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

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↦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 ≥ 2 k − 1, then there are partitions of PG(2 k − 1, q) in normal rational curves of degree 2 k − 1. For smaller q the same construction gives partitions in (q + 1)-tuples of independent points.

Publication year:2014