Birational Geometry of Matroids and Abstract Hyperplane Arrangements

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.