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3 - Examples

Published online by Cambridge University Press:  05 July 2013

János Kollár
Affiliation:
Princeton University, New Jersey
Sándor Kovács
Affiliation:
University of Washington
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Summary

We studied log canonical surface singularities in Section 2.2 and gave examples of typical log terminal 3-fold singularities in Section 2.4. In Section 2.5 we proved that dlt singularities are rational in any dimension.

The aim of this chapter is to show, by many examples, that the above results are by and large optimal: log canonical singularities get much more complicated in dimension 3 and even terminal singularities are likely non-classifiable in dimension 4.

A first set of examples of the various higher dimensional singularities occurring in the minimal model program are given in Section 3.1. These are rather elementary, mostly cones, but they already illustrate how subtle log canonical pairs can be.

We consider in greater detail quotient singularities in Section 3.2. This is a quite classical topic but with many subtle aspects. These are some of the simplest log terminal singularities in any dimension but they are the most likely to come up in applications.

Section 3.3 gives a rather detailed classification of log canonical surface singularities. Strictly speaking, not all of Section 3.3 is needed for the general theory, but it is useful and instructive to have a thorough understanding of a class of concrete examples. Our treatment may be longer than usual, but it applies in positive and mixed characteristics as well.

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Publisher: Cambridge University Press
Print publication year: 2013

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  • Examples
  • János Kollár, Princeton University, New Jersey
  • In collaboration with Sándor Kovács, University of Washington
  • Book: Singularities of the Minimal Model Program
  • Online publication: 05 July 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781139547895.005
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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.

  • Examples
  • János Kollár, Princeton University, New Jersey
  • In collaboration with Sándor Kovács, University of Washington
  • Book: Singularities of the Minimal Model Program
  • Online publication: 05 July 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781139547895.005
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.

  • Examples
  • János Kollár, Princeton University, New Jersey
  • In collaboration with Sándor Kovács, University of Washington
  • Book: Singularities of the Minimal Model Program
  • Online publication: 05 July 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781139547895.005
Available formats
×