The complement of a subspace in a classical polar space

TL;DR

This paper investigates how the ambient polar space can be reconstructed from the complement of a subspace contained in a hyperplane, which resembles a slit or semiaffine space, under certain conditions.

Contribution

It demonstrates that, under specific assumptions, the original polar space can be recovered from the complement of a subspace within a hyperplane.

Findings

The ambient polar space can be reconstructed from the complement under certain conditions.

The complement resembles a slit or semiaffine space, providing a new perspective.

Recovery of the original space is possible with specific assumptions.

Abstract

In a polar space, embeddable into a projective space, we fix a subspace, that is contained in some hyperplane. The complement of that subspace resembles a slit space or a semiaffine space. We prove that under some assumptions the ambient polar space can be recovered in this complement.