Local tropicalizations of splice type surface singularities

Abstract

To Walter Neumann, on the ocassion of his 75 th birthday.

Resolving singularities of curves with one toric morphism

“…The definition and properties of local tropicalizations in a general setting was developed in [38] by Popescu-Pampu and Stepanov. Further applications of local tropicalization are obtained recently by Cueto, Popescu-Pampu, and Stepanov in the case of surface singularities of splice type (see [7]).…”

“…It is a generalization of the notions of hypersurface and complete intersection singularities which are non degenerate with respect to their Newton polyhedra, which was introduced by Khovanskii and Kouchnirenko [26,27]. Cueto, Popescu-Pampu and Stepanov have proven that the ideals defining the natural embeddings of splice type surface singularities are Newton non-degenerate (see [7]). In the second version of the preprint [7], they have deduced the existence of a torific embedding of a reduced complex analytic plane curve singularity, as an application of their results on surface singularities.…”

“…Cueto, Popescu-Pampu and Stepanov have proven that the ideals defining the natural embeddings of splice type surface singularities are Newton non-degenerate (see [7]). In the second version of the preprint [7], they have deduced the existence of a torific embedding of a reduced complex analytic plane curve singularity, as an application of their results on surface singularities. -Lejeune-Jalabert and Reguera proved in [30] that sandwiched surface singularities admit natural torific embeddings, which were called toric environments.…”