Lifting matroid divisors on tropical curves

TL;DR

This paper explores the relationship between tropical geometry, matroid realizability, and the lifting problem for divisors on tropical curves, revealing characteristic-dependent existence and providing explicit bounds related to Mn"ev universality.

Contribution

It establishes a connection between the lifting problem and matroid realizability, providing explicit bounds and demonstrating characteristic-dependent existence of certain tropical curves.

Findings

Existence of tropical curves depends on field characteristic.

Relates lifting problem to matroid realizability.

Provides explicit bounds on matroid size in Mn"ev universality.

Abstract

Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain characteristics or over certain fields of definition. Our examples also apply to the theory of metrized complexes and weighted graphs. These examples arise by relating the lifting problem to matroid realizability. We also give a proof of Mn\"ev universality with explicit bounds on the size of the matroid, which may be of independent interest.