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

We consider a non-degenerate conic in $\PG(2,q^2)$, $q$ odd, that is tangent to $\ell_\infty$ 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 $\PG(2,q^2)$. That is, we define a set $\C$ in $\PG(4,q)$ with $q^2$ points that satisfies certain combinatorial properties. We then show that if $q\ge 7$, 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 $\P(\S)\cong\PG(2,q^2)$ constructed via the Bruck-Bose correspondence.