×

The complement of a subspace in a classical polar space. (English) Zbl 1434.51003

Let us consider a point-line structure \(\mathfrak{M} = \langle S; L\rangle\), where the elements of \(S\) are called points, the elements of \(L\) are called lines, and \(L \subset 2^S\), such a tuple is said to be a partial linear space, or a point-line space, if two distinct lines share at most one point and every line is of size (cardinality) at least \(2\). A partial linear space satisfying that for every line \(\ell\) and a point \(a \not\in \ell\), \(a\) is collinear with one or all points on \(\ell\), will be called a polar space. In the paper under review, the authors study the complement of a subspace in a classical polar space. More specifically, in a polar space, embeddable into a projective space, one fixes a subspace that is contained in some hyperplane, then the complement of that subspace resembles a slit space or a semiaffine space. The main result of the paper tells us that, under some reasonable assumptions, the ambient polar space can be recovered in this complement – see Theorem 3.11 therein.

MSC:

51A50 Polar geometry, symplectic spaces, orthogonal spaces
51A15 Linear incidence geometric structures with parallelism
51A45 Incidence structures embeddable into projective geometries
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Bamberg, J. De Beule and F. Ihringer, New non-existence proofs for ovoids of Hermitian polar spaces and hyperbolic quadrics,Ann. Comb.21(2017), 25-42, doi:10.1007/ s00026-017-0346-0. · Zbl 1365.05037
[2] P. J. Cameron,Projective and Polar Spaces, volume 13 ofQMW Maths Notes, Queen Mary and Westfield College, School of Mathematical Sciences, London, 1992,http://www.maths. qmul.ac.uk/˜pjc/pps/.
[3] A. M. Cohen, Point-line spaces related to buildings, in: F. Buekenhout (ed.),Handbook of Incidence Geometry: Buildings and Foundations, North-Holland, Amsterdam, pp. 647-737, 1995, doi:10.1016/b978-044488355-1/50014-1. · Zbl 0829.51004
[4] A. M. Cohen and E. E. Shult, Affine polar spaces,Geom. Dedicata35(1990), 43-76, doi: 10.1007/bf00147339. · Zbl 0755.51004
[5] H. Karzel and H. Meissner, Geschlitze Inzidenzgruppen und normale Fastmoduln,Abh. Math. Sem. Univ. Hamburg31(1967), 69-88, doi:10.1007/bf02992387. · Zbl 0149.38702
[6] H. Karzel and I. Pieper, Bericht ¨uber geschlitzte Inzidenzgruppen,Jber. Deutsch. Math. Verein. 72(1970), 70-114,http://eudml.org/doc/146588. · Zbl 0202.51001
[7] A. Kreuzer, Semiaffine spaces,J. Comb. Theory Ser. A64(1993), 63-78, doi:10.1016/ 0097-3165(93)90088-p. · Zbl 0806.51003
[8] M. Marchi and S. Pianta, Partial parallelism spaces and slit spaces, in: A. Barlotti, P. V. Ceccherini and G. Tallini (eds.),Combinatorics ’81, North-Holland, Amsterdam-New York, volume 18 ofAnnals of Discrete Mathematics, 1983 pp. 591-600, doi:10.1016/s0304-0208(08) 73337-1, proceedings of the International Conference on Combinatorial Geometries and their Applications held in Rome, June 7 - 12, 1981. · Zbl 0541.51003
[9] K. Petelczyc and M. ˙Zynel, The complement of a point subset in a projective space and a Grassmann space,J. Appl. Logic13(2015), 169-187, doi:10.1016/j.jal.2015.02.002. · Zbl 1386.51002
[10] K.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.