Finite subgroups of $\operatorname{PGL}_2(K)$ arising from configurations of skew lines in $\mathbb{P}^3_K$

TL;DR

This paper classifies finite subgroups of PGL(2,K) arising from configurations of skew lines in projective 3-space, providing explicit constructions and restrictions for abelian and non-abelian groups.

Contribution

It introduces a matrix-based approach to analyze the groups from skew line configurations, explicitly constructing certain groups and ruling out others.

Findings

Realizes cyclic and elementary abelian p-groups as automorphism groups

Proves dihedral groups with n≥3 cannot occur in this context

Constructs examples with groups isomorphic to A4, S4, A5, and certain semidirect products

Abstract

We study finite groups arising from configurations of skew lines in $\mathbb{P}^3_K$. Given a finite set $L$ of pairwise skew lines in $\mathbb{P}^3_K$, one naturally obtains a group $G_{L}\subset \mathrm{Aut}(L_i)\cong \mathrm{PGL}_2(K)$ acting on each line of the configuration. We investigate which finite subgroups of $\mathrm{PGL}_2(K)$ can occur in this way. Our main tool is a matrix description of skew lines in $\mathbb{P}^3_K$, which allows us to express the generators of $G_{L}$ explicitly in terms of a family of matrices $M_i\in \mathrm{GL}_2(K)$. This turns the problem into a concrete linear-algebraic one and makes it possible to analyze both the abelian and the non-abelian cases. In the abelian case, we show that after a change of basis the matrices $M_i$ are simultaneously upper triangular, and we obtain explicit families realizing cyclic groups and elementary abelian $p$-groups. In the non-abelian case, we prove that no dihedral group $D_n$ with $n\ge 3$ can occur, while we construct configurations with $G_{L}\cong A_4$, $S_4$, and $A_5$, describe the corresponding orbit structure, and exhibit affine positive-characteristic examples of groups of the form $C_p\rtimes C_{p-1}$. These results also have applications to the geometry of point sets whose general projection is a complete intersection: they make more explicit the group-theoretic viewpoint on collinearly complete sets and yield new examples of half-grid geproci sets.