Conics in Baer subplanes (Q2419880)

The Bruck-Bose representation of $\mathrm{PG}(2,q^2)$ uses a regular spread in the hyperplane at infinity of $\mathrm{PG}(4,q)$, which has two unique lines $g$ and $g^q$ in the quadratic extension $\mathrm{PG}(4,q^2)$. The authors investigate non-degenerate conics in this representation, showing that in $\mathrm{PG}(4,q^2)$, the extended structure corresponding to a non-degenerate conic $\mathcal{O}$ is the intersection of two quadrics which meet $g$ in the two points corresponding to $\mathcal{O} \cap \ell_\infty$. They characterize the Bruck-Bose representation of conics in Baer subplanes showing that in $\mathrm{PG}(4,q)$, a non-degenerate conic corresponds to a normal rational curve that meets the transversals of the regular spread.