# Sixty-Four Curves of Degree Six

## TL;DR

This paper computationally classifies smooth degree-six curves in the real projective plane, providing explicit examples, empirical distributions, and exploring various geometric properties and topological features.

## Contribution

It introduces software for classifying sextic curves, enumerates explicit representatives, and analyzes their geometric and topological properties through empirical methods.

## Key findings

- Identified 64 rigid isotopy classes of sextic curves.
- Computed probability distributions of geometric features like bitangents and inflection points.
- Explored the topology of the avoidance locus and real tensor rank.

## 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.