Reduced Divisors and Embeddings of Tropical Curves

TL;DR

This paper studies reduced divisors on tropical curves, showing they induce integral affine maps to linear systems, classifying embeddings, and applying these results to prove the existence of Weierstrass points and analyze rank-determining sets.

Contribution

It provides an explicit description of reduced divisors as integral affine maps, classifies tropical curves with very ample canonical divisors, and connects tropical and classical divisor theories.

Findings

Reduced divisors define integral affine maps from tropical curves to linear systems.

Classification of tropical curves with very ample canonical divisors.

Proof of existence of Weierstrass points on tropical curves of genus at least two.

Abstract

Given a divisor $D$ on a tropical curve $\Gamma$, we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system $|D|$. This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into the linear system, and in this way, classify all the tropical curves with a very ample canonical divisor. As an application of the reduced-divisor map, we show the existence of Weierstrass points on tropical curves of genus at least two and present a simpler proof of a theorem of Luo on rank-determining sets of points. We also discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve $C$ and a divisor $D$ of non-negative rank on $C$, reduced divisors equivalent to $D$ define a morphism from $C$ to the complete linear system $|D|$, which is described in terms of Wronskians.