[Paper Review] Birational Geometry of Matroids and Abstract Hyperplane Arrangements

Jaeho Shin|arXiv (Cornell University)|Dec 28, 2019

Polynomial and algebraic computation20 references4 citations

TL;DR

This paper establishes a unified birational matroid geometry framework linking matroids, matroid polytopes, and abstract hyperplane arrangements, resolving long-standing gaps in matroid subdivisions and face structures. It overcomes Mnëv’s universality theorem by proving that for n ≤ 9, every weighted rank-3 matroid tiling extends to a matroid subdivision of the hypersimplex Δ(3,n), and the reduction morphism between moduli spaces of weighted stable line arrangements is surjective, with n = 9 being the sharp bound.

ABSTRACT

A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering hyperplane arrangements, which give rise to convex polytopes called matroid polytopes. Much research has been conducted on these three areas: matroids, matroid polytopes, and hyperplane arrangements. However, substantial gaps in our knowledge remain, and the correspondence diagram between those areas needs to be more extensive. For instance, currently, there is no matroid counterpart of a matroid subdivision, and only some matroid subdivisions are associated with stable hyperplane arrangements. Moreover, we need a deeper understanding of the face structure of a matroid polytope and how to glue or subdivide base polytopes; the latter requires overcoming Mnev's universality theorem. Another interesting question is whether the birational geometry of hyperplane arrangements can be implemented over matroids. In this paper, we develop a theory that integrates the three areas into a trinity relationship and provide solutions to the aforementioned questions while answering as many as possible.

Motivation & Objective

• To close critical gaps in the correspondence between matroids, matroid polytopes, and hyperplane arrangements.
• To establish a matroid counterpart for matroid subdivisions, which was previously missing.
• To resolve the challenge of gluing and subdividing base polytopes despite Mnëv’s universality theorem.
• To implement birational geometry of hyperplane arrangements over matroids through a direct, flats-based approach.

Proposed method

• Introduces new notions: abstract hyperplane arrangements, labeled arrangements with weight structures, and the attaché operation over matroids.
• Develops a binary operation ‘⊛’ on matroidal expressions to model intersections of independence polytopes.
• Uses the lattice ℓ(M) of direct sums of disjoint minor expressions to analyze face structures and matroid intersections.
• Applies a direct approach focused on flats and rank functions, avoiding traditional dual structures like fans or secondary polytopes.
• Establishes equivalence conditions for non-empty intersections of matroid polytopes via permutation-invariant constructions.
• Proves that the intersection of independence polytopes over all permutations of flats equals the independence polytope of the ‘⊛’-constructed matroid.

Experimental results

Research questions

• RQ1 Can a matroid counterpart for matroid subdivisions be constructed, given that only some such subdivisions are linked to stable hyperplane arrangements?
• RQ2 How can the face structure of a matroid polytope be deeply understood and systematically manipulated for gluing or subdivision?
• RQ3 Can Mnëv’s universality theorem be overcome in the context of matroid subdivisions and polyhedral geometry?
• RQ4 Is it possible to implement the birational geometry of hyperplane arrangements over matroids in a coherent and constructive way?
• RQ5 What is the precise condition under which a weighted tiling of a hypersimplex extends to a full matroid subdivision?

Key findings

• For n ≤ 9, every rank-3 and (n-1)-dimensional weighted or admissible matroid tiling in ℝ^n extends to a matroid subdivision of the hypersimplex Δ(3,n), and this bound is tight.
• The reduction morphism ρw,v between moduli spaces of weighted stable n-line arrangements is surjective for all w > v when n ≤ 9, and n = 9 is the sharp bound.
• The intersection of independence polytopes over all permutations of a set of flats equals the independence polytope of the ‘⊛’-constructed matroid.
• A non-empty intersection of matroid polytopes M(Fi) is equivalent to the ‘⊛’-constructed matroid having full rank r(M).
• The equivalence of multiple conditions (e.g., common base existence, permutation invariance, full rank) characterizes when matroid intersections are non-empty.
• The theory successfully overcomes Mnëv’s universality theorem by constructing a direct, flats-based framework that avoids the usual complexity traps.