Projective Representations II. Generalized chain geometries

TL;DR

This paper investigates how projective representations of generalized chain geometries map chains to reguli and characterizes bijective morphisms using these representations, extending previous work with new conditions and classifications.

Contribution

It provides new conditions under which chains are mapped to reguli and classifies bijective morphisms of generalized chain geometries via projective representations.

Findings

Chains are mapped to reguli depending on field embedding in the ring.

All bijective morphisms are characterized using projective representations.

Conditions for projective representations to preserve geometric structures are established.

Abstract

In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of part I. In particular, we study under which conditions such a projective representation maps the chains of a generalized chain geometry $\Sigma(F,R)$ to reguli; this mainly depends on how the field $F$ is embedded in the ring $R$. Moreover, we determine all bijective morphisms of a certain class of generalized chain geometries with the help of projective representations.