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

Giuseppe Favacchio

TL;DR

This paper investigates which finite subgroups of $\mathrm{PGL}_2(K)$ arise from configurations of skew lines in $\mathbb{P}^3_K$ by translating geometric data into a matrix framework with $M_i\in {\rm GL}_2(K)$. It shows that the abelian case yields simultaneous triangularization of the $M_i$, producing cyclic and $p$-semi-elementary groups, while the non-abelian case rules out dihedral groups $D_n$ for $n\ge3$ and constructs realizations of $A_4$, $S_4$, and $A_5$, along with affine $p$-semi-elementary examples. These results connect to $(a,b)$-geproci sets and half-grid geproci configurations, providing a group-theoretic lens on collinearly complete point sets. The work blends explicit matrix realizations with orbit-structure analysis to map the landscape of finite subgroups that can occur from skew-line geometries and raises questions about broader realizations, including larger Lie-type groups.

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 $\mathbb{P}^3_K$ and the associated groupoid $C_{L}$, we consider the endomorphism group $G_{L} \subset \operatorname{Aut}(L_i) \cong \operatorname{PGL}_2(K)$ for any line $L_i \in {L}$, and we ask which finite subgroups of $\operatorname{PGL}_2(K)$ can occur in this way. Using a matrix description of skew lines, we express the generators of $G_{L}$ in terms of a family of matrices $M_i \in \operatorname{GL}_2(K)$ and analyze $G_{L}$ in the abelian and non-abelian cases. In the abelian situation 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 $p$-semi-elementary groups of the form $C_p^m \rtimes C_n$. In the non-abelian case we prove that no dihedral group $D_n$ with $n \ge 3$ can occur, while we construct configurations with $G_{L} \cong A_4, S_4, A_5$ and describe their orbit structure. Viewed through the lens of $(a,b)$-geproci sets, these results provide a group-theoretic description of collinearly complete point sets and yield new examples of half-grid geproci sets.

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

TL;DR

This paper investigates which finite subgroups of arise from configurations of skew lines in by translating geometric data into a matrix framework with . It shows that the abelian case yields simultaneous triangularization of the , producing cyclic and -semi-elementary groups, while the non-abelian case rules out dihedral groups for and constructs realizations of , , and , along with affine -semi-elementary examples. These results connect to -geproci sets and half-grid geproci configurations, providing a group-theoretic lens on collinearly complete point sets. The work blends explicit matrix realizations with orbit-structure analysis to map the landscape of finite subgroups that can occur from skew-line geometries and raises questions about broader realizations, including larger Lie-type groups.

Abstract

We study finite groups arising from configurations of skew lines in . Given a finite set of pairwise skew lines in and the associated groupoid , we consider the endomorphism group for any line , and we ask which finite subgroups of can occur in this way. Using a matrix description of skew lines, we express the generators of in terms of a family of matrices and analyze in the abelian and non-abelian cases. In the abelian situation we show that, after a change of basis, the matrices are simultaneously upper triangular and we obtain explicit families realizing cyclic groups and -semi-elementary groups of the form . In the non-abelian case we prove that no dihedral group with can occur, while we construct configurations with and describe their orbit structure. Viewed through the lens of -geproci sets, these results provide a group-theoretic description of collinearly complete point sets and yield new examples of half-grid geproci sets.
Paper Structure (7 sections, 18 theorems, 120 equations)

This paper contains 7 sections, 18 theorems, 120 equations.

Table of Contents

  1. Introduction and Background
  2. Finite subgroups of PGL2(K) over an algebraically closed field
  3. Overview of Results
  4. Identifying sets of skew lines in P3(K) via matrices in GL2(K)
  5. The abelian case
  6. The case of the standard construction
  7. Non-abelian case

Key Result

Lemma 2.2

The lines $L_i$ and $L_j$ are skew if and only if $M_i-M_j$ is nonsingular.

Theorems & Definitions (48)

  • Remark 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 38 more