Sixty-Four Curves of Degree Six

ABSTRACT

We present a computational study of smooth curves of degree six in the real projective plane. In the Rokhlin–Nikulin classification, there are 56 topological types, refined into 64 rigid isotopy classes. We developed software that determines the topological type of a given sextic and used it to compute empirical probability distributions on the various types. We list 64 explicit representatives with integer coefficients, and we perturb these to draw many samples from each class. This allows us to explore how many of the bitangents, inflection points, and tensor eigenvectors are real. We also study the real tensor rank, the construction of quartic surfaces with prescribed topology, and the avoidance locus, which is the locus of all real lines that do not meet a given sextic. This is a union of up to 46 convex regions, bounded by the dual curve.

KEYWORDS:

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2010 AMS SUBJECT CLASSIFICATION:

• Primary 14P25
• Secondary 14H50, 68W30

Acknowledgments

We are grateful to Paul Breiding, Claus Scheiderer, Anna Seigal, Israel Vainsencher, Emanuele Ventura, and Oleg Viro for their help. This project was completed following a visit by Daniel Plaumann to MPI Leipzig.

Funding

Daniel Plaumann was supported through DFG grant PL 549/3-1. Bernd Sturmfels acknowledges support by the US National Science Foundation (DMS-1419018) and the Einstein Foundation Berlin. Nidhi Kaihnsa and Mahsa Sayyary Namin were funded by the International Max Planck Research School Mathematics in the Sciences (IMPRS).