Toric Fano three-folds with terminal singularities

Abstract

This paper classifies all toric Fano $3$-folds with terminal singularities. This is achieved by solving the equivalent combinatorial problem; that of finding, up to the action of $GL(3,\text{Z})$, all convex polytopes in ${\text{Z}}^3$ which contain the origin as the only non-vertex lattice point. We obtain, up to isomorphism, $233$ toric Fano $3$-folds possessing at worst $\text{Q}$-factorial singularities (of which $18$ are known to be smooth) and $401$ toric Fano $3$-folds with terminal singularities that are not $\text{Q}$-factorial.