An Algorithm for the Tropical Realizability Problem for Families of Curves

Abstract

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

20 pages, 7 figures. Comments welcome!