# Combinatorics and their evolution in resolution of embedded algebroid surfaces

## TL;DR

This paper studies how the characteristic polygon of an embedded algebroid surface evolves during resolution of singularities, providing a clearer understanding and effective bounds on the process.

## Contribution

It offers a detailed explanation of the evolution of the characteristic polygon during resolution, using new techniques, and derives bounds on the number of blow-ups needed.

## Key findings

- Clarifies the evolution of the characteristic polygon during resolution.
- Provides effective bounds on the number of blow-ups required.
- Uses novel techniques to analyze the resolution process.

## Abstract

The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities was not really well understood, as some references had pointed out. The aim of this paper is to explain, in a clear way, how this object changes as the surface gets resolved. In order to get a precise description of the phenomena involved, we need to use different techniques and ideas. Eventually, some effective results regarding the number of blow-ups needed to decrease the multiplicity are obtained as a side product.