2016/03/24 by Ulirsch, Martin
#14A20 #14G22 #14T05 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1603.07589
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 ϕX:X\rightarrowAX 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 X is nothing but the non-Archimedean analytic map associated to ϕX 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.