Overview
Is a unique book that combines birational geometry, traditional local aspects of singularities, and new developments in research on singularities
Shows the origin of recent dramatic developments in research on Du Bois singularities
Introduces and explains the Toric method as a useful way to calculate examples and makes use of mixed Hodge structures to classify singularities
Access this book
Tax calculation will be finalised at checkout
Other ways to access
Table of contents (9 chapters)
-
Front Matter
-
Back Matter
Keywords
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.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)
Authors and Affiliations
Bibliographic Information
Book Title: Introduction to Singularities
Authors: Shihoko Ishii
DOI: https://doi.org/10.1007/978-4-431-55081-5
Publisher: Springer Tokyo
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Japan 2014
Softcover ISBN: 978-4-431-56261-0Published: 23 August 2016
eBook ISBN: 978-4-431-55081-5Published: 19 November 2014
Edition Number: 1
Number of Pages: VIII, 223
Number of Illustrations: 102 b/w illustrations
Topics: Algebraic Geometry, Associative Rings and Algebras, Commutative Rings and Algebras