On algebraic properties of matroid polytopes

TL;DR

This paper classifies matroids based on algebraic properties of their associated polytopes, specifically identifying which are Gorenstein or smooth, advancing the understanding of their algebraic structure.

Contribution

It provides a complete classification of matroids with Gorenstein or smooth polytopes, answering a question posed by Herzog and Hibi.

Findings

Matroid base and independence polytopes are Cohen--Macaulay.

Characterization of Gorenstein matroid polytopes.

Characterization of smooth matroid polytopes.

Abstract

A toric variety is constructed from a lattice polytope. It is common in algebraic combinatorics to carry this way a notion of an algebraic property from the variety to the polytope. From the combinatorial point of view, one of the most interesting constructions of toric varieties comes from the base polytope of a matroid. Matroid base polytopes and independence polytopes are Cohen--Macaulay. We study two natural stronger algebraic properties -- Gorenstein and smooth. We provide a full classifications of matroids whose independence polytope or base polytope is smooth or Gorenstein. The latter answers to a question raised by Herzog and Hibi.