Annales de l’Institut Henri Poincaré D

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Volume 2, Issue 3, 2015, pp. 309–333
DOI: 10.4171/AIHPD/20

Published online: 2015-08-28

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

Metod Saniga[1], Hans Havlicek[2], Frédéric Holweck[3], Michel Planat[4] and Petr Pracna[5]

(1) Vienna University of Technology, Wien, Austria
(2) TU Wien, Austria
(3) Université de Bourgogne Franche-Comté, Belfort, France
(4) Institut FEMTO-ST, Besançon, France
(5) National Information Centre for European Research, Prague, Czech Republic

Let $S_{(N)} \equiv \operatorname{PG}(1,\,2) \times \operatorname{PG}(1,\,2) \times \cdots \times \operatorname{PG}(1,\,2)$ 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 $\{H', H'', \overline{H' \Delta H''}\}$ 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 {\it 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 $\operatorname{PG}(2^N-1,2)$, we shall further describe which types of geometric hyperplanes of $S_{(N)}$ lie on a certain hyperbolic quadric $\mathcal{Q}_0^+(2^N-1,2) \subset \operatorname{PG}(2^N-1,2)$ 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 $\mathcal{W}(7,2)$.

Keywords: Binary Segre varietes, Veldkamp spaces, hyperbolic quadrics

Saniga Metod, Havlicek Hans, Holweck Frédéric, Planat Michel, Pracna Petr: Veldkamp-space aspects of a sequence of nested binary Segre varieties. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 2 (2015), 309-333. doi: 10.4171/AIHPD/20