The pure condition for incidence geometries

TL;DR

This paper introduces a pure condition for incidence geometries that determines when they can be realized in d-dimensional space with distinct points and hyperplanes, extending Whiteley's earlier work on generic realizations.

Contribution

It defines a new pure condition for incidence geometries, analogous to frameworks, using invariant theory and bracket polynomials, to characterize realizability with specific normals.

Findings

The pure condition is expressed as a bracket polynomial.

Explicit computations of the pure condition are provided for planar examples.

The condition helps identify when incidence geometries can be realized with given normals.

Abstract

The space of \emph{parallel redrawings} of an incidence geometry $(P,H,I)$ with an assigned set of normals is the set of points and hyperplanes in $\mathbb{R}^d$ satisfying the incidences given by $(P,H,I)$, such that the hyperplanes have the assigned normals. In 1989, Whiteley characterized the incidence geometries that have d-dimensional realizations with generic hyperplane normals such that all points and hyperplanes are distinct. However, some incidence geometries can be realized as points and hyperplanes in d-dimensional space, with the points and hyperplanes distinct, but only for specific choices of normals. Such incidence geometries are the topic of this article. In this article, we introduce a pure condition for parallel redrawings of incidence geometries, analogous to the pure condition for bar-and-joint frameworks, introduced by White and Whiteley. The d-dimensional pure condition of an incidence geometry (P,H,I) imposes a condition on the normals assigned to the hyperplanes of (P,H,I) required for d-dimensional realizations of (P,H,I) with distinct points. We use invariant theory to show that is a bracket polynomial. We will also explicitly compute the pure condition as a bracket polynomial for some examples in the plane.