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Published December 2015 | Accepted Version
Journal Article Open

Tropicalization of the moduli space of stable maps

Abstract

Let X be an algebraic variety and let S be a tropical variety associated to X. We study the tropicalization map from the moduli space of stable maps into X to the moduli space of tropical curves in S. We prove that it is a continuous map and that its image is compact and polyhedral. Loosely speaking, when we deform algebraic curves in X, the associated tropical curves in S deform continuously; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions, vanishing cycles and quantifier elimination for rigid subanalytic sets.

Additional Information

© Springer-Verlag Berlin Heidelberg 2015. Received 24 March 2015. Accepted 06 July 2015. Published 31 July 2015. Issue Date: December 2015. I am very grateful to Maxim Kontsevich and Antoine Chambert-Loir for inspirations and support. Special thanks to Antoine Ducros from whom I learned model theory and its applications to tropical geometry. I appreciate valuable discussions with Vladimir Berkovich, Pierrick Bousseau, Ilia Itenberg, François Loeser, Florent Martin, Johannes Nicaise, Sam Payne and Michael Temkin. Comments given by the referees helped greatly improve the paper.

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August 20, 2023
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