A lifting partition theorem for tropical tritangent classes to smooth space sextic curves

TL;DR

This paper studies the structure of tropical tritangent classes to smooth space sextic curves, revealing how classical tritangent planes lift from tropical configurations and identifying specific partition patterns for generic curves.

Contribution

It establishes a lifting partition theorem showing that only six of ten possible lift multiplicity partitions occur for generic curves, linking these to tropical combinatorial types.

Findings

15 connected components of tritangent classes in tropical space sextics.

Each tropical class contains 8 tritangent planes of the algebraic curve.

Only six out of ten possible partitions of 8 into powers of 2 arise for generic curves.

Abstract

The set of tritangent planes to smooth tropical space sextic curves has 15 connected components, recording continuous displacements of planes preserving the tritangency condition. These 15 tritangent classes are polyhedral complexes in $\mathbf{R}^3$, and each of them contains the tropicalization of precisely eight tritangent planes to any smooth space sextic curve with the given tropicalization. Prior joint work of the authors with Len confirms that each tropical tritangent plane has 0, 1, 2, 4 or 8 lifts to classical tritangent planes defined over the algebraic closure of the field over which the original algebraic curve is defined. Our main theorem states that when the input classical curve is generic, then only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes. Furthermore, we show that these partitions are completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type.