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A geometric connection between the split first and second rows of the Freudenthal–Tits magic square

Anneleen De Schepper and Magali Victoor

Vol. 20 (2023), No. 1, 1–53
Abstract

A projective representation ${G}_{1}$ of a variety of the first row of the Freudenthal–Tits magic square can be obtained as the absolute geometry of a (symplectic) polarity $\rho$ of the projective representation ${G}_{2}$ of a variety one cell below. In this paper, we extend this geometric connection between ${G}_{1}$ and ${G}_{2}$ by showing that any nondegenerate quadric $Q$ of maximal Witt index containing ${G}_{2}$ gives rise to a variety isomorphic to ${G}_{1}$, in the sense that the symplecta of ${G}_{2}$ contained in totally isotropic subspaces of $Q$ are the absolute symplecta of a unique (symplectic) polarity $\rho$ of ${G}_{2}$. Except for the smallest case, we also show that any nondegenerate quadric containing ${G}_{2}$ has maximal Witt index; and in the largest case, we obtain that there are only three kinds of possibly degenerate quadrics containing the Cartan variety ${\mathsc{ℰ}}_{6}\left(\mathbb{𝕂}\right)$.

Keywords
Veronese variety, spherical buildings, embeddings, geometric hyperplanes
Primary: 51E24