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

2022/03/28 by Soohyun Park, Park, Soohyun
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2203.14953

openalex publication_date 2022/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Matroidal Cayley-Bacharach and independence/dependence of geometric properties of matroids
• The number of rank-k flats in a matroid with no U_2,n-minor
• Unavoidable flats in matroids representable over prime fields
• Extending matroid classes closed under flats
• Characterizing real-representable matroids with large average hyperplane-size