A note on algebraic rank, matroids, and metrized complexes

TL;DR

This paper explores the relationship between algebraic rank, matroids, and metrized complexes, revealing field-dependent behaviors and challenging assumptions about divisor smoothability in algebraic geometry.

Contribution

It demonstrates that algebraic rank can depend on the ground field and shows that equality of algebraic and combinatorial rank does not guarantee smoothability of divisors.

Findings

Algebraic rank varies with the ground field in certain graphs.

Equality of algebraic and combinatorial rank is not sufficient for smoothability.

Provides examples linking matroid realizability to algebraic rank.

Abstract

We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory of metrized complexes to show that equality between the algebraic and combinatorial rank is not a sufficient condition for smoothability of divisors, thus giving a negative answer to a question posed by Caporaso, Melo, and the author.