Abstract
A geometric hyperplane of a point--line geometry is a proper subspace which meets each line non-trivially. If H is a hyperplane of a projective space P, and the point line geometry Γ has an embedding in P , then the pullback from H is a geometric hyperplane of Γ. We show that all geometric hyperplanes arise in this way for polar spaces of typeD n , the Grassmann space of lines, and the exceptional geometry E 1,6 . The actual geometric hyperplanes are studied in several cases.
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References
Aschbacher, M.: The 27-dimensional module for E6: I, Invent. Math. 89 (1987), 159–195.
Behrends, R.E., Dreitlein, J., Fronsdal, C. and Lee, B.W.: Simple groups and strong interaction symmetries, Rev. Modern Phys. 34(1962), 1–40.
Blok, R. and Brouwer, A.: Spanning point-line geometries in buildings of spherical type (Preprint).
Buekenhout, F.: Cooperstein's theory, Simon Stevin 57 (1983), 125–140.
Cohen, A.M.: Point-line characterizations of buildings, in L.A. Rosati (ed.) Buildings and the Geometry of Diagrams: Como 1984. Lecture Notes in Maths 1181, Springer, 1986, 191–206.
Cohen, A.M. and Cooperstein, B.N.: The 2-spaces of the standard E6 (q)-module, Geom. Dedicata 25 (1988), 467–480.
Cohen, A.M. and Shult, E.E.: Affine polar spaces, Geom. Dedicata 35 (1990), 43–76.
Cooperstein, B.N.: A characterization of some Lie incidence structures, Geom. Dedicata 6 (1977), 205–258.
Cooperstein, B.N.: A four-form for half-spin groups of type D6 (Preprint).
Cooperstein, B.N. and Shult, E.E.: Frames and bases of Lie incidence geometries (accepted by J. Geometry).
Hall, J.I. and Shult, E.E.: Locally cotriangular graphs, Geom. Dedicata 18 (1985), 113–159.
Igusa, J.-I.: A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997–1028.
Ronan, M.: Embeddings and hyperplanes of discrete geometries. European J. Combin. 8 (1987), 179–185.
Shult, E.E.: Lie incidence geometries, in E. Keith Lloyd (ed.), Surveys in Combinatorics, London Math. Soc. Lecture Note Series 82, Cambridge University Press, 1983, pp. 157–184.
Shult, E.E. and Yanushka, A.: Near n-gons and line systems, Geom. Dedicata 9 (1980) 1–72.
Veldkamp, F.D.: Polar geometry I-V, Proc. Kon. Ned. Akad. Wet. A62 (1959), 512–551; A63 (1960), 207–212 (= Indag. Math. 21, 22).
Wells, A.L. Jr.: Universal projective embeddings of the Grassmannian, half spinor and dual orthogonal geometries, Quart. J. Math. Oxford (2), 34 (1983), 375–386.
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Cooperstein, B.N., Shult, E.E. Geometric Hyperplanes of Lie Incidence Geometries. Geometriae Dedicata 64, 17–40 (1997). https://doi.org/10.1023/A:1004969708687
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DOI: https://doi.org/10.1023/A:1004969708687