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

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.

arXiv commentSimplified and clarified main results and their proofs, adjusted definition to be closer to the geometric analogue; 19 pages