Tropicalization of 1-tacnodal curves on toric surfaces

TL;DR

This paper introduces a tropicalization approach to classify 1-tacnodal curves with an $A_3$ singularity on toric surfaces, extending previous work on 1-cuspidal curves.

Contribution

It defines a tropical analogue of 1-tacnodal curves and classifies the tropical curves corresponding to these singularities.

Findings

Classified tropical curves corresponding to 1-tacnodal curves.

Extended tropicalization methods to include $A_3$ singularities.

Provided a framework for studying complex singularities via tropical geometry.

Abstract

A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by E. Shustin. He classified the degeneration of 1-cuspidal curves using polyhedral complexes called tropical curves. In this paper, we define a tropical version of a 1-tacnodal curve, that is, a curve having exactly one singular point whose topological type is $A_3$, and by applying the tropicalization method, we classify tropical curves which correspond to 1-tacnodal curves.