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

Abstract

Let $S_{(N)}\equiv \mathrm{PG}(1,2)\times \mathrm{PG}(1,2)\times \cdots \times \mathrm{PG}(1,2)$ be a Segre variety that is an $N$-fold direct product of projective lines of size three. Given two geometric hyperplanes $H^{\prime }$ and $H^{\prime \prime }$ of $S_{(N)}$, let us call the triple $\{H^{\prime },H^{\prime \prime },\overline{H^{\prime }\Delta H^{\prime \prime }}\}$ the Veldkamp line of $S_{(N)}$. We shall demonstrate, for the sequence $2\le N\le 4$, 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 $2\le N\le 4$, the (ordinary part of) Veldkamp space of $S_{(N)}$ is $\mathrm{PG}(2^N-1,2)$, we shall further describe which types of geometric hyperplanes of $S_{(N)}$ lie on a certain hyperbolic quadric ${\mathcal{Q}}_0^{+}(2^N-1,2)\subset \mathrm{PG}(2^N-1,2)$ 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 $\mathcal{W}(7,2)$.