An asymptotic rigidity property from the realizability of chirotope extensions

TL;DR

The paper studies when chirotopes realized by adding points to a base configuration P can be realized on another configuration Q. It proves a strong rigidity result: P and Q are directly affinely equivalent iff every finite generic extension of P has its chirotope realizable on top of Q, and it provides a quantitative version with extensions of size O(log(1/ε)) guaranteeing ε-approximate affine proximity. The approach combines Von Staudt constructions to encode arithmetic in incidences, corank-1 reductions to lower dimension, scattering methods to propagate constraints, and careful perturbation to handle nongeneric cases, culminating in a uniform, dimensionally robust rigidity theorem. This advances understanding of realization spaces of chirotopes/oriented matroids by linking extension realizability to direct affine equivalence and yields a practical method to distinguish configurations via bounded-size finite extensions.

Abstract

Let $P$ be a finite full-dimensional point configuration in $\mathbb{R}^d$. We show that if a point configuration $Q$ has the property that all finite chirotopes realizable by adding (generic) points to $P$ are also realizable by adding points to $Q$, then $P$ and $Q$ are equal up to a direct affine transform. We also show that for any point configuration $P$ and any $\varepsilon>0$, there is a finite, (generic) extension $\widehat P$ of $P$ with the following property: if another realization $Q$ of the chirotope of $P$ can be extended so as to realize the chirotope of $\widehat P$, then there exists a direct affine transform that maps each point of $Q$ within distance $\varepsilon$ of the corresponding point of $P$.

Figures (11)

• Figure 1: Two (projectively equivalent) point configurations $P=(p_1,p_2,p_3,p_4)$ and $Q=(q_1,q_2,q_3,q_4)$. The chirotope of any $1$-point extension of $Q$ is realizable on top of $P$, but the chirotope of $P \sqcup \{p\}$ is not realizable on top of $Q$.
• Figure 2: Two point configurations with the same chirotope.
• Figure 3: The construction of Von Staudt. In each configuration, the parameterization is relative to the projective basis $B=(b_0,b_1,b_\infty)$ formed by the red points (in the addition and substraction constructions, the point $b_1$ plays no role so it is omitted). In each construction, the parameters of the blue points are chosen freely and the parameter of the green point is determined by the incidences.
• Figure 4: The maps $b(q,\cdot)$ and $b(p,\cdot)$ map $\mathbb{R}\cup\{\infty\}$ to the projective line $\bar{\ell}$. Restricted respectively to $\mathbb{R}\cup\{\infty\}\setminus \{\omega(q)\}$ and $\mathbb{R}\cup\{\infty\}\setminus \{\omega(p)\}$, they provide a map to the affine line $\ell$ (with the image of $\infty$ being respectively $p$ and $q$). This figure represents these maps schematically.
• Figure 5: The pair $\{2,3\}$ is good for this configuration, which allows for the construction of $P_{\downarrow \{2,3\}}$.

Theorems & Definitions (42)

• Theorem 1
• Theorem 2
• Conjecture 3
• Conjecture 4
• Lemma 2.1
• proof
• Lemma 3.1: Von Staudt Staudt
• proof
• Lemma 4.1
• proof