On the Veldkamp Space of GQ(4, 2)

TL;DR

This paper investigates the structure of the Veldkamp space of GQ(4, 2), revealing its non-linear nature, subspace isomorphisms, and orbit classifications of ovoids, with detailed analysis of V-line configurations.

Contribution

It provides the first detailed analysis of the Veldkamp space of GQ(4, 2), including its subspaces, V-line types, and ovoid orbit classifications, demonstrating its complex non-linear structure.

Findings

Veldkamp space is not a linear space, with V-lines sharing multiple points.

Identified a subspace isomorphic to PG(3, 4) with specific V-points and V-lines.

Classified ovoids into two orbits with distinct stabilizer group actions.

Abstract

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.