A moduli stack of tropical curves

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.

Metadaten
Author:	Renzo Cavalieri, Melody Chan, Martin Ulirsch ORCiD GND, Jonathan Wise
URN:	urn:nbn:de:hebis:30:3-552983
DOI:	https://doi.org/10.1017/fms.2020.16
ISSN:	2050-5094
Parent Title (English):	Forum of mathematics. Sigma
Publisher:	Cambridge University Press
Place of publication:	Cambridge
Document Type:	Article
Language:	English
Date of Publication (online):	2020/04/24
Date of first Publication:	2020/04/24
Publishing Institution:	Universitätsbibliothek Johann Christian Senckenberg
Release Date:	2020/08/06
Volume:	8
Issue:	e23
Page Number:	93
First Page:	1
Last Page:	93
Note:	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.
HeBIS-PPN:	47098385X
Institutes:	Informatik und Mathematik / Mathematik
Dewey Decimal Classification:	5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:	14-XX ALGEBRAIC GEOMETRY / 14Axx Foundations / 14A20 Generalizations (algebraic spaces, stacks)
	14-XX ALGEBRAIC GEOMETRY / 14Txx Tropical geometry [See also 12K10, 14M25, 14N10, 52B20] / 14T05 Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]
Sammlungen:	Universitätspublikationen
Open-Access-Publikationsfonds:	Informatik und Mathematik
Licence (German):	Creative Commons - Namensnennung 4.0

Open Access