Obstructions to approximating tropical curves in surfaces via intersection theory

TL;DR

This paper introduces new local obstructions to approximating tropical curves in smooth tropical surfaces, linking tropical and complex intersection theories, with applications in classifying tropical curves and analyzing tropical surfaces of specific degrees.

Contribution

It establishes new obstructions based on intersection theory, classifies certain tropical curves, and analyzes the presence of tropical lines in surfaces of degrees 3 and higher.

Findings

Classified all locally irreducible approximable 3-valent fan tropical curves in non-singular tropical planes.

Proved that degree 3 tropical surfaces contain finitely many approximable tropical lines.

Showed that degree 4 or higher tropical surfaces contain no approximable tropical lines.

Abstract

We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on the relation between tropical and complex intersection theories which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a non-singular fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.