Semifield flocks, eggs, and ovoids of Q(4, q)

In this article we consider the connection between semifield flocks of a quadratic cone in $\PG(3,q^n)$, eggs in $\PG(4n-1,q)$ and ovoids of $Q(4,q^n)$, when $q$ is odd. Starting from a semifield flock of a quadratic cone in $\PG(3,q^n)$, $q$ odd, $\cal F$ one can obtain an ovoid $\cal O(F)$ of $Q(4,q^n)$ using the construction of Thas \cite{THAS1997(2)}. With a semifield flock there also corresponds a good egg $\cal E$ of $\PG(4n-1,q)$ (see, e.g., \cite{LAVRAUWPENTTILA200*}) and the TGQ $T(\cal E)$ contains at least $q^{3n}+q^{2n}$ subquadrangles all isomorphic to $Q(4,q^n)$ (Thas \cite{THAS1994}). Hence by subtending one can obtain ovoids of $Q(4,q^n)$ (consider the set of points in the subquadrangle collinear with a point not in the subquadrangle). Here we prove that all the ovoids subtended from points of type (ii) are isomorphic to $\cal O(F)$, and that in at least $2q^n$ subGQ's the ovoids subtended from points of type (i) are isomorphic to the ovoids subtended from points of type (ii).