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On line covers of finite projective and polar spaces

  • Antonio Cossidente
  • Francesco PaveseEmail author
Article
  • 35 Downloads

Abstract

An m-cover of lines of a finite projective space \(\mathrm{PG}(r,q)\) (of a finite polar space \({\mathcal {P}}\)) is a set of lines \({\mathcal {L}}\) of \(\mathrm{PG}(r,q)\) (of \({\mathcal {P}}\)) such that every point of \(\mathrm{PG}(r,q)\) (of \({\mathcal {P}}\)) contains m lines of \({\mathcal {L}}\), for some m. Embed \(\mathrm{PG}(r,q)\) in \(\mathrm{PG}(r,q^2)\). Let \({{\bar{{\mathcal {L}}}}}\) denote the set of points of \(\mathrm{PG}(r,q^2)\) lying on the extended lines of \({\mathcal {L}}\). An m-cover \({\mathcal {L}}\) of \(\mathrm{PG}(r,q)\) is an \((r-2)\)-dual m-cover if there are two possibilities for the number of lines of \({\mathcal {L}}\) contained in an \((r-2)\)-space of \(\mathrm{PG}(r,q)\). Basing on this notion, we characterize m-covers \({\mathcal {L}}\) of \(\mathrm{PG}(r,q)\) such that \({{\bar{{\mathcal {L}}}}}\) is a two-character set of \(\mathrm{PG}(r,q^2)\). In particular, we show that if \({\mathcal {L}}\) is invariant under a Singer cyclic group of \(\mathrm{PG}(r,q)\) then it is an \((r-2)\)-dual m-cover. Assuming that the lines of \({\mathcal {L}}\) are lines of a symplectic polar space \({\mathcal {W}}(r,q)\) (of an orthogonal polar space \({\mathcal {Q}}(r,q)\) of parabolic type), similarly to the projective case we introduce the notion of an \((r-2)\)-dual m-cover of symplectic type (of parabolic type). We prove that an m-cover \({\mathcal {L}}\) of \({\mathcal {W}}(r,q)\) (of \({\mathcal {Q}}(r,q)\)) has this dual property if and only if \({\bar{{\mathcal {L}}}}\) is a tight set of an Hermitian variety \({\mathcal {H}}(r,q^2)\) or of \({\mathcal {W}}(r,q^2)\) (of \({\mathcal {H}}(r,q^2)\) or of \({\mathcal {Q}}(r,q^2)\)). We also provide some interesting examples of \((4n-3)\)-dual m-covers of symplectic type of \({\mathcal {W}}(4n-1,q)\).

Keywords

Finite projective space Finite polar space m-cover Two-character set Tight set 

Mathematics Subject Classification

Primary 51E12 Secondary 51E20 51A50 

Notes

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Authors and Affiliations

  1. 1.Dipartimento di Matematica, Informatica ed EconomiaUniversità degli Studi della BasilicataPotenzaItaly
  2. 2.Dipartimento di Meccanica, Matematica e ManagementPolitecnico di BariBariItaly

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