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Hironaka’s characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition (Pol) holds, or a mild condition (*) on the singularity holds. For example, the latter is fulfilled if the residue field is perfect.

We show that given a face of Hironaka’s characteristic polyhedron, it does only depend on the singularity and a flag defined by the linear form determining the face. As a consequence we get that certain numerical data obtained from the characteristic polyhedron are invariants of the singularity. In particular, they do not depend on an embedding.

We prove Local Uniformization for arbitrary excellent hypersurface threefolds of multiplicity smaller than the residue characteristic. This article is part of the authors’ Resolution of Singularities program for arithmetic varieties of dimension three. The proof builds upon Hironaka’s characteristic polyhedron and invariants.

Abstract:: Given a rational surface singularity (S,P), we relate S-isomorphism classes of normal blowing-ups of S and properties of semifactorization for complete ideals in . This relation is understood via the natural cell decomposition of the characteristic cone P˜(X/S), where X is a desingularization of S.