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 $GL(3,\boldsymbol{Z})$, all convex polytopes in $\boldsymbol{Z}^3$ which contain the origin as the only non-vertex lattice point. We obtain, up to isomorphism, 233 toric Fano 3-folds possessing at worst $\boldsymbol{Q}$-factorial singularities (of which 18 are known to be smooth) and 401 toric Fano 3-folds with terminal singularities that are not $\boldsymbol{Q}$-factorial.