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MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY

Part of: Curves Families, fibrations

Published online by Cambridge University Press:  03 June 2016

RENZO CAVALIERI ,
SIMON HAMPE ,
HANNAH MARKWIG  and
DHRUV RANGANATHAN
RENZO CAVALIERI
Affiliation:
Department of Mathematics, Colorado State University, USA; renzo@math.colostate.edu
SIMON HAMPE
Affiliation:
Institut für Mathematik, Technische Universitaät Berlin, Germany; hampe@math.tu-berlin.de
HANNAH MARKWIG
Affiliation:
Fachbereich Mathematik, Eberhard Karls Universität Tübingen, Germany; hannah@mathematik.uni-tuebingen.de
DHRUV RANGANATHAN
Affiliation:
Department of Mathematics, Yale University, USA; dhruv.ranganathan@yale.edu

Abstract

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We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

MSC classification

Primary: 14T05: Tropical geometry
Secondary: 14D22: Fine and coarse moduli spaces 14H10: Families, moduli (algebraic)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e9
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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) 2016

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