On the Veldkamp Space of GQ(4, 2)

Metod Saniga (1)

1 Astronomical Institute of the Slovak Academy of Sciences

The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply giving several examples of Veldkamp lines (V-lines) having two or even three Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size five, one also finds V-lines of cardinality three and two. There, however, exists a subspace of the Veldkamp space isomorphic to PG(3, 4) having 45 perps and 40 plane ovoids as its 85 V-points, with its 357 V-lines being of four distinct types. A V-line of the first type consists of five perps on a common line (altogether 27 of them), the second type features three perps and two ovoids sharing a tricentric triad (240 members), whilst the third and fourth type each comprises a perp and four ovoids in the rosette centered at the (common) center of the perp (90). It is also pointed out that 160 non-plane ovoids (tripods) fall into two distinct orbits -- of sizes 40 and 120 -- with respect to the stabilizer group of a copy of GQ(2, 2); a tripod of the first/second orbit sharing with the GQ(2, 2) a tricentric/unicentric triad, respectively. Finally, three remarkable subconfigurations of V-lines represented by fans of ovoids through a fixed ovoid are examined in some detail.

Mots clés: GQ(4; 2); Geometric Hyperplane; Veldkamp Space; PG(3; 4)

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Géométrie algébrique [math.AG]; Physique des Hautes Energies - Théorie [hep-th]; Physique Quantique [quant-ph]

Metod Saniga. On the Veldkamp Space of GQ(4, 2). International Journal of Geometric Methods in Modern Physics, 2011, pp.39-47. ⟨10.1142/S0219887811004951⟩. ⟨hal-00443909v2⟩