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

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