Abstract
We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both to the corresponding theories on algebraic varieties and to previous work on graphs and abstract tropical curves. In addition, we establish conditions for the divisor-curve intersection numbers on a tropical complex to agree with the generic fiber of a degeneration.
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Acknowledgements
Throughout this project, I’ve benefited from conversations with Matt Baker, Spencer Backman, Alex Fink, Christian Haase, Paul Hacking, June Huh, Eric Katz, Madhusudan Manjunath, Farbod Shokrieh, Bernd Sturmfels, Yu-jong Tzeng, and Josephine Yu. I’d especially like to thank Sam Payne for his many insightful suggestions and thoughtful comments on an early draft of this paper. I was supported by the National Science Foundation Award No. DMS-1103856 and National Security Agency Award H98230-16-1-0019. I also thank the University of Georgia for its hospitality during the first round of revisions on this paper.
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