# Tritangent planes to space sextics: the algebraic and tropical stories

## TL;DR

This paper explores the classical problem of counting tangent planes to space sextic curves, analyzing real solutions, and connecting algebraic and tropical geometry to understand tritangent planes.

## Contribution

It provides new counts of real tritangents for canonical sextic curves and links algebraic and tropical approaches to study their properties.

## Key findings

- Number of real tritangents varies with curve real structure
- Tropicalization of sextic curves has exactly 15 tritangent planes
- Constructs examples with maximum and minimal real tritangents

## Abstract

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 such a curve is real. We then 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.