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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

A projective representation G1 of a variety of the first row of the Freudenthal–Tits magic square can be obtained as the absolute geometry of a (symplectic) polarity ρ of the projective representation G2 of a variety one cell below. In this paper, we extend this geometric connection between G1 and G2 by showing that any nondegenerate quadric Q of maximal Witt index containing G2 gives rise to a variety isomorphic to G1, in the sense that the symplecta of G2 contained in totally isotropic subspaces of Q are the absolute symplecta of a unique (symplectic) polarity ρ of G2. Except for the smallest case, we also show that any nondegenerate quadric containing G2 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 6(𝕂).

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Veronese variety, spherical buildings, embeddings, geometric hyperplanes
Mathematical Subject Classification
Primary: 51E24
Received: 2 November 2021
Accepted: 6 September 2022
Published: 15 February 2023
Anneleen De Schepper
Department of Mathematics: Algebra and Geometry
Ghent University
Magali Victoor
Department of Mathematics: Algebra and Geometry
Ghent University