Divisors on matroids and their volumes

TL;DR

This paper introduces a volume polynomial for matroids, providing a combinatorial formula and connecting it to algebraic geometry, specifically the volumes of divisors on blow-ups of projective spaces.

Contribution

It defines a new volume polynomial for matroids, offers a complete combinatorial formula, and links matroid invariants to algebraic geometric concepts.

Findings

Explicit formula for volume polynomial of realizable matroids

Introduction of matroid volume as a new invariant

Connections between matroid volume and algebraic geometry

Abstract

The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, and give a complete combinatorial formula. For a realizable matroid, we thus obtain an explicit formula for the classical volume polynomial of the associated wonderful compactification. We then introduce a new invariant called the volume of a matroid as a particular specialization of its volume polynomial, and discuss its algebro-geometric and combinatorial properties in connection to graded linear series on blow-ups of projective spaces.