How to Repair Tropicalizations of Plane Curves Using Modifications

ABSTRACT

Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this article is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input curve. Our approach is based on the close connection between analytic curves (in the sense of Berkovich) and tropical curves. We investigate the effect of these tropical modifications on the tropicalization map defined on the analytification of the given curve.

Our study is motivated by the case of plane elliptic cubics, where good embeddings are characterized in terms of the j-invariant. Given a plane elliptic cubic whose tropicalization contains a cycle, we present an effective algorithm, based on non-Archimedean methods, to linearly re-embed the curve in dimension 4 so that its tropicalization reflects the j-invariant. We give an alternative elementary proof of this result by interpreting the initial terms of the A-discriminant of the defining equation as a local discriminant in the Newton subdivision.

Keywords:

• tropical geometry
• tropical modifications
• Berkovich spaces
• elliptic curves
• discriminants
• 2010 AMS Subject Classification
• 14T05
• 51M20
• 14H52
• 14G22

Funding

The first author was supported by an Alexander von Humboldt Postdoctoral Research Fellowship (Germany) and by an NSF postdoctoral fellowship DMS-1103857 (USA). The second author was supported by DFG-grant 4797/5-1 and by GIF-grant 1174-197.6/2011.

Acknowledgments

We wish to thank Erwan Brugallé, Arne Buchholz, Ilia Itenberg, Diane Maclagan, Thomas Markwig, Ralph Morrison, Bernd Sturmfels, Till Wagner, and Annette Werner for very fruitful conversations. We also want to thank the anonymous referee for his/her careful reading and various suggestions that helped us to improve this paper. All the computations in this paper were done using the tropical.lib library for Singular [Jensen et al. 07]. Part of this project was carried out during the 2013 program on Tropical Geometry and Topology at the Max-Planck Institut für Mathematik in Bonn, where the second author was in residence. We thank MPI for their hospitality.

Notes

¹The authors refer to them as skeletons associated to strictly semistable pairs (see [Gubler et al. 14, Section 4]).