Tropical Moduli Spaces of Rational Graphically Stable Curves

Abstract

The tropical moduli space $\mathcal{M}_{0,n}^{\textrm{trop}}$ is a cone complex which parameterizes leaf-labeled metric trees called tropical curves. We introduce graphic stability and describe a refinement of the cone complex given by radial alignment. We prove that given a complete multipartite graph $\Gamma$, the moduli space of radially aligned $\Gamma$-stable tropical curves can be given the structure of a balanced fan. This fan structure coincides with the Bergman fan of the cycle matroid of $\Gamma$.