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We give a new proof of the main theorem of Thas (J. Combin. Theory Ser. A 79 (1997) 223–254) concerning the connection between good eggs in PG(4n-1,q), q odd, and Veronese varieties, using the model for good eggs in PG(4n-1,q), q odd, from Lavrauw and Penttila (J. Combin. Theory Ser. A 96(2) (2001) 303–315).
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 of PG(4n-1,q), q odd, is isomorphic to an ovoid of PG(3,qn) seen over GF(q) if and only if there exists a (3n-1)-space skew from the good element and containing at least five egg elements. There are only three...
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