Combinatorics and their evolution in resolution of embedded algebroid surfaces

The seminal concept of characteristic polygon of an embedded algebroid surface, developed by Hironaka expanding on an idea of Newton, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities is not really well understood, as some references had pointed out.

The aim of this paper is to study 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.

Keywords

resolution of surface singularities, Newton polygon, equimultiple locus, blowing-up

2010 Mathematics Subject Classification

14H20, 32S25

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The first author was supported by Project Métodos Computacionales en Álgebra, D-módulos, Teoría de la Representación y Optimización (MTM2016-75024-P) (Ministerio de Economía y Competitividad). The second and third authors were supported by Project Geometría Aritmética, D-módulos y Singularidades (MTM2016–75027–P) (Ministerio de Economía y Competitividad) and Project Singularidades, Geometría Algebraica Aritmética y Teoría de Representaciones: Estructuras y Métodos Diferenciales, Cohomológicos, Combinatorios y Computacionales (P12–FQM–2696) (Junta de Andalucía and FEDER).

Received 13 February 2018

Accepted 15 January 2020

Published 10 March 2021