Resolving singularities of curves with one toric morphism

Abstract

We give an explicit positive answer, in the case of reduced curve singularities, to a question of B. Teissier about the existence of a toric embedded resolution after reembedding. In the case of a curve singularity pC,Oq contained in a non singular surface S such a reembedding may be defined in terms of a sequence of maximal contact curves associated to C. We prove that there exists a toric modification, after reembedding, which provides an embedded resolution of C. We use properties of the semivaluation space of S at O to describe how the the dual graph of the minimal embedded resolution of C may be seen on the local tropicalization of S associated to this reembedding.