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Resolution of Singularities: An Introduction

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Handbook of Geometry and Topology of Singularities I

Abstract

The problem of resolution of singularities and its solution in various contexts can be traced back to I. Newton and B. Riemann. This paper is an attempt to give a survey of the subject starting with Newton till the modern times, as well as to discuss some of the main open problems that remain to be solved. The main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski’s approach via valuations, Hironaka’s celebrated result in characteristic zero and all dimensions and its subsequent strengthenings and simplifications, existing results in positive characteristic (mostly up to dimension three), de Jong’s approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singularities of vector fields and foliations. In many places, we have tried to summarize the main ideas of proofs of various results without getting too much into technical details.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR CNRS 5219, Toulouse, Cedex, France

    Mark Spivakovsky

  2. Unidad Mixta Internacional 2001 CNRS, Laboratorio Solomon Lefschetz, Cuernavaca, Mexico

    Mark Spivakovsky

  3. Unidad Cuernavaca, Instituto de Matemáticas, UNAM, Cuernavaca, Mexico

    Mark Spivakovsky

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  1. Unidad Cuernavaca Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, Mexico

    José Luis Cisneros Molina

  2. Centre de Mathématiques et Informatique, Université d’Aix-Marseille, Marseille, France

    Dũng Tráng Lê

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    José Seade

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Spivakovsky, M. (2020). Resolution of Singularities: An Introduction. In: Cisneros Molina, J.L., Lê, D.T., Seade, J. (eds) Handbook of Geometry and Topology of Singularities I. Springer, Cham. https://doi.org/10.1007/978-3-030-53061-7_3

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