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One of the famous open problems in Finite Geometry is the classification of ovoids in a projective 3-space PG(3,  q ) over the finite field with  q  elements,  q  even. For  q  odd, this classification was obtained in 1955 by A. Barlotti [ A. Barlotti, Un'estensione del teorema di Segre-Kustaanheimo.  Boll. Un. Mat. Ital.  (3)  10  (1955), 498–506. MR0075606 (17,776b) Zbl 0066.38901 ] and G. Panella [ G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito.  Boll. Un. Mat. Ital.  (3)  10  (1955), 507–513. MR0075607 (17,776c) Zbl 0066.38902 ]. A breakthrough result was a recent one of M. R. Brown, who showed in [ M. R. Brown, Ovoids of PG (3, q), q even, with a conic section. J. London Math. Soc. (2)  62  (2000), 569–582. MR1783645 (2001i:51012) Zbl 1038.51008 ] that when such an ovoid has a conic plane section, the ovoid must be an elliptic quadric. Even more recently, M. R. Brown and M. Lavrauw [ M. R. Brown, M. Lavrauw, Eggs in PG(4 n  − 1,  q ), q even, containing a pseudo-conic. Bull. London Math. Soc. 36 (2004), 633–639. MR2070439 (2005k:51009) Zbl 1064.51005 ] generalized this theorem by obtaining a similar result for higher dimensions. Both results have equivalent statements in the theory of (translation) generalized quadrangles.

Elation generalized quadrangles of order ( q , q 2 ), q even, with a classical subquadrangle of order q