Veldkamp-space aspects of a sequence of nested binary Segre varieties

Let S-(N) = PG(1, 2) x PG(1, 2) x ... x PG(1, 2) be a Segre variety that is an N-fold direct product of projective lines of size three. Given two geometric hyperplanes H' and H `' of S-(N), let us call the triple {H', H `', (H' Delta H `') over bar} the Veldkamp line of S-(N). We shall demonstrate, for the sequence 2 <= N <= 4, that the properties of geometric hyperplanes of S-(N) are fully encoded in the properties of Veldkamp lines of S(N-1). Using this property, a complete classification of all types of geometric hyperplanes of S(4) is provided. Employing the fact that, for 2 <= N <= 4, the (ordinary part of) Veldkamp space of S-(N) is PG(2(N) - 1, 2), we shall further describe which types of geometric hyperplanes of S-(N) lie on a certain hyperbolic quadric Q(0)(+)(2(N) - 1, 2) subset of PG(2(N) - 1, 2) that contains the S-(N) and is invariant under its stabilizer group; in the N = 4 case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type LG(4, 8), to the set of 2295 maximal subspaces of the symplectic polar space W(7, 2).