Introduction to Singularities

  • Shihoko Ishii

Table of contents

  1. Front Matter
    Pages i-viii
  2. Shihoko Ishii
    Pages 1-5
  3. Shihoko Ishii
    Pages 27-54
  4. Shihoko Ishii
    Pages 81-97
  5. Shihoko Ishii
    Pages 99-117
  6. Shihoko Ishii
    Pages 119-160
  7. Shihoko Ishii
    Pages 161-199
  8. Shihoko Ishii
    Pages 201-214
  9. Back Matter
    Pages 215-223

About this book


This book is an introduction to singularities for graduate students and researchers.

It is said that algebraic geometry originated in the seventeenth century with the famous work Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences by Descartes. In that book he introduced coordinates to the study of geometry. After its publication, research on algebraic varieties developed steadily. Many beautiful results emerged in mathematicians’ works. Most of them were about non-singular varieties. Singularities were considered “bad” objects that interfered with knowledge of the structure of an algebraic variety. In the past three decades, however, it has become clear that singularities are necessary for us to have a good description of the framework of varieties. For example, it is impossible to formulate minimal model theory for higher-dimensional cases without singularities. Another example is that the moduli spaces of varieties have natural compactification, the boundaries of which correspond to singular varieties. A remarkable fact is that the study of singularities is developing and people are beginning to see that singularities are interesting and can be handled by human beings. This book is a handy introduction to singularities for anyone interested in singularities. The focus is on an isolated singularity in an algebraic variety. After preparation of varieties, sheaves, and homological algebra, some known results about 2-dim

ensional isolated singularities are introduced. Then a classification of higher-dimensional isolated singularities is shown according to plurigenera and the behavior of singularities under a deformation is studied.


Canonical divisors Classification of singularities Mixed Hodge structure Resolution Singularities

Authors and affiliations

  • Shihoko Ishii
    • 1
  1. 1.Graduate School of Mathematical Sciences, The University of TokyoTokyoJapan

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