Minimal matroids in dependency posets: algorithms and applications to computing irreducible decompositions of circuit varieties
Emiliano Liwski, Fatemeh Mohammadi
TL;DR
This work develops an algorithmic framework to identify minimal matroids of point-line configurations with respect to the dependency poset and uses these to construct irreducible decompositions of circuit varieties. By partitioning the search into $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ and combining the results, the authors enable a principled decomposition strategy that reduces complex varieties to manageable, irreducible components, especially for symmetric configurations such as the Fano plane, MacLane, and the affine plane of order $3$. They demonstrate the approach on several classical $v_3$ configurations, reveal new irreducible decompositions, and connect their framework to the theory of $\mathcal{X}$-matroids, including conjectures about uniqueness and a refined structural view via extended rank bounds. The paper also develops redundancy-detection methods using realization spaces and perturbations to prune unnecessary components, providing a practical toolkit for studying circuit and matroid varieties in combinatorial geometry.
Abstract
We study point-line configurations, their minimal matroids, and their associated circuit varieties. We present an algorithm for identifying the minimal matroids of these configurations with respect to dependency order, or equivalently, the maximal matroids with respect to weak order, and use it to determine the irreducible decomposition of their corresponding circuit varieties. Our algorithm is applied to several classical configurations, including the Fano matroid, affine plane of order three, MacLane, and Pappus configurations. Additionally, we explore the connection to a conjecture by Jackson and Tanigawa, which provides a criterion for the uniqueness of the minimal matroids.