Characterising pointsets in PG(4,q) that correspond to conics

Original · EN

We consider a non-degenerate conic in (2,q²), q odd, that is tangent to ℓ∞ and look at its structure in the Bruck-Bose representation in (4,q). We determine which combinatorial properties of this set of points in (4,q) are needed to reconstruct the conic in (2,q²). That is, we define a set in (4,q) with q² points that satisfies certain combinatorial properties. We then show that if q≥ 7, we can use to construct a regular spread § in the hyperplane at infinity of (4,q), and that corresponds to a conic in the Desarguesian plane ¶(§)(2,q²) constructed via the Bruck-Bose correspondence.