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# 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, Hans Havlicek, Frédéric Holweck, Michel Planat and Petr Pracna

(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