Applications of line geometry, III: The quadric Veronesean and the chords of a twisted cubic

The chords of a twisted cubic in PG(3, q) are mapped via their Plucker coordinates to the points of a Veronese surface lying on the Klein quadric in PG(5, q). This correspondence over a finite field gives a cap in PG(5, q), that is, a set of points no three of which are collinear. The dual structure, namely the axes of the osculating developable, is also mapped to a Veronese surface. The two surfaces can be combined to give a larger cap. The constructions can be extended to the chords and axes of an arbitrary (q+ I)-arc in PG(3, q) when q is even. An alternative construction for the cap associated to a twisted cubic is given for q odd. Research supported by G.N.S.A.G.A. of C.N.R. and the Italian Ministry for Research and Technology. The third author is a senior research assistant of the Belgian National Fund for Scientific Research. Australasian Journal of Combinatorics 16(1997). pp.99-111