Varieties of Lines in 3-Space

TL;DR

This paper investigates the geometric and algebraic properties of line configurations in 3-space constrained by graph incidences, connecting to rigidity theory and applications in physics, and provides computational tools for their analysis.

Contribution

It characterizes the dimension, irreducibility, and multidegree of incidence varieties of lines in 3-space, introducing spanning-tree coordinates for efficient computation.

Findings

Determined the dimension of incidence varieties for any graph.

Characterized when these varieties are irreducible or complete intersections.

Developed computational methods including symbolic and numerical decompositions.

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.