Lifting divisors with imposed ramifications on a generic chain of loops

TL;DR

This paper demonstrates that divisors with specified ramification on a generic chain of loops can be lifted to algebraic curves, extending previous work and addressing tropical intersection lifting within polytopal domains.

Contribution

It extends the lifting results of divisors with ramification from metric graphs to algebraic curves, particularly for chains of loops with generic edge lengths.

Findings

Divisors with ramification on a chain of loops lift to divisors on curves.

Provides a positive answer to lifting tropical intersections in polytopal domains.

Extends Cartwright-Jensen-Payne's work on divisor lifting.

Abstract

Proves that if a curve has totally split reduction and the corresponding skeleton (as a metric graph) is a chain of loops with generic edge lengths, then every divisor on the graph with imposed ramification at the rightmost vertex $p$ of the skeleton lifts to a divisor class on the curve with the same ramification at some point $P$ that retracts to $p$, extending the work of Cartwright-Jensen-Payne. Gives a positive answer to lifting proper tropical intersections within a polytopal domain in an analytic torus.