Skip to main content Accessibility help
×
Hostname: page-component-6bb9c88b65-t28k2 Total loading time: 0 Render date: 2025-07-25T10:31:12.738Z Has data issue: false hasContentIssue false

Resolution of Singularities

Published online by Cambridge University Press:  25 June 2025

Michael Schneider
Affiliation:
Universität Bayreuth, Germany
Yum-Tong Siu
Affiliation:
Harvard University, Massachusetts
Get access

Summary

This article is an exposition of our algorithm for canonical resolution of singularities in characteristic zero (Invent. Math. 128 (1997), 207-302), with an essentially complete proof of the theorem in the hypersurface case. We define a local invariant for desingularization whose values are finite sequences that can be compared lexicographically. Our invariant takes only finitely many maximum values (at least locally), and we get an algorithm for canonical desingularization by successively blowing up its maximum loci. The invariant can be described by a local construction that provides equations for the centres of blowing up. Our construction is presented here in parallel with a worked example.

1. Introduction

Resolution of singularities has a long history that goes back to Newton in the case of plane curves. For higher-dimensional singular spaces, the problem was formulated toward the end of the last century, and it was solved in general, for algebraic varieties defined over fields of characteristic zero, by Hironaka in his famous paper [1964]. (That paper includes the case of real-analytic spaces; Hironaka's theorem for complex-analytic spaces is proved in [Hironaka 1974; Aroca et al. 1975; 1977].) But Hironaka's result is highly non-constructive. His proof is one of the longest and hardest in mathematics, and it seems fair to say that only a handful of mathematicians have fully understood it. We are not among them! Resolution of singularities is used in many areas of mathematics, but even certain aspects of the theorem (for example, canonicity: see 1.11 below) have remained unclear.

Information

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

Accessibility standard: Unknown

Accessibility compliance for the PDF of this book is currently unknown and may be updated in the future.

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×