Tropical Linear Series and Tropical Independence

TL;DR

This paper introduces a new definition of tropical linear series that captures key combinatorial features of tropicalizations of algebraic curves, emphasizing properties like rank and independence, and explores their structural aspects.

Contribution

It proposes a novel definition of tropical linear series combining rank and independence, and studies their properties, including finite generation for rank 1 series.

Findings

Tropical linear series of rank 1 are finitely generated as tropical modules.

The restriction of a tropical linear series to a connected subgraph preserves the rank.

Open problems are identified for higher rank tropical linear series.

Abstract

We propose a definition of tropical linear series that isolates some of the essential combinatorial properties of tropicalizations of not-necessarily-complete linear series on algebraic curves. The definition combines the Baker-Norine notion of rank with the notion of tropical independence and has the property that the restriction of a tropical linear series of rank r to a connected subgraph is a tropical linear series of rank r. We show that tropical linear series of rank 1 are finitely generated as tropical modules and state a number of open problems related to algebraic, combinatorial, and topological properties of higher rank tropical linear series