Enumerative geometry of skew lines in $\mathbb P^3$ with a given associated finite group

Abstract: For any finite set $\mathcal L$ of 3 or more skew lines in $\mathbb P<sup>3_{\overline{K}}$ over an algebraically closed field $\overline{K}$ of arbitrary characteristic, there is a canonical associated subgroup $G_{\mathcal L}$ of ${\rm PGL}<em>2(\overline{K})$. Given a finite subgroup $G\subset{\rm PGL}_2(\overline{K})$ we study which configurations of lines have $G</em>{\mathcal L}=G$. We derive an upper bound on the number $|\mathcal L|$ of lines in terms of the order $|G|$ of the group $G$ and as an application we classify up to projective equivalence which sets $\mathcal L$ in $\mathbb P^3_{\mathbb C}$ have $G_{\mathcal L}=G$ for certain finite nonabelian groups $G$.

• The paper establishes a classification of skew lines in ℙ³ by linking configurations to finite subgroups of PGL₂ through a novel groupoid construction.
• It derives an explicit upper bound |₍G₎| ≤ (|G|-2)(|G|-1)² + 2, ensuring finiteness of configurations for any prescribed finite group.
• The study provides detailed classification results for nonabelian groups like A₄ and S₄, connecting geometric configuration with combinatorial and group theoretic methods.

Enumerative Geometry of Skew Lines in $\mathbb{P}^3$ with a Given Associated Finite Group

Problem Formulation and Context

The paper introduces a classification problem within the framework of enumerative algebraic geometry, focusing on finite configurations of skew lines in $\mathbb{P}^3$ and their canonical association to finite subgroups $G \subset \mathrm{PGL}_2(K)$, where $K$ is an algebraically closed field of arbitrary characteristic. The central question is to determine, up to projective equivalence, which sets of skew lines correspond to a prescribed subgroup $G$ via the groupoid construction developed in prior works such as [POLITUS3].

A key insight is the transferability of combinatorial results on "spreads" (maximal sets of skew lines) over finite fields to the context of arbitrary fields by imposing group-theoretic constraints on the set of lines. Over fields such as $\mathbb{C}$, the classification problem is infinite already for four skew lines due to the cross-ratio variation, but for configurations constrained by a finite $G$, the classification becomes finite and computable.

Groupoid Association and Matrix Parametrization

Given $s \geq 3$ pairwise skew lines $\mathcal{L} = \{L_1, \ldots, L_s\}$ in $\mathbb{P}^3$, a groupoid $\mathbb{P}^3$0 is constructed whose morphisms correspond to invertible geometric transitions defined using triple intersections. The automorphism group for any line in $\mathbb{P}^3$1, denoted $\mathbb{P}^3$2, sits inside $\mathbb{P}^3$3 and is invariant under projective transformations of the configuration.

After normalizing coordinates so that $\mathbb{P}^3$4 are part of the configuration, every additional line is uniquely represented by a matrix $\mathbb{P}^3$5. The group $\mathbb{P}^3$6 is generated by both the conjugacy classes $\mathbb{P}^3$7 and the differences $\mathbb{P}^3$8 for all $\mathbb{P}^3$9, a geometric subtlety that arises from the condition of pairwise skewness (requiring invertibility of $G \subset \mathrm{PGL}_2(K)$0).

Finiteness Result and Explicit Upper Bounds

The main theorem establishes that for any fixed finite subgroup $G \subset \mathrm{PGL}_2(K)$1, there exists a unique minimal finite set $G \subset \mathrm{PGL}_2(K)$2 of lines in the normalized model such that any configuration $G \subset \mathrm{PGL}_2(K)$3 with $G \subset \mathrm{PGL}_2(K)$4 must satisfy $G \subset \mathrm{PGL}_2(K)$5. An explicit upper bound is derived:

$G \subset \mathrm{PGL}_2(K)$6

This holds regardless of field characteristic (under mild restrictions), and the bound is tight for small groups but grows rapidly for groups such as $G \subset \mathrm{PGL}_2(K)$7, $G \subset \mathrm{PGL}_2(K)$8, and $G \subset \mathrm{PGL}_2(K)$9.

Further, configurations with more than 201900 skew lines in characteristic zero must correspond either to infinite or to cyclic groups, since for $K$0, $K$1, and $K$2 the bound is always below this threshold.

Classification for Nonabelian Groups: $K$3 and $K$4

The paper applies the general finiteness theorem to classify, up to projective equivalence, configurations with $K$5 or $K$6 over $K$7:

$K$8 case:

• There is a unique configuration (up to projective equivalence) of five skew lines with $K$9. The group action on candidate matrices is described via explicit generators from the binary tetrahedral group, and admissibility conditions are encoded in a compatibility graph derived from the group structure.

$G$0 case:

• Generic configurations with $G$1 have between five and ten lines. The paper enumerates cliques in the compatibility graph and identifies maximal and minimal $G$2-generating subsets, showing uniqueness for extremal cases and constructing explicit representatives for conjugacy classes. Notably, not all configurations arise from subconfigurations of maximal ten-line examples.

The paper provides constructive methods, including projective transformations and conjugacy action in $G$3 and $G$4, to establish projective equivalence classes. In all cases, the group $G$5 and the geometry of the line configuration are tightly intertwined.

Implications and Connections

The groupoid perspective links the algebraic structure of finite group actions with geometric properties of line configurations. These results have implications for the classification of "geproci sets" (finite point sets whose general projection yields a complete intersection), as such sets often arise as groupoid orbits in skew-line configurations. The explicit classification for nonabelian cases fills a gap left by prior studies focused on abelian structures.

The theory suggests that combinatorial and algebraic methods—such as compatibility graphs and cycle decompositions—are essential tools for understanding the enumerative geometry of line spreads beyond finite fields. Furthermore, the correspondence between projective equivalence and conjugacy classes in group actions provides a pathway for generalizing these results to other configurations of subspaces.

Theoretical and Practical Outlook

The methodology and results enable systematic study of the interplay between algebraic groups and geometric configurations, particularly in algebraic geometry and combinatorial design theory. The techniques could inspire further research on higher-dimensional analogues, classifications in positive characteristic, and connections to moduli spaces.

Future developments might include automation of compatibility graph enumeration for larger or more complex groups, application to intersection theory of higher-dimensional subspaces, and exploration of links to geometric representation theory. The groupoid construction may also find roles in the study of point-line incidence structures, coding theory, and quantum information geometry.

Conclusion

The paper rigorously classifies finite configurations of skew lines in $G$6 associated to prescribed finite subgroups of $G$7, deriving explicit bounds, structural properties, and projective equivalence classes with special attention to nonabelian cases. The results elucidate the role of groupoids and compatibility graphs in enumerative geometry, provide powerful combinatorial tools for configuration classification, and open avenues for further research within algebraic geometry and related fields.