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. A 86, 49-62) from the Pen ttila-Williams ovoid, we obtain the dual egg, which is not isomorphic to an y of the previous known examples. Furthermore we give a new proof of a conj ecture of J. A. Thas (1994, J. Combin. Theory Ser. A 67, 140-160) using our model and classify all eggs of PG(7, 2) which is equivalent to the classif ication of all translation generalised quadrangles of order (4, 16). (C) 20 01 Academic Press.