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 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. A 67, 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). (C) 2001 Academic Press.