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arXiv · 2511.21333

Varieties of Lines in 3-Space

Benjamin Hollering·Elia Mazzucchelli·Matteo Parisi·Bernd Sturmfels
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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.

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BibTeXRIS

Benjamin Hollering, Elia Mazzucchelli, Matteo Parisi, Bernd Sturmfels. 2025-11-26. Varieties of Lines in 3-Space. https://arxiv.org/abs/2511.21333

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