Relating Different Definitions of Linear Series on Tropical Curves

TL;DR

This paper compares two definitions of tropical linear series, proves their relationship, provides counterexamples, and explores their extensions and realizability on tropical curves.

Contribution

It establishes that strongly recursive tropical linear series are a subset of combinatorial limit linear series and introduces extensions and realizability results.

Findings

Strongly recursive tropical linear series are combinatorial limit linear series of the same rank.

Counterexample showing the converse does not hold.

Extensions include simplified definitions and relationships with other tropical linear series.

Abstract

We compare strongly recursive tropical linear series as defined by Farkas, Jensen, and Payne with combinatorial limit linear series as defined by Amini and Gierczak. We show that strongly recursive tropical linear series of rank $r$ are combinatorial limit linear series of rank $r$ and construct a counterexample to the converse. We also present several extensions of our main result, including a simplification of the definition of combinatorial limit linear series and an investigation of their relationship with tropical linear series in the sense of Chang et al. Finally, we address the realizability of permutation arrays as local arrays of linear series on tropical curves. Finally, discuss the realizability of permutation arrays as local arrays of linear series on tropical curves.