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 ${\mathcal{ℰ}}_6(\mathbb{𝕂})$.

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