Introduction to Singularities

• ​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 κ7. 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)