Imbrex geometries

TL;DR

This paper introduces a new axiom for strong parapolar spaces of diameter 2, characterizing various geometries including Hjelmslev-Moufang planes and Segre varieties, and offers a unified framework for their study.

Contribution

It provides a novel axiom that characterizes key geometries and generalizes a lemma used in parapolar space research, linking local tangent space conditions to classical varieties.

Findings

Characterization of Hjelmslev-Moufang planes and related geometries.

Unified framework for studying Segre varieties and Grassmannians.

A local tangent space condition characterizes classical varieties.

Abstract

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a Lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety $\mathcal{E}_{6,1}$ by means of a local condition on tangent spaces.