Combinatorial Algebraic Geometry pp 21-45 | Cite as

# From Curves to Tropical Jacobians and Back

## Abstract

For a curve over an algebraically closed field that is complete with respect to a nontrivial valuation, we study its tropical Jacobian. We first tropicalize the curve and then use the weighted metric graph to compute the tropical Jacobian. Finding the abstract tropicalization of a general curve defined by polynomial equations is difficult, because an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction. We solve these problems for hyperelliptic curves by using admissible covers. We also calculate the period matrix from a weighted metric graph, which gives the tropical Jacobian and tropical theta divisor. Lastly, we look at how to compute a curve that has a given period matrix.

## MSC 2010 codes:

14T05## Notes

### Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. We heartily thank Bernd Sturmfels for leading the Apprenticeship Weeks and for providing many valuable insights, ideas and comments. We would also like to thank Melody Chan, for providing several useful directions which were encoded in the paper and for suggesting Example 3.5. We are grateful to Renzo Cavalieri, for a long, illuminating conversation about admissible covers, and Sam Payne and Martin Ulirsch for suggesting references and for clarifying some obscure points. We also thank Achill Schürmann and Mathieu Dutour Sikirić for their input on software for working with Delaunay subdivisions. The first author was supported by the Fields Institute for Research in Mathematical Sciences, the second author was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1106400 and the Max Planck Institute for Mathematics in the Sciences, Leipzig, and the third author was supported by a UC Berkeley University Fellowship and the Max Planck Institute for Mathematics in the Sciences, Leipzig.

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