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

Abstract

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