Geometry of Matroids and Hyperplane Arrangements

Jaeho Shin

Published 2019-12-28Version 1

There is a trichotomic relation between hyperplane arrangements, convex polytopes and matroids. We expand this theory while resolving the complexity issue expected by Mnev's universality theorem. In particular, we invent a combinatorial apparatus for the geometry of hyperplane arrangements in terms of semilattices and their operations, for instance, puzzle-pieces and the matroidal MMP. We also investigate matroid tilings and their extensions. As an algebro-geometric application, we answer Alexeev's question.