Characterising pointsets in $\mathrm{{PG}}(4,q)$ that correspond to conics

Abstract

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