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 ≥ 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.