Rigid Gorenstein toric Fano varieties arising from directed graphs

A directed edge polytope ${\mathcal{A}}_G$ is a lattice polytope arising from root system $A_n$ and a finite directed graph $G$. If every directed edge of $G$ belongs to a directed cycle in $G$, then ${\mathcal{A}}_G$ is terminal and reflexive, that is, one can associate this polytope to a Gorenstein toric Fano variety $X_G$ with terminal singularities. It is shown by Totaro that a toric Fano variety which is smooth in codimension $2$ and $\mathbb{Q}$-factorial in codimension $3$ is rigid. In the present paper, we classify all directed graphs $G$ such that $X_G$ is a toric Fano variety which is smooth in codimension $2$ and $\mathbb{Q}$-factorial in codimension $3$.