Resolution of Curve and Surface Singularities in Characteristic Zero / Edition 1by K. Kiyek, J.L. Vicente
Pub. Date: 12/20/2004
Publisher: Springer Netherlands
This book covers the beautiful theory of resolutions of surface singularities in characteristic zero. The primary goal is to present in detail, and for the first time in one volume, two proofs for the existence of such resolutions. One construction was introduced by H.W.E. Jung, and another is due to O. Zariski. Jung's approach uses quasi-ordinary singularities and… See more details below
This book covers the beautiful theory of resolutions of surface singularities in characteristic zero. The primary goal is to present in detail, and for the first time in one volume, two proofs for the existence of such resolutions. One construction was introduced by H.W.E. Jung, and another is due to O. Zariski. Jung's approach uses quasi-ordinary singularities and an explicit study of specific surfaces in affine three-space. In particular, a new proof of the Jung-Abhyankar theorem is given via ramification theory. Zariski's method, as presented, involves repeated normalisation and blowing up points. It also uses the uniformization of zero-dimensional valuations of function fields in two variables, for which a complete proof is given. Despite the intention to serve graduate students and researchers of Commutative Algebra and Algebraic Geometry, a basic knowledge on these topics is necessary only. This is obtained by a thorough introduction of the needed algebraic tools in the two appendices.
Table of Contents
Preface. Note to the Reader. Terminology. I: Valuation Theory. 1. Marot Rings. 2. Manis Valuation Rings. 3. Valuation Rings and Valuations. 4. The Approximate Theorem for Independent Valuations. 5. Extensions of Valuations. 6. Extending Valuations to Algebraic Overfields. 7. Extensions of Discrete Valuations. 8. Ramification Theory of Valuations. 9. Extending Valuations to Non-Algebraic Overfields. 10. Valuations of Algebraic Function Fields. 11. Valuations Dominating a Local Domain. II: One-Dimensional Semilocal Cohen-Macaulay Rings. 1. Transversal Elements. 2. Integral Closure of One-Dimensional Semilocal Cohen-Macaulay Rings. 3. One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings. 4. Blowing up Ideals. 5. Infinitely Near Rings. III: Differential Modules and Ramification. 1. Introduction. 2. Norms and Traces. 3. Formally Unramified and Ramified Extensions. 4. Unramified Extensions and Discriminants. 5. Ramification for Quasilocal Rings. 6. Integral Closure and Completion. IV: Formal and Convergent Power Series Rings. 1. Formal Power Series Rings. 2. Convergent Power Series Rings. 3. Weierstraß Preparation Theorem. 4. The Category of Formal and Analytic Algebras. 5. Extensions of Formal and Analytic Algebras. V: Quasiordinary Singularities. 1. Fractionary Power Series. 2. The Jung-Abhyankar Theorem: Formal Case. 3. The Jung-Abhyankar Theorem: Analytic Case. 4. Quasiordinary Power Series. 5. A Generalized Newton Algorithm. 6. Strictly Generated Semigroups. VI: The Singularity Zq = XYp. 1. Hirzebruch-Jung Singularities. 2. Semigroups and Semigroup Rings. 3. Continued Factions. 4. Two-Dimensional Cones. 5. Resolution of Singularities. VII: Two-Dimensional Regular Local Rings. 1. Ideal Transform. 2. Quadratic Transforms and Ideal Transforms. 3. Complete Ideals. 4. Factorization of Complete Ideals. 5. The Predecessors of a Simple Ideal. 6. Uniformization. 7. Resolution of Surface Singularities II: Blowing up and Normalizing. Appendices. A: Results from Classical Algebraic Geometry. 1. Generalities. 2. Affine and Finite Morphisms. 3. Products. 4. Proper Morphisms. 5. Algebraic Cones and Projective Varieties. 6. Regular and Singular points. 7. Normalization of a Variety. 8. Desingularization of a Variety. 9. Dimension of Fibres. 10. Quasifinite Morphisms and Ramification. 11. Divisors. 12. Some Results on Projections. 13. Blowing up. 14. Blowing up: the Local Rings. B: Miscellaneous Results. 1. Ordered Abelian Groups. 2. Localization. 3. Integral Extensions. 4. Some Results on Graded Rings and Modules. 5. Properties of the Rees Ring. 6. Integral Closure of Ideals. 7. Decomposition Group and Inertia Group. 8. Decomposable Rings. 9. The Dimension Formula. 10. Miscellaneous Results. Bibliography. Index of Symbols. Index.
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