Characterizations and properties ofgood eggs in PG(4n − 1 ,q ), q odd

Using the explicit description given in [ M. Lavrauw, T. Penttila, On eggs and translation generalised quadrangles, J. Combin. Theory Ser. A. 96 (2) (2001) 303–315] we prove that a good egg ofPG (4n − 1 ,q) , q odd, is isomorphic to an ovoid ofPG (3 ,q n ) seen over GF(q) ifand only ifthere exists a (3n − 1)-space skew from the good element and containing at least five egg elements. There are only three classes ofknown examples ofeggs in PG (4n − 1 ,q) , which are not elementary. For each of these classes we obtain strong restraints on the intersection ofegg elements with (3n − 1)-spaces spanned by three different egg elements. © 2005 Elsevier B.V. All rights reserved.