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Synthetic and projective properties of embeddable polar spaces

Antonio Pasini

Vol. 20 (2023), No. 2-3, 519–542

A projective property of an embeddable polar space 𝒮 is a property of the family of its projective embeddings, such as the existence of an embedding of vector dimension twice the rank of 𝒮 or the fact that all embeddings of 𝒮 have the same dimension, while properties such as the fact that all pairs of opposite points are regular or all triads of points are centric, are synthetic properties. We prove that all pairs of opposite points of an embeddable polar space 𝒮 of rank n are regular if and only if 𝒮 admits a 2n-dimensional embedding; we also prove that all embeddings of 𝒮 have the same dimension if and only if the subspace spanned by two opposite singular subspaces is closed under taking hyperbolic lines. Moreover, we characterize the fact that all triads of points of 𝒮 are centric by means of suitable properties of the universal embedding of 𝒮.

Dedicated to the memory of Jacques Tits

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polar spaces, embeddings, regularity, centric triads
Mathematical Subject Classification
Primary: 51A50
Received: 31 May 2022
Revised: 1 December 2022
Accepted: 20 December 2022
Published: 13 September 2023
Antonio Pasini
Department of Information Engineering and Mathematics
University of Siena