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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.
_0. Preliminaries: Variations of making singularities0.1. By cutting-hypersurface singularities, hyperplane section of singularities0.2. By taking quotients-quotient singularities, quotient of singularities0.3. By lifting up-covering singularities0.4. By contractions 1. Sheaves, algebraic varieties and analytic spaces1.1. Preliminaries on sheaves1.2. Sheaves on a topological space1.3. Analytic space and Algebraic variety1.4. Coherent sheaves2. Homological algebra and duality2.1. Injective resolutions2.2. i-th derived functors2.3. Ext2.4. Cohomologies with the coefficients on sheaves2.5. Derived functors and duality2.6. Spectral sequence3. Singularities, algebraization and resolutions of singularities3.1. Definition of a singularity3.2. Algebraization theorem3.3. Blowups and resolutions of the singularities3.4. Toric resolutions of the singularities4. Divisors on a variety and the corresponding sheaves4.1. Locally free sheaves, invertible sheaves and divisorial sheaves4.2. Divisors4.3. The canonical sheaves and a canonical divisor4.4. Intersections of divisors5. Differential forms around the singularities5.1. Ramification formula5.2. Canonical singularities, terminal singularities and rational singularities6. Two dimensional singularities6.1. Resolutions of two-dimensional singularities6.2. The fundamental cycle6.3. Rational singularities6.4. Quitient singularities6.5. Rational double points6.6. Elliptic singularities6.7. Two-dimensional Du Bois singularities6.8. Classification of two-dimensional singularities by _7. Higher dimensional singularities7.1. Mixed Hodge structures and Du Bois singularities7.2. Minimal model problem7.3. Higher dimensional canonical singularities and terminal singularities7.4. Higher dimensional 1-Gorenstein singularities8. Deformations of singularities8.1. Change of properties under deformations8.2. Versal deformationsAppendix: Recent resultsReferences
"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)