Combinatorics of skew lines in $\mathbb P^3$ with an application to algebraic geometry

Abstract

This article introduces a previously unrecognized combinatorial structure underlying configurations of skew lines in $\mathbb{P}^3$, and reveals its deep and surprising connection to the algebro-geometric concept of geproci sets. Given any field $\mathbb{K}$ and a finite set $\mathcal L$ of 3 or more skew lines in $\mathbb{P}^3_\mathbb{K}$, we associate to it a group $G_{\mathcal L}$ and a groupoid $C_{\mathcal L}$ whose action on the union $\cup_{L\in\mathcal L}L$ provides orbits which have a rich combinatorial structure. We characterize when $G_{\mathcal L}$ is abelian and give partial results on its finiteness. The notion of \emph{collinearly complete} subsets is introduced and shown to correspond exactly to unions of groupoid orbits. In the case where $\mathbb{K}$ is a finite field and $\mathcal L$ is a full spread in $\mathbb{P}^3_\mathbb{K}$ (i.e., every point of $\mathbb{P}^3_\mathbb{K}$ lies on a line in $\mathcal{L}$), we prove that $G_{\mathcal L}$ being abelian characterizes the classical spread given by the fibers of the Hopf fibration. Over any algebraically closed field, we establish that finite unions of $C_{\mathcal L}$-orbits are geproci sets - that is, finite sets whose general projections to a plane are complete intersections. Furthermore, we prove a converse: if $\mathbb{K}$ is algebraically closed and $Z \subset \mathbb{P}^3_\mathbb{K}$ is a geproci set consisting of $m$ points on each of $s \geq 3$ skew lines $\mathcal L$ where the general projection of $Z$ is a complete intersection of type $(m, s)$, then $Z$ is a finite union of orbits of $C_{\mathcal L}$. This work thus uncovers a profound combinatorial framework governing geproci sets, providing a new bridge between incidence combinatorics and algebraic geometry.