Elation generalized quadrangles of order (q, q²), q even, with a classical subquadrangle of order q

Abstract

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(4n − 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.

In this note, we improve these results by showing that an elation generalized quadrangle of order (q, q²), q even, with a subGQ containing the elation point arises from a non-singular elliptic quadric in PG(5, q).

The theorem itself arises as a corollary of a more general observation which works for all characteristics. There is a wealth of consequences.