Geometries arising from trilinear forms on low-dimensional vector spaces
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Ilaria Cardinali
und
Luca Giuzzi
Abstract
Let 𝓖k(V) be the k-Grassmannian of a vector space V with dim V = n. Given a hyperplane H of 𝓖k(V), we define in [3] a point-line subgeometry of PG(V) called the geometry of poles ofH. In the present paper, exploiting the classification of alternating trilinear forms in low dimension, we characterize the possible geometries of poles arising for k = 3 and n ≤ 7 and propose some new constructions. We also extend a result of [6] regarding the existence of line spreads of PG(5, 𝕂) arising from hyperplanes of 𝓖3(V).
Communicated by: A. Pasini
References
[1] A. Barlotti, J. Cofman, Finite Sperner spaces constructed from projective and affine spaces. Abh. Math. Sem. Univ. Hamburg40 (1974), 231–241. MR0335305 Zbl 0271.5000710.1007/BF02993602Suche in Google Scholar
[2] R. H. Bruck, R. C. Bose, Linear representations of projective planes in projective spaces. J. Algebra4 (1966), 117–172. MR0196590 Zbl 0141.3680110.1016/0021-8693(66)90054-8Suche in Google Scholar
[3] I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating k-linear forms. J. Algebraic Combin. 45 (2017), 931–963. MR3641973 Zbl 0673194110.1007/s10801-016-0730-6Suche in Google Scholar
[4] A. M. Cohen, A. G. Helminck, Trilinear alternating forms on a vector space of dimension 7. Comm. Algebra16 (1988), 1–25. MR921939 Zbl 0646.1501910.1080/00927878808823558Suche in Google Scholar
[5] B. De Bruyn, Hyperplanes of embeddable Grassmannians arise from projective embeddings: a short proof. Linear Algebra Appl. 430 (2009), 418–422. MR2460527 Zbl 1161.5100210.1016/j.laa.2008.08.003Suche in Google Scholar
[6] J. Draisma, R. Shaw, Singular lines of trilinear forms. Linear Algebra Appl. 433 (2010), 690–697. MR2653833 Zbl 1213.1502010.1016/j.laa.2010.03.040Suche in Google Scholar
[7] J. Draisma, R. Shaw, Some noteworthy alternating trilinear forms. J. Geom. 105 (2014), 167–176. MR3176345 Zbl 1310.1504110.1007/s00022-013-0202-2Suche in Google Scholar
[8] J. Harris, Algebraic geometry. Springer 1995. MR1416564 Zbl 0779.14001Suche in Google Scholar
[9] J. Harris, L. W. Tu, On symmetric and skew-symmetric determinantal varieties. Topology23 (1984), 71–84. MR721453 Zbl 0534.5501010.1016/0040-9383(84)90026-0Suche in Google Scholar
[10] H. Havlicek, Zur Theorie linearer Abbildungen. I, II. J. Geom. 16 (1981), 152–167, 168–180. MR642264 Zbl 0463.5100310.1007/BF01917585Suche in Google Scholar
[11] H. Havlicek, C. Zanella, Incidence and combinatorial properties of linear complexes. Results Math. 51 (2008), 261–274. MR2400168 Zbl 1143.5100310.1007/s00025-007-0275-zSuche in Google Scholar
[12] P. Revoy, Trivecteurs de rang 6. Bull. Soc. Math. France Mém. no. 59 (1979), 141–155. MR532012 Zbl 0405.1502410.24033/msmf.256Suche in Google Scholar
[13] E. Shult, Geometric hyperplanes of embeddable Grassmannians. J. Algebra145 (1992), 55–82. MR1144658 Zbl 0751.5100210.1016/0021-8693(92)90176-MSuche in Google Scholar
[14] J. Tits, Sur la trialité et certains groupes qui s’en déduisent. Inst. Hautes Études Sci. Publ. Math. no. 2 (1959), 13–60. MR1557095 Zbl 0088.3720410.1007/BF02684706Suche in Google Scholar
[15] H. Van Maldeghem, Generalized polygons. Birkhäuser 1998. MR1725957 Zbl 0914.5100510.1007/978-3-0348-0271-0Suche in Google Scholar
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Artikel in diesem Heft
- Frontmatter
- On the cells in a stationary Poisson hyperplane mosaic
- Conical geodesic bicombings on subsets of normed vector spaces
- Variance estimates and almost Euclidean structure
- Special cubic Cremona transformations of â„™6 and â„™7
- Lipschitz–Killing curvatures and polar images
- On the connectivity of the hyperbolicity region of irreducible polynomials
- Biharmonic hypersurfaces in 5-dimensional non-flat space forms
- On the principal Ricci curvatures of a Riemannian 3-manifold
- On the curve Yn = Xâ„“(Xm + 1) over finite fields
- Geometries arising from trilinear forms on low-dimensional vector spaces
Artikel in diesem Heft
- Frontmatter
- On the cells in a stationary Poisson hyperplane mosaic
- Conical geodesic bicombings on subsets of normed vector spaces
- Variance estimates and almost Euclidean structure
- Special cubic Cremona transformations of â„™6 and â„™7
- Lipschitz–Killing curvatures and polar images
- On the connectivity of the hyperbolicity region of irreducible polynomials
- Biharmonic hypersurfaces in 5-dimensional non-flat space forms
- On the principal Ricci curvatures of a Riemannian 3-manifold
- On the curve Yn = Xâ„“(Xm + 1) over finite fields
- Geometries arising from trilinear forms on low-dimensional vector spaces
