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Finite groupoids of configurations of lines in $\mathbb{P}^{3}_{\mathbb{C}}$

Jake Kettinger

TL;DR

The paper develops a framework to study groupoids induced by line configurations in $\mathbb{P}^3_\mathbb{C}$ via projection maps using an auxiliary line, and analyzes finite-automorphism group behavior for key geproci configurations. It constructs groupoids from the $D_4$, Penrose, and Klein configurations, computes their automorphism groups ($S_3$, $S_4$, and related subgroups such as $A_4$ for subconfigurations), and investigates related subconfigurations (half- and quasi-Penrose) and their symmetry structures. It also identifies configurations (e.g., $F_4$, the $120$-cell, and the Schäfli double six) with infinite automorphism groups and outlines extensions to higher dimensions, positive characteristic, and Grassmannian lifts, aiming to connect geproci geometry with finite subgroups of $\mathrm{PGL}(2,\mathbb{C})$ and to develop a unifying groupoid-theoretic perspective. The work provides a concrete computational framework (via Macaulay2) and establishes a basis for future exploration of more complex configurations and their symmetries, including potential octahedral and tetrahedral actions in related settings.

Abstract

In this paper, we investigate groupoids coming from configurations of lines in three-dimensional space. Given a point and two skew lines in $\mathbb{P}^{3}_{K}$ over a field $K$, there exists a unique line containing the given point and meeting the two given lines. We use this construction to define a projection function from one line to another by using a skew line as an auxiliary. This way, we may create a groupoid whose objects are lines in a configuration, and whose morphisms are induced by these projection functions. We look at specific configurations for $K=\mathbb{C}$ that yield groupoids with finite automorphism groups.

Finite groupoids of configurations of lines in $\mathbb{P}^{3}_{\mathbb{C}}$

TL;DR

The paper develops a framework to study groupoids induced by line configurations in via projection maps using an auxiliary line, and analyzes finite-automorphism group behavior for key geproci configurations. It constructs groupoids from the , Penrose, and Klein configurations, computes their automorphism groups (, , and related subgroups such as for subconfigurations), and investigates related subconfigurations (half- and quasi-Penrose) and their symmetry structures. It also identifies configurations (e.g., , the -cell, and the Schäfli double six) with infinite automorphism groups and outlines extensions to higher dimensions, positive characteristic, and Grassmannian lifts, aiming to connect geproci geometry with finite subgroups of and to develop a unifying groupoid-theoretic perspective. The work provides a concrete computational framework (via Macaulay2) and establishes a basis for future exploration of more complex configurations and their symmetries, including potential octahedral and tetrahedral actions in related settings.

Abstract

In this paper, we investigate groupoids coming from configurations of lines in three-dimensional space. Given a point and two skew lines in over a field , there exists a unique line containing the given point and meeting the two given lines. We use this construction to define a projection function from one line to another by using a skew line as an auxiliary. This way, we may create a groupoid whose objects are lines in a configuration, and whose morphisms are induced by these projection functions. We look at specific configurations for that yield groupoids with finite automorphism groups.

Paper Structure

This paper contains 13 sections, 14 theorems, 69 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction & History
  2. Background
  3. The $D_4$ Configuration
  4. The Penrose Configuration
  5. The Groupoid of the half-Penrose Spread
  6. The Groupoid of the Penrose Spread
  7. The Double Penrose Configuration
  8. The Klein Configuration
  9. Future Work
  10. Groupoids with Infinite and Icosahedral Automorphism Group
  11. Higher-Dimensional Configurations
  12. Configurations in Positive Characteristic
  13. Lifts to the Grassmannian

Key Result

Proposition 1

Let $\mathcal{L}$ be a set of lines in $\mathbb{P}^3$. Denote by $\Pi$ the set of all compositions of functions of the form $\pi(L_1,L_2,L_3)$, with $L_1,L_2,L_3\in\mathcal{L}$ and $L_1\cap L_2=L_2\cap L_3=\varnothing$. Then $(\mathcal{L},\Pi)$ is a groupoid.

Figures (1)

  • Figure 1: A labeling of the $D_4$ configuration

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Proposition 1
  • Definition 4
  • Definition 5
  • Example 1
  • Example 2
  • Theorem 1
  • ...and 26 more