Tropical Lines on Cubic Surfaces

TL;DR

This paper classifies all possible combinatorial configurations of tropical lines on smooth tropical cubic surfaces, providing bounds on their number and confirming the existence of exactly 27 lines on a generic tropical cubic surface.

Contribution

It introduces primal and dual motifs to encode tropical line positions and classifies all such motifs on smooth tropical surfaces, establishing bounds and exact counts for tropical lines.

Findings

Classified all motifs of tropical lines on smooth tropical surfaces.

Provided an upper bound for the number of tropical lines.

Confirmed that a general tropical cubic surface contains exactly 27 lines.

Abstract

Given a tropical line $L$ and a smooth tropical surface $X$, we look at the position of $L$ on $X$. We introduce its primal and dual motif which are respectively a decorated graph and a subcomplex of the dual triangulation of $X$. They encode the combinatorial position of $L$ on $X$. We classify all possible motifs of tropical lines on general smooth tropical surfaces. This classification allows to give an upper bound for the number of tropical lines on a general smooth tropical surface with a given subdivision. We focus in particular on surfaces of degree three. As a concrete example, we look at tropical cubic surfaces dual to a fixed honeycomb triangulation, showing that a general surface contains exactly $27$ tropical lines.