The group law on the tropical Hesse pencil

TL;DR

This paper explores the addition law on the tropical Hesse curve, demonstrating its realization through tropical line intersections, deriving the addition formula via ultradiscretization, and identifying the automorphism group as a dihedral group.

Contribution

It introduces a tropical analogue of the Hessian group and derives the addition formula for the tropical Hesse curve using ultradiscretization.

Findings

Addition of points on the tropical Hesse curve via tropical line intersections.

Derivation of the addition formula through ultradiscretization.

Identification of the automorphism group as the dihedral group of degree three.

Abstract

We show that the addition of points on the tropical Hesse curve can be realized via the intersection with a tropical line. Then the addition formula for the tropical Hesse curve is reduced from those for the level-three theta functions through the ultradiscretization procedure. A tropical analogue of the Hessian group, the group of linear automorphisms acting on the Hesse pencil, is also investigated; it is shown that the dihedral group of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.