Realizability of tropical pluri-canonical divisors

TL;DR

This paper provides a combinatorial criterion to determine when a tropical divisor arises from a smooth algebraic curve with a pluri-canonical divisor, extending previous work on tropical canonical divisors and incorporating recent advances in moduli space compactifications.

Contribution

It introduces tropical normalized covers and reduces the realizability problem for pluri-canonical divisors to the realizability of these covers, generalizing prior results and integrating recent developments.

Findings

Provides a combinatorial criterion for realizability

Introduces tropical normalized covers as a key concept

Generalizes previous realizability results for canonical divisors

Abstract

Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to $k$ times the canonical divisor for $k \in \mathbb{Z}_{\geq 1}$. In this article we give a purely combinatorial criterion to determine if such a pair arises as the tropicalization of a pair consisting of a smooth algebraic curve over a non-Archimedean field with algebraically closed residue field of characteristic 0 together with an effective pluri-canonical divisor. To do so, we introduce tropical normalized covers as special instances of tropical Hurwitz covers and reduce the realizability problem for pluri-canonical divisors to the realizability problem for normalized covers. Our main result generalizes the work of M\"oller-Ulirsch-Werner on realizability of tropical canonical divisors and incorporates the recent progress on compactifications of strata of $k$-differentials in the work of Bainbridge-Chen-Gendron-Grushevsky-M\"oller.