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

TL;DR

This paper constructs and analyzes finite groupoids arising from line configurations in complex projective three-space, revealing new algebraic structures and symmetries in geometric arrangements.

Contribution

It introduces a novel method to define groupoids from line configurations using projection functions, highlighting cases with finite automorphism groups over the complex numbers.

Findings

Constructed groupoids from line configurations in $ ext{P}^3_{ ext{C}}$

Identified configurations with finite automorphism groups

Provided a framework linking geometric configurations to algebraic structures

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.