Non-Archimedean geometry of Artin fans

TL;DR

This paper explores the role of Artin fans in non-Archimedean and tropical geometry, revealing their connection to tropicalization maps and moduli spaces of curves through a stack-theoretic framework.

Contribution

It establishes a link between Artin fans, tropicalization, and non-Archimedean analytic maps, providing a new perspective on the structure of tropical and algebraic moduli spaces.

Findings

Tropicalization map corresponds to non-Archimedean analytic map via Artin fans.

Reinterpretation of the moduli space of tropical curves as a non-Archimedean skeleton.

Artin fans are described completely in terms of combinatorial Kato stacks.

Abstract

The purpose of this article is to study the role of Artin fans in tropical and non-Archimedean geometry. Artin fans are logarithmic algebraic stacks that can be described completely in terms of combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato's notion of a fan. Every logarithmic algebraic stack admits a tautological strict morphism $\phi_\mathcal{X}:\mathcal{X}\rightarrow\mathcal{A}_\mathcal{X}$ to an associated Artin fan. The main result of this article is that, on the level of underlying topological spaces, the natural functorial tropicalization map of $\mathcal{X}$ is nothing but the non-Archimedean analytic map associated to $\phi_\mathcal{X}$ by applying Thuillier's generic fiber functor. Using this framework, we give a reinterpretation of the main result of Abramovich-Caporaso-Payne identifying the moduli space of tropical curves with the non-Archimedean skeleton of the corresponding algebraic moduli space.