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

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.

The complement of a subspace in a classical polar space

“…They prove that from such an affine reduct the ambient polar space can be recovered. In [9] we prove something similar for the complement of a subset in a projective space. Looking at the results of these two papers one sees that an interesting case has been set aside: the complement of a subspace in a polar space.…”

“…As this paper is closely related to [3] and [9], it borrows some concepts, notations and reasonings from these two works. There are however new difficulties in this case.…”

Affinization of Segre products of partial linear spaces

“…i , so by 3.1 the following is evident. We say that µ is non-zero on i-th segment when for all u ∈ V such that (18). Moreover, the system u j must be linearly independent to have u j ∈ Sub k j (V j ) in (20).…”

“…A problem that is closely related to the removal of a point subset or a line subset or both is reconstruction of the ambient space from the remainder. This is addressed in [18] for projective spaces and for Grassmann spaces, while [21] deals with the Segre product of Grassmann spaces.…”