Tropical Convexity and Canonical Projections

TL;DR

This paper introduces tropical convexity and canonical projections in the space of effective divisors on metric graphs, extending the concept of reduced divisors and exploring their properties using potential theory.

Contribution

It develops a framework for tropical convexity on divisor spaces, defining canonical projections and extending reduced divisors beyond classical linear systems.

Findings

Defined a natural metric on divisor spaces

Extended the notion of reduced divisors to tropical convex sets

Analyzed properties of tropical convex sets using reduced divisors

Abstract

Using a potential theory on metric graphs "Gamma", we introduce the notion of tropical convexity to the space "RDiv^d(Gamma)" of effective R-divisors of degree d on "Gamma" and show that a natural metric can be defined on "RDiv^d(Gamma)". In addition, we extend the notion of reduced divisors which is conventionally defined in a complete linear system |D| with respect to a single point in "Gamma". In our general setting, a reduced divisor is defined uniquely as an R-divisor in a compact tropical convex subset "T" of "RDiv^d(Gamma)" with respect to a certain R-divisor "E" of the same degree d. In this sense, we consider reduced divisors as canonical projections onto "T". We also investigate some basic properties of tropical convex sets using techniques developed from general reduced divisors.