On the addition formula for the tropical Hesse pencil

TL;DR

This paper derives the addition formula for the tropical Hesse pencil by ultradiscretizing level-three theta functions, establishing a link between classical and tropical geometries, and providing explicit formulas and configurations.

Contribution

It introduces the addition formula for the tropical Hesse pencil via ultradiscretization of level-three theta functions, connecting classical and tropical geometries explicitly.

Findings

Derived the addition formula for the tropical Hesse pencil.

Established a correspondence between the amoeba of the Hesse cubic and the tropical Hesse curve.

Provided the tropical counterpart of the Hesse configuration.

Abstract

We give the addition formula for the tropical Hesse pencil, which is the tropicalization of the Hesse pencil parametrized by the level-three theta functions, via those for the ultradiscrete theta functions. The ultradiscrete theta functions are reduced from the level-three theta functions through the procedure of ultradiscretization by choosing their parameters appropriately. The parametrization of the level-three theta functions firstly introduced in \cite{KKNT09} gives an explicit correspondence between the amoeba of the real part of the Hesse cubic curve and the tropical Hesse curve. Moreover, through the parametrization, we obtain the subtraction-free forms of the addition formulae for the level-three theta functions, which lead to the addition formula for the tropical Hesse pencil in terms of the ultradiscretization. Using the parametrization, the tropical counterpart of the Hesse configuration is also given.