Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T16:15:38.314Z Has data issue: false hasContentIssue false
  • English
  • Français

A COMBINATORIAL CHARACTERIZATION OF THE LAGRANGIAN GRASSMANNIAN LG(3,6)(${\mathbb{K}}$)

Part of: Linear incidence geometry Geometry Finite geometry and special incidence structures Real and complex geometry

Published online by Cambridge University Press:  21 July 2015

J. SCHILLEWAERT  and
H. VAN MALDEGHEM
J. SCHILLEWAERT
Affiliation:
Department of Mathematics, Imperial College, South Kensington Campus, London SW7-2AZ, United Kingdom e-mail: jschillewaert@gmail.com
H. VAN MALDEGHEM
Affiliation:
Department of Mathematics, Ghent University Krijgslaan 281-S22, B-9000 Ghent, Belgium e-mail: hvm@cage.ugent.be
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide a combinatorial characterization of LG(3,6)(${\mathbb{K}}$) using an axiom set which is the natural continuation of the Mazzocca–Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields, Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal–Tits magic square.

MSC classification

Primary: 51A45: Incidence structures imbeddable into projective geometries
Secondary: 51A50: Polar geometry, symplectic spaces, orthogonal spaces 51E24: Buildings and the geometry of diagrams 51M35: Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 293 - 311
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Blokhuis, A. and Brouwer, A. E., The universal embedding dimension of the binary symplectic dual polar space, in The 2000 Com2MaC Conference on Association Schemes, Codes and Designs (Pohang). Discrete Math. 264 (2003), 3–11.Google Scholar
2.Cardinali, I. and De Bruyn, B., The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces, Adv. Geom. 8 (2008), 111–137.CrossRefGoogle Scholar
3.De Bruyn, B. and Van Maldeghem, H., Dual polar spaces of rank 3 defined over quadratic alternative division algebras, J. Reine Angew. Math., to appear.Google Scholar
4.Cooperstein, B., On the generation of dual polar spaces of symplectic type over finite fields, Eur. J. Comb. 18 (1997), 849–856.CrossRefGoogle Scholar
5.De Bruyn, B. and Pasini, A., Generating symplectic and Hermitian dual polar spaces over arbitrary fields nonisomorphic to $\mathbb{F}_{2}$, Electron. J. Comb. 14 (2009), #R54, 17pp.Google Scholar
6.Krauss, O., Geometrische Charakterisierung von Veronesemannigfaltigkeiten, PhD Thesis (Braunschweig, 2014).Google Scholar
7.Li, P., On the universal embedding of the Sp 2n(2) dual polar space, J. Combin. Theory Ser. A 94 (1) (2001), 100–117.Google Scholar
8.Schillewaert, J. and Van Maldeghem, H., Quadric Veronesean caps, Bull. Belgian Math. Soc. Simon Stevin 20 (2013), 19–25.Google Scholar
9.Schillewaert, J. and Van Maldeghem, H., On the varieties of the second row of the split Freudenthal-Tits magic square. Available at http://arxiv.org/abs/1308.0745.Google Scholar
10.Shult, E. E., Points and Lines, Characterizing the Classical Geometries (Universitext, Springer-Verlag, Berlin, Heidelberg, 2011).Google Scholar
11.Tits, J., Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles, Indag. Math. 28 (1966), 223–237.Google Scholar
12.Tits, J., A local approach to buildings, in The Geometric vein. The Coxeter Festschrift, Coxeter Symposium, University of Toronto, 21–25 May 1979 (Springer-Verlag, New York 1982), 519–547.Google Scholar
13.Yoshiara, S., Embeddings of flag-transitive classical locally polar geometries of rank 3, Geom. Dedicata 43 (1992), 121–165.Google Scholar
14.Zak, F. L., Tangents and secants of algebraic varieties, in Translations of Mathematical Monographs, vol 127 (American Mathematical Society, Providence, RI, 1993). viii+164 pp.Google Scholar