Minimality of a toric embedded resolution of rational triple points after bouvier-gonzalez-sprinberg

Abstract

Nash's problem concerning arcs poses the question of whether it is possible to construct a bijective relationship between the minimal resolution of a surface singularity and the irreducible components within its arcs space. As a reverse question, one might inquire whether it is possible to derive a resolution from the arcs space of the given singularity. This paper focuses on non-isolated hypersurface singularities in $\mathbb{C}^3$ whose normalisations are surface in $\mathbb{C}^4$ having rational singularities of multiplicity 3. For each of these singularities, we construct a non singular refinement of its dual Newton polyhedron with valuations attached to specific irreducible components of its jet schemes. Subsequently, we get a toric embedded resolution of these singularities. To establish the minimality of this resolution, we generalize the notion of a profile of a simplicial cone, as introduced in [6]. As a corollary, we obtain that the Hilbert basis of the dual Newton polyhedron of a rational singularity with multiplicity 3 provides a minimal toric embedded resolution for our singularities.