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Logarithmic structures, Artin fans, and the moduli stack of tropical curves

Banff International Research Station for Mathematical Innovation and Discovery

Artin fans are logarithmic algebraic stacks that are logarithmically \'etale over the base field. Despite their seemingly abstract definition, the geometry of Artin fans 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. In this talk, following a rapid introduction to logarithmic geometry from a modular point of view, I am going to give an expository account of the theory of Artin fans and explain how Thuillier's non-Archimedean skeleton of a toroidal embedding can be understood as an analytification of the associated Artin fan. In the special case of a toric variety, this simply reduces to the fact that the Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient. Finally, Artin fans also provide the motivation for an ongoing joint project with R. Cavalieri, M. Chan, and J. Wise, in which we develop a stack-theoretic framework for the study of tropical moduli spaces.

Mathematics; Algebraic geometry; Combinatorics

video/mp4

Author affiliation: University of Bonn

2017-02-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59605

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/