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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Wissenschaftlicher Beirat: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

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Band 19, Heft 2

Hefte

Geometries arising from trilinear forms on low-dimensional vector spaces

Ilaria Cardinali
  • Korrespondenzautor
  • Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, 53100, Siena, Italy
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/ Luca Giuzzi
Online erschienen: 09.04.2019 | DOI: https://doi.org/10.1515/advgeom-2018-0027

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 of H. 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).

Keywords: Grassmann geometry; hyperplanes; multilinear forms

MSC 2010: 15A75; 14M15; 15A69

References

  • [1]

    A. Barlotti, J. Cofman, Finite Sperner spaces constructed from projective and affine spaces. Abh. Math. Sem. Univ. Hamburg 40 (1974), 231–241. MR0335305 Zbl 0271.50007CrossrefGoogle Scholar

  • [2]

    R. H. Bruck, R. C. Bose, Linear representations of projective planes in projective spaces. J. Algebra 4 (1966), 117–172. MR0196590 Zbl 0141.36801CrossrefGoogle 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 06731941CrossrefGoogle Scholar

  • [4]

    A. M. Cohen, A. G. Helminck, Trilinear alternating forms on a vector space of dimension 7. Comm. Algebra 16 (1988), 1–25. MR921939 Zbl 0646.15019CrossrefGoogle 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 1161.51002Web of ScienceCrossrefGoogle Scholar

  • [6]

    J. Draisma, R. Shaw, Singular lines of trilinear forms. Linear Algebra Appl. 433 (2010), 690–697. MR2653833 Zbl 1213.15020CrossrefWeb of ScienceGoogle Scholar

  • [7]

    J. Draisma, R. Shaw, Some noteworthy alternating trilinear forms. J. Geom. 105 (2014), 167–176. MR3176345 Zbl 1310.15041CrossrefGoogle Scholar

  • [8]

    J. Harris, Algebraic geometry. Springer 1995. MR1416564 Zbl 0779.14001Google Scholar

  • [9]

    J. Harris, L. W. Tu, On symmetric and skew-symmetric determinantal varieties. Topology 23 (1984), 71–84. MR721453 Zbl 0534.55010CrossrefGoogle Scholar

  • [10]

    H. Havlicek, Zur Theorie linearer Abbildungen. I, II. J. Geom. 16 (1981), 152–167, 168–180. MR642264 Zbl 0463.51003CrossrefGoogle Scholar

  • [11]

    H. Havlicek, C. Zanella, Incidence and combinatorial properties of linear complexes. Results Math. 51 (2008), 261–274. MR2400168 Zbl 1143.51003CrossrefWeb of ScienceGoogle Scholar

  • [12]

    P. Revoy, Trivecteurs de rang 6. Bull. Soc. Math. France Mém. no. 59 (1979), 141–155. MR532012 Zbl 0405.15024Google Scholar

  • [13]

    E. Shult, Geometric hyperplanes of embeddable Grassmannians. J. Algebra 145 (1992), 55–82. MR1144658 Zbl 0751.51002CrossrefGoogle 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 0088.37204Google Scholar

  • [15]

    H. Van Maldeghem, Generalized polygons. Birkhäuser 1998. MR1725957 Zbl 0914.51005Google Scholar

Artikelinformationen

Erhalten: 21.03.2017

Revidiert: 17.08.2017

Online erschienen: 09.04.2019

Erschienen im Druck: 24.04.2019


Communicated by: A. Pasini


Quellenangabe: Advances in Geometry, Band 19, Heft 2, Seiten 269–290, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0027.

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Alexander I. Suciu
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2019, Seite 1

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