orthogonal spaces

Keywords
Polar spaces projective spaces weak embeddings

Citation

Pasini, Antonio. Dominant lax embeddings of polar spaces. Bull. Belg. Math. Soc. Simon Stevin 12 (2006), no. 5, 871--882. doi:10.36045/bbms/1136902622. https://projecteuclid.org/euclid.bbms/1136902622


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