Geometry of the inversion in a finite field and partitions of PG(2k - 1, q) in normal rational curves

被引：3
作者：
Lavrauw M. [1 ]
Zanella C. [1 ]
机构：
[1] Dipartimento di Tecnica e Gestione dei Sistemi Industriali
来源：
Journal of Geometry | 2014年 / 105卷 / 1期
关键词：
finite field; finite projective space; normal rational curve; partition; Spread;
D O I：
10.1007/s00022-013-0197-8
中图分类号：
学科分类号：
摘要：
Let L = Fqn be a finite field and let F= Fq be a subfield of L. Consider L as a vector space over F and the associated projective space that is isomorphic to PG(n - 1, q). The properties of the projective mapping induced by x {mapping} x-1 have been studied in Csajbók (Finite Fields Appl. 19:55-66, 2013), Faina et al. (Eur. J. Comb. 23:31-35, 2002), Havlicek (Abh. Math. Sem. Univ. Hamburg 53:266-275, 1983), Herzer (Abh. Math. Sem. Univ. Hamburg 55:211-228 1985, Handbook of Incidence Geometry. Buildings and Foundations. Elsevier, Amsterdam, 1995). The image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer k, if q ≥ 2k - 1, then there are partitions of PG(2k - 1, q) in normal rational curves of degree 2k - 1. For smaller q the same construction gives partitions in (q + 1)-tuples of independent points. © 2013 Springer Basel.
引用
收藏
页码：103 / 110
页数：7
相关论文
共 13 条
  • [1] Blunck A., Herzer A., Kettengeometrien-Eine Einführung, (2005)
  • [2] Bonisoli A., Cossidente A., Mixed partitions of projective geometries, Des. Codes Cryptogr., 20, pp. 143-154, (2000)
  • [3] Cossidente A., Mixed partitions of PG(3, q), J. Geom., 68, pp. 48-57, (2000)
  • [4] Cossidente A., Funk M., Labbate D., Graphs arising from mixed partitions of PG(5,q), Bull. Inst. Comb. Appl., 33, pp. 57-67, (2001)
  • [5] Csajbok B., Linear subspaces of finite fields with large inverse-closed subsets, Finite Fields Appl., 19, pp. 55-66, (2013)
  • [6] Faina G., Kiss G., Marcugini S., Pambianco F., The cyclic model for PG(n, q) and a construction of arcs, Eur. J. Comb., 23, pp. 31-35, (2002)
  • [7] Hartshorne R., Algebraic Geometry, (1977)
  • [8] Havlicek H., Eine affine Beschreibung von Ketten, Abh. Math. Sem. Univ. Hamburg, 53, pp. 266-275, (1983)
  • [9] Herzer A., Über die Darstellung affiner Ketten als Normkurven, Abh. Math. Sem. Univ. Hamburg, 55, pp. 211-228, (1985)
  • [10] Herzer A., Chain geometries, Handbook of Incidence Geometry., (1995)
12