Affine Spaces within Projective Spaces

TL;DR

This paper explores the structure of affine spaces formed within projective spaces by complements of a fixed subspace, characterizing certain lines as reguli and applying these ideas to dual spreads without finite-dimensional or pappian assumptions.

Contribution

It introduces a novel affine space structure within projective spaces and characterizes lines as reguli, extending the theory beyond finite-dimensional or pappian spaces.

Findings

Certain lines are affine reguli or cones over reguli

The affine space structure aids in describing dual spreads

Results hold without finite-dimensional or pappian assumptions

Abstract

We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite-dimensional or pappian.