Introduction to Singularities

  • Shihoko Ishii

Table of contents

  1. Front Matter
    Pages i-x
  2. Shihoko Ishii
    Pages 1-5
  3. Shihoko Ishii
    Pages 27-54
  4. Shihoko Ishii
    Pages 81-98
  5. Shihoko Ishii
    Pages 99-117
  6. Shihoko Ishii
    Pages 119-159
  7. Shihoko Ishii
    Pages 161-198
  8. Shihoko Ishii
    Pages 199-212
  9. Back Matter
    Pages 227-236

About this book


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

Algebraic geometry is said to have originated in the seventeenth century with the famous workDiscours 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. First, mostly non-singular varieties were studied. 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. 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-dimensional 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. In the second edition, brief descriptions about recent remarkable developments of the researches are added as the last chapter.


Canonical divisors Classification of singularities Mixed Hodge structure Resolution Singularities

Authors and affiliations

  • Shihoko Ishii
    • 1
  1. 1.The University of TokyoTokyoJapan

Bibliographic information