Tritangent Planes to Space Sextics: The Algebraic and Tropical Stories

  • Corey HarrisEmail author
  • Yoav Len
Part of the Fields Institute Communications book series (FIC, volume 80)


We discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We determine the number of real tritangents when the curve is real. We also revisit a curve constructed by Emch with the greatest known number of real tritangents and, conversely, construct a curve with very few real tritangents. Using recent results on the relation between algebraic and tropical theta characteristics, we show that the tropicalization of a canonical sextic curve has 15 tritangent planes.

MSC 2010 codes:

14T05 14H50 14N10 14P25 



This article was initiated during the Apprenticeship Weeks (22 August 2016–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. We thank Lars Kastner, Sara Lamboglia, Steffen Marcus, Kalina Mincheva, Evan Nash, and Bach Tran who participated with us in the morning and evening sessions that led to this project. A great thanks goes to Kristin Shaw and Frank Sottile for many helpful discussions and insightful suggestions. We thank the editors Greg Smith, Bernd Sturmfels, and the liaison committee for always being vigilant, and making sure that this book sees the light of day. Finally, we thank the anonymous referees for their valuable comments and suggestions. Corey Harris was supported by the Fields Institute for Research in Mathematical Sciences, by the Clay Mathematics Institute and by NSA award H98230-16-1-0016.


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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Combinatorics and Optimization DepartmentUniversity of WaterlooWaterlooCanada

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