Nearly all subspaces of a classical polar space arise from its universal embedding

TL;DR

This paper proves that in most cases, subspaces of a classical polar space correspond exactly to projective subspaces under its universal embedding, revealing a fundamental geometric structure.

Contribution

It establishes that nearly all subspaces of a classical polar space are preimages of projective subspaces via the universal embedding, clarifying their geometric relationship.

Findings

Subspaces with non-degenerate rank ≥ 2 are preimages of projective subspaces.

Universal embedding maps subspaces to projective subspaces in most cases.

Results apply to all embeddable polar spaces except specific exceptions.

Abstract

Let $\Gamma$ be an embeddable non-degenerate polar space of finite rank $n \geq 2$. Assuming that $\Gamma$ admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least $5$ and certain generalized quadrangles defined over quaternion division rings), let $\varepsilon:\Gamma\to\mathrm{PG}(V)$ be the universal embedding of $\Gamma$. Let $\cal S$ be a subspace of $\Gamma$ and suppose that $\cal S$, regarded as a polar space, has non-degenerate rank at least $2$. We shall prove that $\cal S$ is the $\varepsilon$-preimage of a projective subspace of $\mathrm{PG}(V)$.