Varieties of Lines in 3-Space

This page contains supplementary files for the paper Varieties of Lines in 3-Space by Benjamin Hollering, Elia Mazzucchelli, Matteo Parisi, and Bernd Sturmfels. 

In this paper 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 also study its multidegree, showing that it encodes a family of Schubert problems. We further introduce spanning-tree coordinates that enable efficient symbolic computations, and provide full numerical irreducible decompositions for incidence varieties with up to eight lines. These constructions play a key role in the Landau analysis of scattering amplitudes in particle physics.

Code:

In Section 2 of our paper, we present a survey of line incidence varieties on 5 nodes. These corresponding ideals are small enough to be handled symbolically, especially after passing to the spanning tree coordinates discussed in Section 6. Code for creating the line incidence ideal is available in both Julia and Macaulay2. In Macaulay2 there is code available for creating the ideal in every coordinate system we describe throughout the paper while in Julia this is limited to the original Plücker coordinates, the affine coordinates, and the Elekes-Sharir framework. Various helper functions are also provided such as making the concurrent lines ideal or checking the rank of a given graph in the rigidity matroid. This code can be found in the GitHub repository linked below. 

Data:

In Section 7 we present a summary of the properties of incidence varieties of triangle-free graphs on $\ell = 5, 6, 7, 8$ nodes and all graphs on $\ell = 5,6,7$ nodes. To do this, we computed a numerical irreducible decomposition of the incidence variety using the function numerical_irreducible_decomposition in HomotopyContinuation.jl. From this, we then determine if a given incidence variety, is irreducible or a complete intersection by simply checking if our numerical irreducible decomposition consists of one component and has correct codimension respectively. The last property we report on is whether or not a graph is realizable, meaning $W_G \neq \emptyset$. To check this, for every component we determine which nonedges $ij \notin G$ vanish numerically by which we mean

$${\rm max}_{p \in W} |A_iA_j(p)|  < \varepsilon \, , \quad
\qquad \hbox{for some tolerance $\varepsilon>0$} \, .$$

In the files below for each graph we provide a more detailed breakdown of our numerical irreducible decomposition which consists of the following data:

• A drawing of the graph
• the edge set of the graph, the number of edges of the graph, and its rank in the rigidity matroid
• a table which consists of the dimension, codimension, and degree of each component of our variety which appears in our witness set
• whether or not the incidence variety is irreducible, a complete intersection, or realizable
• whether or not our witness set is complete
• for each component of our witness set, a list of non-edges of the graph which vanish on that component

This more detailed breakdown of the data can be found in the pdfs below and is available for incidence varieties of triangle-free graphs on $\ell = 5, 6, 7, 8$ nodes and all graphs on $\ell = 5,6 $ nodes. We note that this data comes with one main caveat which concerns whether or not our witness set is complete. In the case that we detect that our graph is a complete intersection, we know the degree of our incidence variety and thus can correctly detect when we have the correct number of points in our witness set. When the incidence variety is not a complete intersection, our witness set might be incomplete and we would not know. We know this happens for some incidence varieties which are not reduced, such as the graph $K_{2,3}'$ which is the first example of a graph where $W_G \neq \emptyset$ but $Y_G = \emptyset$. $W_G$ is a non-reduced primary component of $I_G$ which does not appear in our numerical irreducible decomposition due to a known error in the implementation. Thus if our witness set is incomplete, our results on realizability may be flawed, especially when $I_G$ is not radical.