\(U_\sigma\) is smooth exactly when the minimal generators of \(\sigma\) are part of a \(\ZZ\)-basis of \(N\); such a cone is called smooth. \(X_\Sigma\) is smooth exactly when every cone of \(\Sigma\) is. It is \(\QQ\)-factorial exactly when every cone is simplicial.
(Toric resolution)
Every fan admits a refinement into smooth cones, obtained by repeatedly subdividing along lattice points, and the induced morphism \(X_{\Sigma'} \to X_\Sigma\) is a proper birational resolution.
Resolution of singularities is a hard theorem in general; here it is a subdivision algorithm: repeatedly inserting lattice points refines every fan into smooth cones, and the induced morphism is a proper birational resolution. The dictionary turns each hard statement into a finite computation with lattice points, and supplies examples on demand for questions asked elsewhere: a normal variety that is not smooth, a Weil divisor that is not Cartier, a class group with torsion.
Dependencies
PR-O8V3IThe orbit-cone correspondenceuses
Backlinks
FE-TORCDThe cone over the rational normal curve, and a class group with torsionrelated-toD-FULSIMPLESimple and simplicial polytopes and simplicial normal fansusesPR-FULCRITSmooth, complete, projective, Fano, read off the fanusesFE-FULQUOTThe cone over the rational normal curve as a cyclic quotient singularityusesFE-TORMINRESThe minimal resolution of a toric surface singularityuses