On Eggs and Translation Generalised Quadrangles

We study eggs in PG(4n−1, q). A new model for eggs is presented in which all known examples are given. We calculate the general form of the dual egg for eggs arising from a semifield flock. Applying this to the egg obtained by L. Bader et al. (1999, J. Combin. Theory Ser. A86, 49–62) from the Penttila–Williams ovoid, we obtain the dual egg, which is not isomorphic to any of the previous known examples. Furthermore we give a new proof of a conjecture of J. A. Thas (1994, J. Combin. Theory Ser. A67, 140–160) using our model and classify all eggs of PG(7, 2) which is equivalent to the classification of all translation generalised quadrangles of order (4, 16).