Matroids satisfying the matroidal Cayley--Bacharach property and ranks of covering flats

TL;DR

This paper investigates matroids with a Cayley-Bacharach-like property, showing that for certain coverings, ranks of flats are unbounded, and explores their relation to generalized permutohedra and graphic matroids.

Contribution

It demonstrates that matroids satisfying the matroidal Cayley-Bacharach condition can have unbounded ranks of flats covering the set, addressing a question about their structural properties.

Findings

No bound on ranks of flats covering the set for certain matroids.

Connections between matroidal Cayley-Bacharach property and generalized permutohedra.

Relation between this property and the structure of graphic matroids.

Abstract

Let $M$ be a matroid satisfying a matroidal analogue of the Cayley-Bacharach condition. Given a number $k \ge 2$, we show that there is no nontrivial bound on ranks of a $k$-tuple of flats covering the underlying set of $M$. This addresses a question of Levinson-Ullery motivated by earlier results which show that bounding the number of points satisfying the Cayley-Bacharach condition forces them to lie on low-dimensional linear subspaces. We also explore the general question what matroids satisfy the matroidal Cayley-Bacharach condition of a given degree and its relation to the geometry of generalized permutohedra and graphic matroids.