Varieties of Lines in 3-Space

TL;DR

This work develops a geometric framework for configurations of lines in $\mathbb{P}^3$ governed by edge-incidence graphs $G$. By embedding line configurations into $V_G\subseteq {\rm Gr}(2,4)^\ell$ and connecting to rigidity theory, the authors obtain a comprehensive picture of dimension, irreducibility, and complete-intersection criteria via contraction stability, plus a multidegree (Schubert-calculus) interpretation that encodes leading-singularity data through augmented graphs $G_u$. A practical spanning-tree coordinate system is introduced to enable efficient symbolic and numerical computations, and extensive numerical decompositions are produced for up to eight lines, revealing intricate component structures and realisability phenomena. The framework directly informs Landau analysis in scattering amplitudes, linking algebraic geometry of line incidences to physics through momentum-twistor variables and Schubert problems. Overall, the paper provides both foundational theory and computational tools for incidence varieties of lines with broad applications in mathematics and high-energy physics.

Abstract

We consider configurations of lines in 3-space with incidences prescribed by a graph. This defines a subvariety in a product of Grassmannians. Leveraging a connection with rigidity theory in the plane, for any graph, we determine the dimension of the incidence variety and characterize when it is irreducible or a complete intersection. We study its multidegree and the family of Schubert problems it encodes. Our spanning-tree coordinates enable efficient symbolic computations. We also provide numerical irreducible decompositions for incidence varieties with up to eight lines. These constructions with lines play a key role in the Landau analysis of scattering amplitudes in particle physics.

Figures (4)

• Figure 1: The four irreducible components in the incidence variety for the graph $G$ in Example \ref{['ex:whiteandblack']}. The last row shows the configurations of four lines given by each component.
• Figure 2: Bases and spanning circuits of the rigidity matroid for $\ell = 4, 5, 6$.
• Figure 3: This graph with $\ell \!=\! 10$ is $(2,4)$-sparse, $K_{2,4}$-free, and not strictly contraction-stable.
• Figure 4: Histogram for the number of components of $V_G$ for connected graphs with $\ell=6,7$.

Theorems & Definitions (75)

• Example 1.1: $\ell=3$
• Example 2.1: Complete graphs
• Example 2.2: $\ell = 4$
• Example 2.3: $\ell=5, |G| = 7$
• Example 2.4: $\ell=5$, not CI
• Example 2.5: $\ell=3$
• Proposition 2.6
• proof : Proof and Discussion
• Example 3.1: $\ell=1$
• Example 3.2: $\ell=3$