Hostname: page-component-6bf8c574d5-5ws7s Total loading time: 0 Render date: 2025-03-08T05:16:03.231Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERIZING HERMITIAN VARIETIES IN THREE- AND FOUR-DIMENSIONAL PROJECTIVE SPACES

Part of: Finite geometry and special incidence structures Theory of error-correcting codes and error-detecting codes Designs and configurations

Published online by Cambridge University Press:  29 October 2018

ANGELA AGUGLIA
ANGELA AGUGLIA*
Affiliation:
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, I-70126 Bari, Italy email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize Hermitian cones among the surfaces of degree $q+1$ of $\text{PG}(3,q^{2})$ by their intersection numbers with planes. We then use this result and provide a characterization of nonsingular Hermitian varieties of $\text{PG}(4,q^{2})$ among quasi-Hermitian ones.

Keywords

Hermitian varietyquasi-Hermitian varietyalgebraic variety

MSC classification

Primary: 05B25: Finite geometries
Secondary: 51E20: Combinatorial structures in finite projective spaces 94B05: Linear codes, general
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 1 , August 2019 , pp. 1 - 8
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The author was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA–INdAM).

References

Aguglia, A., ‘Quasi-Hermitian varieties in PG(r, q 2), q even’, Contrib. Discrete Math. 8(1) (2013), 31–37.Google Scholar
Aguglia, A., Bartoli, D., Storme, L. and Weiner, Zs., ‘A characterization of Hermitian varieties as codewords’, Electron. J. Combin. 25(1) (2018), P1.71.Google Scholar
Aguglia, A., Cossidente, A. and Korchmáros, G., ‘On quasi-Hermitian varieties’, J. Combin. Des. 20(10) (2012), 433–447.Google Scholar
Bose, R. C. and Burton, R. C., ‘A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes’, J. Combin. Theory 1 (1966), 96–104.Google Scholar
Buekenhout, F., ‘Existence of unitals in finite translation planes of order q 2 with a kernel of order q ’, Geom. Dedicata 5 (1976), 189–194.Google Scholar
De Winter, S. and Schillewaert, J., ‘A note on quasi-Hermitian varieties and singular quasi-quadrics’, Bull. Belg. Math. Soc. Simon Stevin 17(5) (2010), 911–918.Google Scholar
De Winter, S. and Schillewaert, J., ‘Characterizations of finite classical polar spaces by intersection numbers with hyperplanes and spaces of codimension 2’, Combinatorica 30(1) (2010), 25–45.Google Scholar
Hirschfeld, J. W. P., Finite Projective Spaces of Three Dimensions (Oxford University Press, Oxford, 1985).Google Scholar
Hirschfeld, J. W. P., Projective Geometries over Finite Fields, 2nd edn (Oxford University Press, Oxford, 1998).Google Scholar
Hirschfeld, J. W. P., Storme, L., Thas, J. A. and Voloch, J. F., ‘A characterization of Hermitian curves’, J. Geom. 41(1–2) (1991), 72–78.Google Scholar
Hirschfeld, J. W. P. and Thas, J. A., General Galois Geometries (Springer, London, 1991).Google Scholar
Homma, M. and Kim, S. J., ‘Around Sziklai’s conjecture on the number of points of a plane curve over a finite field’, Finite Fields Appl. 15(4) (2009), 468–474.Google Scholar
Homma, M. and Kim, S. J., ‘Sziklai’s conjecture on the number of points of a plane curve over a finite field III’, Finite Fields Appl. 16(5) (2010), 315–319.Google Scholar
Homma, M. and Kim, S. J., ‘The characterization of Hermitian surfaces by the number of points’, J. Geom. 107 (2016), 509–521.Google Scholar
Innamorati, S., Zanetti, M. and Zuanni, F., ‘On two character (q 7 + q 5 + q 2 + 1)-sets in PG(4, q 2)’, J. Geom. 106(2) (2015), 287–296.Google Scholar
Masaaki, H. and Kim, S. J., ‘Sziklai’s conjecture on the number of points of a plane curve over a finite field II’, in: Finite Fields: Theory and Applications, Contemporary Mathematics, 518 (American Mathematical Society, Providence, RI, 2010), 225–234.Google Scholar
Metz, R., ‘On a class of unitals’, Geom. Dedicata 8 (1979), 125–126.Google Scholar
Pavese, F., ‘Geometric constructions of two-character sets’, Discrete Math. 338(3) (2015), 202–208.Google Scholar
Schillewaert, J. and Thas, J. A., ‘Characterizations of Hermitian varieties by intersection numbers’, Des. Codes Cryptogr. 50 (2009), 41–60.Google Scholar
Segre, B., ‘Le geometrie di Galois’, Ann. Mat. Pura Appl. 4(48) (1959), 1–96.Google Scholar
Segre, B., ‘Forme e geometrie Hermitiane, con particolare riguardo al caso finito’, Ann. Mat. Pura Appl. 70(4) (1965), 1–201.Google Scholar