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# Dual embeddings of dense near polygons

Bart De Bruyn (UGent)
(2012) 103. p.33-54
Author
Organization
Abstract
Let e: S -> Sigma be a full polarized projective embedding of a dense near polygon S, i.e., for every point p of S, the set H(p) of points at non-maximal distance from p is mapped by e into a hyperplane Pi(p) of Sigma. We show that if every line of S is incident with precisely three points or if S satisfies a certain property (P(de)) then the map p bar right arrow Pi p defines a full polarized embedding e* (the so-called dual embedding of e) of S into a subspace of the dual Sigma* of Sigma. This generalizes a result of [6] where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P(de)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding.
Keywords
polarized embedding, GEOMETRIES, hyperplane, dual embedding, near polygon

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## Citation

Chicago
De Bruyn, Bart. 2012. “Dual Embeddings of Dense Near Polygons.” Ars Combinatoria 103: 33–54.
APA
De Bruyn, B. (2012). Dual embeddings of dense near polygons. ARS COMBINATORIA, 103, 33–54.
Vancouver
1.
De Bruyn B. Dual embeddings of dense near polygons. ARS COMBINATORIA. 2012;103:33–54.
MLA
De Bruyn, Bart. “Dual Embeddings of Dense Near Polygons.” ARS COMBINATORIA 103 (2012): 33–54. Print.
```@article{3101091,
abstract     = {Let e: S -{\textrangle} Sigma be a full polarized projective embedding of a dense near polygon S, i.e., for every point p of S, the set H(p) of points at non-maximal distance from p is mapped by e into a hyperplane Pi(p) of Sigma. We show that if every line of S is incident with precisely three points or if S satisfies a certain property (P(de)) then the map p bar right arrow Pi p defines a full polarized embedding e* (the so-called dual embedding of e) of S into a subspace of the dual Sigma* of Sigma. This generalizes a result of [6] where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P(de)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding.},
author       = {De Bruyn, Bart},
issn         = {0381-7032},
journal      = {ARS COMBINATORIA},
language     = {eng},
pages        = {33--54},
title        = {Dual embeddings of dense near polygons},
url          = {http://cage.ugent.be/geometry/Files/280/dual\_emb.pdf},
volume       = {103},
year         = {2012},
}

```
Web of Science
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