Abstract
Using the notion of special tangent it is shown that there are precisely two kinds of representation on a cone with planar vertex of a projective plane of order q3 in PG(6, q). The Desarguesian plane PG(2, q3) admits both kinds of representation. Non-spread representations of PG(2, qi) in PG(2i, q) can be found for all i ≥ 3, and we give three methods, one algebraic and two geometric, to construct these. Some indications of the applications of such representations are given, including the representation of subplanes and conics of PG(2, q3) in PG(6, q). These non-standard representations can be generalised to representations of PG(n, q), and are also valid for projective geometries over infinite fields.
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