Innovations in Incidence Geometry

On the intersection of Hermitian surfaces

Nicola Durante and Gary Ebert

Full-text: Open access

Abstract

Giuzzi (2006) and Donati and Duranti (2008) determined the structure of the intersection of two Hermitian surfaces of PG ( 3 , q 2 ) under the hypotheses that in the pencil they generate there is at least one degenerate surface. Aguglia, Cossidente and Ebert (2005) and Donati and Duranti (2008) showed that under suitable hypotheses the intersection of two Hermitian surfaces generating a non-degenerate pencil is a pseudo-regulus. Here we completely determine all possible intersection configurations for two Hermitian surfaces of PG ( 3 , q 2 ) generating a non-degenerate pencil.

Article information

Source
Innov. Incidence Geom., Volume 6-7, Number 1 (2007), 153-167.

Dates
Received: 7 November 2007
Accepted: 15 February 2008
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323176

Digital Object Identifier
doi:10.2140/iig.2008.6.153

Mathematical Reviews number (MathSciNet)
MR2515264

Zentralblatt MATH identifier
1183.51001

Subjects
Primary: 05B25: Finite geometries [See also 51D20, 51Exx] 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx]

Keywords
Hermitian curve Hermitian surface

Citation

Durante, Nicola; Ebert, Gary. On the intersection of Hermitian surfaces. Innov. Incidence Geom. 6-7 (2007), no. 1, 153--167. doi:10.2140/iig.2008.6.153. https://projecteuclid.org/euclid.iig/1551323176


Export citation

References

  • A. Aguglia, A. Cossidente and G. L. Ebert, On pairs of permutable Hermitian surfaces, Discrete Math. 301 (2005), 28–33.
  • G. Donati and N. Durante, A subset of the Hermitian surface, Innov. Incidence Geom. 3 (2006), 13–23.
  • ––––, On the intersection of Hermitian curves and Hermitian surfaces, Discrete Math. 308 (2008), no. 22, 5196–5203.
  • G. L. Ebert and J. W. P. Hirschfeld, Complete systems of lines on a Hermitian surface over a finite field, Des. Codes Cryptogr. 9 (1999), 253–268.
  • J. W. Freeman, Reguli and pseudo-reguli in ${\rm PG}(3,\,s\sp{2})$, Geom. Dedicata 9 (1980), 267–280.
  • L. Giuzzi, On the intersection of Hermitian surfaces, J. Geom. 85 (2006), 49–60.
  • J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Clarendon Press, Oxford, 1991.
  • B. C. Kestenband, Unital intersections in finite projective planes, Geom. Dedicata 11 (1981), 107–117.
  • ––––, Degenerate unital intersections in finite projective planes, Geom. Dedicata 13 (1982), 101–106.
  • B. Segre, Lectures on Modern Geometry, Cremonese, Roma, 1961.
  • D. E. Taylor, The Geometry of the Classical Groups, Haldermann Verlag Berlin, Berlin, 1992.
  • O. Veblen, A system of axioms for geometry, Trans. Amer. Math. Soc. 5 (1904), no. 3 343–384. \end thebibliography