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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: 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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Volume 19, Issue 2


Geometries arising from trilinear forms on low-dimensional vector spaces

Ilaria Cardinali
  • Corresponding author
  • Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, 53100, Siena, Italy
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/ Luca Giuzzi
Published Online: 2019-04-09 | DOI: https://doi.org/10.1515/advgeom-2018-0027


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


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About the article

Received: 2017-03-21

Revised: 2017-08-17

Published Online: 2019-04-09

Published in Print: 2019-04-24

Communicated by: A. Pasini

Citation Information: Advances in Geometry, Volume 19, Issue 2, Pages 269–290, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0027.

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