The complement of a point subset in a projective space and a Grassmann space

TL;DR

This paper investigates how the ambient structure of a projective space and Grassmann spaces can be reconstructed from the complement of a fixed point subset, revealing new insights into their geometric properties.

Contribution

It demonstrates that under certain conditions, the entire projective space and Grassmann spaces can be recovered from the complement of a point set, extending previous understanding of geometric reconstruction.

Findings

Reconstruction of projective space from its point complement

Extension of reconstruction results to Grassmann spaces

Conditions on line sizes for successful recovery

Abstract

In a projective space we fix some set of points, a horizon, and investigate the complement of that horizon. We prove, under some assumptions on the size of lines, that the ambient projective space, together with its horizon, both can be recovered in that complement. Then we apply this result to show something similar for Grassmann spaces.