Matroidal Cayley-Bacharach and independence/dependence of geometric properties of matroids

TL;DR

This paper explores the relationship between a matroidal Cayley-Bacharach property and geometric properties of matroids, focusing on nestohedra, paving matroids, and supersolvable arrangements, revealing structural and combinatorial insights.

Contribution

It establishes how the matroidal Cayley-Bacharach property relates to matroid structures like nestohedra and hyperplane arrangements, offering new connections between combinatorics and geometry.

Findings

MCB(a) is determined by building set structures in nestohedra.

Connections between minimal degrees and matroid geometry are identified.

Analysis of supersolvable arrangements links independence conditions to recursive matroid properties.

Abstract

We consider the relationship between a matroidal analogue of the degree $a$ Cayley-Bacharach property (finite sets of points failing to impose independent conditions on degree $a$ hypersurfaces) and geometric properties of matroids. If the matroid polytopes in question are nestohedra, we show that the minimal degree matroidal Cayley-Bacharach property denoted $MCB(a)$ is determined by the structure of the building sets used to construct them. This analysis also applies for other degrees $a$. Also, it does not seem to affect the combinatorial equivalence class of the matroid polytope. However, there are close connections to minimal nontrivial degrees $a$ and the geometry of the matroids in question for paving matroids (which are conjecturally generic among matroids of a given rank) and matroids constructed out of supersolvable hyperplane arrangements. The case of paving matroids is still related to with properties of building sets since it is closely connected to (Hilbert series of) Chow rings of matroids, which are combinatorial models of the cohomology of wonderful compactifications. Finally, our analysis of supersolvable line and hyperplane arrangements give a family of matroids which are natrually related to independence conditions imposed by points one plane curves or can be analyzed recursively.