Rigid Gorenstein toric Fano varieties arising from directed graphs

TL;DR

This paper classifies directed graphs that produce rigid Gorenstein toric Fano varieties with specific smoothness and factoriality properties, linking graph structure to algebraic geometric features.

Contribution

It provides a complete classification of directed graphs G for which the associated toric Fano variety X_G is rigid, smooth in codimension 2, and Q-factorial in codimension 3.

Findings

Classified all directed graphs G with the specified properties.

Connected graph structure to the rigidity of the associated toric Fano varieties.

Established criteria for the smoothness and factoriality conditions in terms of graph cycles.

Abstract

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 $3$.