Introduction to Singularities

by Shihoko Ishii

Overview

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 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. 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 bookis 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.

Product Details

ISBN-13:	9784431562610
Publisher:	Springer Japan
Publication date:	08/23/2016
Edition description:	Softcover reprint of the original 1st ed. 2014
Pages:	223
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

About the Author

Shihoko Ishii is a professor emeritus at the University of Tokyo and a professor at the Tokyo Woman's Christian University.

Table of Contents

1. Sheaves, algebraic varieties and analytic spaces.- 2. Homological algebra and duality.- 3. Singularities, algebraization and resolutions of singularities.- 4. Divisors on a variety and the corresponding sheaves.- 5. Differential forms around the singularities.- 6. Two dimensional singularities.- 7. Higher dimensional singularities.- 8. Deformations of singularities.

Editorial Reviews

This book presents some results of the study of singularities within the field of algebraic geometry. … The text is clear and well written and it is accessible to non-expert readers. … the author defines all the concepts and the results are either proved or a reference is provided. This makes the book self-contained in a broad sense. It is a useful volume both for readers studying singular varieties for the first time and for experts on classification of singularities.” (Santiago Encinas, Mathematical Reviews, August, 2015)

From the Publisher