Smooth tropical complete intersection curves of genus $3$ in $\mathbb{R}^3$

TL;DR

This paper introduces a method to describe tropical complete intersections in three-dimensional space and applies it to classify smooth genus 3 curves, showing certain topologies are impossible.

Contribution

It develops a new approach for analyzing tropical intersections in 3 and classifies smooth genus 3 tropical curves, addressing a problem posed by Morrison.

Findings

No smooth tropical complete intersection curves in 3 have lollipop graph skeletons.

The method determines the topological type of tropical intersections using polyhedral complexes.

Provides partial classification of genus 3 tropical curves in 3.

Abstract

We develop a method for describing the tropical complete intersection of a tropical hypersurface and a tropical plane in $\mathbb{R}^3$. This involves a method for determining the topological type of the intersection of a tropical plane curve and $\mathbb{R}_{\leq 0}^2$ by using a polyhedral complex. As an application, we study smooth tropical complete intersection curves of genus $3$ in $\mathbb{R}^3$. In particular, we show that there are no smooth tropical complete intersection curves in $\mathbb{R}^3$ whose skeletons are the lollipop graph of genus $3$. This gives a partial answer to a problem of Morrison.