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Tropicalization of theta characteristics, double covers, and Prym varieties

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Abstract

We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of \(2^{g-1}\) even theta characteristics and \(2^{g-1}\) odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.

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Acknowledgements

The bulk of this paper was written during a Research in Pairs stay at Oberwolfach. We would like to thank the institute for providing ideal working conditions for exploring these ideas. The first author’s travel was supported by an AMS Simons travel grant, and the second author was partially support by DFG Grant MA 4797/6-1. We are grateful to Matt Baker for insightful remarks on a previous version of this manuscript, and thank Sam Payne, Joe Rabinoff, Dhruv Ranganathan, and Farbod Shokrieh for fielding our questions. Finally, we thank the referees for their insightful remarks.

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Authors and Affiliations

  1. University of Kentucky, Lexington, KY, USA

    David Jensen

  2. University of Waterloo, Waterloo, Canada

    Yoav Len

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  1. David Jensen
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  2. Yoav Len
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Correspondence to Yoav Len.

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Jensen, D., Len, Y. Tropicalization of theta characteristics, double covers, and Prym varieties. Sel. Math. New Ser. 24, 1391–1410 (2018). https://doi.org/10.1007/s00029-017-0379-6

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