TY  - JOUR
A1  - Cavalieri, Renzo
A1  - Chan, Melody
A1  - Ulirsch, Martin
A1  - Wise, Jonathan
T1  - A moduli stack of tropical curves
T2  - Forum of mathematics. Sigma
N2  - We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.
Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.
Y1  - 2020
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/55298
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-552983
SN  - 2050-5094
N1  - This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
VL  - 8
IS  - e23
SP  - 1
EP  - 93
PB  - Cambridge University Press
CY  - Cambridge
ER  -