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Introduction to Singularities

Autor Shihoko Ishii
en Limba Engleză Paperback – 25 dec 2018
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.
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Specificații

ISBN-13: 9784431568711
ISBN-10: 4431568719
Pagini: 236
Ilustrații: X, 236 p. 4 illus.
Dimensiuni: 155 x 235 x 13 mm
Greutate: 0.35 kg
Ediția:Softcover reprint of the original 2nd ed. 2018
Editura: Springer
Colecția Springer
Locul publicării:Tokyo, Japan

Cuprins

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.

Notă biografică

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

Textul de pe ultima copertă

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.

Caracteristici

Presents a unique combination of birational geometry, traditional local aspects of singularities, and new developments in research on singularities Shows concise introductions of tools for study of singularities such as homological algebra, spectral sequence, toric varieties, and Hodge theory, which are also useful for study of other subjects in algebra and algebraic geometry Describes briefly the recent dramatic developments of the three directions in research on singularities

Recenzii

“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)