Resolving singularities of curves with one toric morphism

TL;DR

This paper demonstrates that for reduced curve singularities, a toric embedded resolution can be explicitly constructed after reembedding, linking resolution properties to tropicalization and semivaluation spaces.

Contribution

It provides an explicit construction of toric embedded resolutions for reduced curve singularities via reembedding and analyzes their properties using tropical geometry.

Findings

Existence of toric modifications after reembedding for curve singularities.

Description of the dual graph of resolution in terms of local tropicalization.

Connection between semivaluation space properties and resolution structure.

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 $(C,O)$ contained in a non singular surface $S$ such a reembedding may be defined in terms of a sequence of maximal contact curves of the minimal embedded resolution of $C$. 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 dual graph of the minimal embedded resolution of $C$ may be seen on the local tropicalization of $S$ associated to this reembedding.