Arxiv:1409.3503V1 [Math.AG] 11 Sep 2014 Htddntcm Rmasto Etr Vrapriua field Particular a Over Vectors T of That Set Noticed a Whitney from Matroid
MATROID THEORY FOR ALGEBRAIC GEOMETERS ERIC KATZ Abstract. This article is a survey of matroid theory aimed at algebraic geometers. Ma- troids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be representable. Still, one may apply linear algebraic constructions to non-representable matroids. There are a number of different definitions of matroids, a phenomenon known as cryptomorphism. In this survey, we begin by reviewing the classical definitions of matroids, develop operations in matroid theory, summarize some results in representability, and construct polynomial invariants of matroids. Afterwards, we focus on matroid polytopes, introduced by Gelfand- Goresky-MacPherson-Serganova, which give a cryptomorphic definition of matroids. We explain certain locally closed subsets of the Grassmannian, thin Schubert cells, which are labeled by matroids, and which have applications to representability, moduli problems, and invariants of matroids following Fink-Speyer. We explain how matroids can be thought of as cohomology classes in a particular toric variety, the permutohedral variety, by means of Bergman fans, and apply this description to give an exposition of the proof of log-concavity of the characteristic polynomial of representable matroids due to the author with Huh. 1. Introduction This survey is an introduction to matroids for algebraic geometry-minded readers. Ma- troids are a combinatorial abstraction of linear subspaces of a vector space with distinguished basis or, equivalently, a set of labeled set of vectors in a vector space. Alternatively, they are a generalization of graphs and are therefore amenable to a structure theory similar to that of graphs.
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