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A MODULI STACK OF TROPICAL CURVES

Part of: Algebraic geometry: Foundations

Published online by Cambridge University Press:  24 April 2020

RENZO CAVALIERI ,
MELODY CHAN ,
MARTIN ULIRSCH  and
JONATHAN WISE
RENZO CAVALIERI
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, Colorado80523-1874, USA; renzo@math.colostate.edu
MELODY CHAN
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island02912, USA; melody_chan@brown.edu
MARTIN ULIRSCH
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, 60325Frankfurt am Main, Germany; ulirsch@math.uni-frankfurt.de
JONATHAN WISE
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Boulder,Colorado80309-0395, USA; jonathan.wise@math.colorado.edu

Abstract

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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.

MSC classification

Primary: 14T05: Tropical geometry
Secondary: 14A20: Generalizations (algebraic spaces, stacks)
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e23
Creative Commons
Creative Common License - CCCreative Common License - BY
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.
Copyright
© The Author(s) 2020

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