Characterizations of symplectic polar spaces

TL;DR

This paper characterizes symplectic polar spaces based on their incidence properties without referencing their embeddings, especially relevant over fields of characteristic 2 where multiple embeddings exist.

Contribution

It provides a new characterization of symplectic polar spaces purely through incidence properties, independent of their embeddings.

Findings

Characterization of symplectic polar spaces via incidence properties

Applicable to fields of characteristic 2 with multiple embeddings

No reliance on embedding-specific properties

Abstract

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when S admits different (non isomorphic) embeddings, as it is the case (precisely) when S is defined over a field of characteristic 2.