Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties

Metod Saniga (1, 2) , Hans Havlicek (2) , Frédéric Holweck (3) , Michel Planat (4) , Petr Pracna (5)

1 Astronomical Institute of the Slovak Academy of Sciences
2 TUW - Institut fuer Diskrete Mathematik und Geometrie
3 M3M - Laboratoire Mécatronique 3M - Méthodes, Modèles , Métiers
4 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
5 J. Heyrovsky institute of Physical Chemistry

Let $S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times PG(1,\,2)$ be a Segre variety that is $N$-fold direct product of projective lines of size three. Given two geometric hyperplanes $H'$ and $H''$ of $S_{(N)}$, let us call the triple $\{H', H'', \overline{H' \Delta H''}\}$ the Veldkamp line of $S_{(N)}$. We shall demonstrate, for the sequence $2 \leq N \leq 4$, 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 $2 \leq N \leq 4$, 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 $\mathcal{Q}_0^+(2^N-1,2) \subset PG(2^N-1,2)$ 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 $\mathcal{W}(7,2)$.

Domaines: Combinatoire [math.CO]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]

Metod Saniga, Hans Havlicek, Frédéric Holweck, Michel Planat, Petr Pracna. Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties. Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions, 2015, 2 (3), pp.309-333. ⟨hal-00966480⟩