An Algorithm for the Tropical Realizability Problem for Families of Curves

TL;DR

This paper introduces an algorithm to determine the algebraic conditions under which a family of algebraic curves tropicalizes to a given tropical fan curve, aiding in solving the tropical realizability problem.

Contribution

The paper presents the first algorithmic approach to describe the Zariski closure of the realization locus for tropical fan curves within algebraic families.

Findings

Algorithm effectively computes algebraic conditions for tropicalization.

Provides a systematic method to describe the realization locus.

Advances understanding of tropical realizability in algebraic geometry.

Abstract

Given a tropical fan curve $\Sigma$ and a family of algebraic curves $X \rightarrow \mathbb{A}^k$ we define the realization locus $\mathop{Real}_\Sigma \subseteq \mathbb{A}^k$ as the set of fibers $X_a$ whose tropicalization is $\Sigma$. We produce an algorithm that describes the Zariski closure of $\mathop{Real}_\Sigma$ by imposing algebraic conditions for each ray of $\Sigma$.