Algebraic Geometry of Cactus, Pascal, and Pappus Matroids

TL;DR

This paper investigates rank-three matroids, especially classical configurations like Pascal and Pappus, providing explicit equations for their realization spaces and proving realizability and irreducibility of cactus matroids.

Contribution

It introduces a family of cactus graph-based matroids, computes explicit defining equations for their varieties, and proves key properties like realizability and irreducibility.

Findings

Explicit generating sets for matroid ideals of Pascal, Pappus, and cactus matroids.

Proof that all cactus matroids are realizable.

Matroid varieties are irreducible.

Abstract

We study rank-three matroids, known as point-line configurations, and their associated matroid varieties, defined as the Zariski closures of their realization spaces. Our focus is on determining finite generating sets of defining equations for these varieties, up to radical, and describing the irreducible components of the corresponding circuit varieties. We generalize the notion of cactus graphs to matroids, introducing a family of point-line configurations whose underlying graphs are cacti. Our analysis includes several classical matroids, such as the Pascal, Pappus, and cactus matroids, for which we provide explicit finite generating sets for their associated matroid ideals. The matroid ideal is the ideal of the matroid variety, whose construction involves a saturation step with respect to all the independence relations of the matroid. This step is computationally very expensive and has only been carried out for very small matroids. We provide a complete generating set of these ideals for the Pascal, Pappus, and cactus matroids. The proofs rely on classical geometric techniques, including liftability arguments and the Grassmann--Cayley algebra, which we use to construct so-called bracket polynomials in these ideals. In addition, we prove that every cactus matroid is realizable and that its matroid variety is irreducible.