GR¨OBNER FANS AND MINIMAL EMBEDDED TORIC RESOLUTIONS OF RATIONAL DOUBLE POINT SINGULARITIES

Mourtada and Pl´enat (2018) give minimal embedded toric resolutions of ADEsingularities in C3 by constructing regular refinements of their dual Newton polyhedrons with the elements of their embedded valuation sets derived from the jet schemes constructed by the first author in 2014. On the other hand, in the works by Aroca et al., the authors represent the Gr¨obner fan of a Newton non-degenerate variety and prove that a regular refinement of the Gr¨obner fan of such a singularity yields an embedded toric resolution. In this paper, we reconstruct embedded toric resolutions of ADE-singularities. We give the explicit constructions of their Gr¨obner fans and refine them using the concept of profile. This provides an alternative and computationally effective framework to the one based on jet schemes.