Geometry of Matroids and Hyperplane Arrangements

Abstract

There is a trinity relationship between hyperplane arrangements, matroids and convex polytopes. We expand and combinatorialize it as settling the complexity issue expected by Mnev's universality theorem. Based on this theory, we show that for n less than or equal to 9 every matroid tiling in the hypersimplex Delta(3,n) associated to a weighted stable hyperplane arrangement extends to a matroid subdivision of Delta(3,n) and that the bound 9 for n is sharp. As a straightforward application, we completely answer Alexeev's algebro-geometric question.Comment: 46 pages, 24 figures; v3: minor improvement