Minimality of a toric embedded resolution of rational triple points after Bouvier-Gonzalez-Sprinberg (Q6634431)

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. The authors focus on non-isolated hypersurface singularities in \(\mathbb{C}^3\) whose normalizations are surface in \(\mathbb{C}^4\) having rational singularities of multiplicity \(3\). For each of these singularities, they construct a nonsingular refinement of its dual Newton polyhedron with valuations attached to specific irreducible components of its jet schemes. They obtain a toric embedded resolution of these singularities. To establish the minimality of this resolution, they generalize the notion of a profile of a simplicial cone, as well as obtain that the Hilbert basis of the dual Newton polyhedron of a rational singularity with multiplicity \(3\) provides a minimal toric embedded resolution for the singularities.