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Tropicalizing the space of admissible covers

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Abstract

We study the relationship between tropical and classical Hurwitz moduli spaces. Following recent work of Abramovich, Caporaso and Payne, we outline a tropicalization for the moduli space of generalized Hurwitz covers of an arbitrary genus curve. Our approach is to appeal to the geometry of admissible covers, which compactify the Hurwitz scheme. We study the relationship between a combinatorial moduli space of tropical admissible covers and the skeleton of the Berkovich analytification of the classical space of admissible covers. We use techniques from non-archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization, and that the degree of the classical branch map can be recovered from the tropical side. As a consequence, we obtain a proof, at the level of moduli spaces, of the equality of classical and tropical Hurwitz numbers.

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Notes

  1. The notion of a tropical Hurwitz cover for discrete graphs was introduced by Caporaso in [11], in order to study the gonality of graphs. In particular she characterizes which graphs covering a tree are dual graphs of a classical admissible cover.

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Acknowledgments

R. Cavalieri acknowledges with gratitude the support by NSF grant DMS-1101549, NSF RTG grant 1159964, and H. Markwig is support by DFG grant MA 4797/6-1. D. Ranganathan acknowledges many fruitful conversations with Tyler Foster, Dave Jensen, Douglas Ortego, Yoav Len, and Martin Ulirsch. The authors thank Sam Payne for introducing them, as well as for several insightful comments. We also thank an anonymous referee for helpful comments on an earlier version.

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Authors and Affiliations

  1. Department of Mathematics, Colorado State University, Fort Collins, USA

    Renzo Cavalieri

  2. Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany

    Hannah Markwig

  3. Department of Mathematics, Yale University, New Haven, USA

    Dhruv Ranganathan

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  1. Renzo Cavalieri
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  2. Hannah Markwig
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  3. Dhruv Ranganathan
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Correspondence to Hannah Markwig.

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Cavalieri, R., Markwig, H. & Ranganathan, D. Tropicalizing the space of admissible covers. Math. Ann. 364, 1275–1313 (2016). https://doi.org/10.1007/s00208-015-1250-8

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