Abstract
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain characteristics or over certain fields of definition. Our examples also apply to the theory of metrized complexes and weighted graphs. These examples arise by relating the lifting problem to matroid realizability. We also give a proof of Mnëv universality with explicit bounds on the size of the matroid, which may be of independent interest.
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Acknowledgements
I would like to thank Spencer Backman, Melody Chan, Alex Fink, Eric Katz, Yoav Len, Diane Maclagan, Sam Payne, Kristin Shaw, and Ravi Vakil for helpful comments on this project. The project was begun while the author was supported by NSF Grant DMS-1103856L.
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Cartwright, D. Lifting matroid divisors on tropical curves. Mathematical Sciences 2, 23 (2015). https://doi.org/10.1186/s40687-015-0041-x
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DOI: https://doi.org/10.1186/s40687-015-0041-x