The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noether's works on plane curves [cf. [148], [149]]. And probably the origin of the problem was to have a formula to compute the genus of a plane curve. The genus is the most useful birational invariant of a curve in classical projective geometry. It was long known that, for a plane curve of degree n having l m ordinary singular points with respective multiplicities ri, i E {1, . . . , m}, the genus p of the curve is given by the formula = (n - l)(n - 2) _ ~ "r. (r. _ 1) P 2 2 L. . ,. •• . Of course, the problem now arises: how to compute the genus of a plane curve having some non-ordinary singularities. This leads to the natural question: can we birationally transform any (singular) plane curve into another one having only ordinary singularities? The answer is positive. Let us give a flavor (without proofs) 2 on how Noether did it • To solve the problem, it is enough to consider a special kind of Cremona trans formations, namely quadratic transformations of the projective plane. Let ~ be a linear system of conics with three non-collinear base points r = {Ao, AI, A }, 2 and take a projective frame of the type {Ao, AI, A ; U}.
Author(s): K. Kiyek, J. L. Vicente
Series: Algebra and Applications 4
Edition: 2004
Publisher: Springer
Year: 2004
Language: English
Pages: 506
Preface
The Curves
The Point of View of Max Noether
The First Resolution of Singularities of Curves
The Controlled Process
The Surfaces
Jung Methods
The Zariski Method
Prerequisites
Main Sources
Acknowledgements
Note to the Reader
Terminology
Chapter I: Valuation Theory
1. Marot Rings
1.1 Marot Rings
1.2 Large Quotient Ring
1.3 Rings With Large Jacobson Radical
2. Manis Valuation Rings
2.1 Manis Valuation Rings
2.2 Manis Valuations
2.3 The Approximation Theorem For Discrete Manis Valuations
3. Valuation Rings and Valuations
3.1 Valuation Rings
3.2 Subrings and Overrings of Valuation Rings
3.3 Valuations
3.4 Composite Valuations
3.5 Discrete Valuations
3.6 Existence of Valuations of the Second Kind
4. The Approximation Theorem For Independent Valuations
5. Extensions of Valuations
5.1 Existence of Extensions
5.2 Reduced Ramification Index and Residue Degree
5.3 Extension of Composite Valuations
6. Extending Valuations to Algebraic Overfields
6.1 Some General Results
6.2 The Formula ef ≤ n
6.3 The Formula Σe_i f_i ≤ n
6.4 The Formula Σe_i f_i = n
7. Extensions of Discrete Valuations
7.1 Intersections of Discrete Valuation Rings
7.2 Extensions of Discrete Valuations
7.3 Some Classes of Extensions
7.4 Quadratic Number Fields
8. Ramification Theory of Valuations
8.1 Generalities
8.2 The Value Groups Γ, Γ_z, Γ_T
8.3 The Ramification Group
9. Extending Valuations to Non-Algebraic Overfields
10. Valuations of Algebraic Function Fields
11. Valuations Dominating a Local Domain
Chapter II: One-Dimensional Semilocal Cohen-Macaulay Rings
1. Transversal Elements
1.1 Adic topologies
1.2 The Hilbert Polynomial
1.3 Transversal Elements
2. Integral Closure of One-Dimensional Semilocal Cohen-Macaulay Rings
2.1 Invertible Modules
2.2 The Integral Closure
2.3 Integral Closure and Manis Valuation Rings
3. One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings
3.1 Two Length Formulae
3.2 Divisible Modules
3.3 Compatible Extensions
3.4 Criteria for One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings
4. Blowing up Ideals
4.1 The Blow-up Ring Rᵃ
4.2 Integral Closure
4.3 Stable Ideals
5. Infinitely Near Rings
Chapter III: Differential Modules and Ramification
1. Introduction
2. Norms and Traces
2.1 Some Linear Algebra
2.2 Determinant and Characteristic Polynomial
2.3 The Trace Form
3. Formally Unramified and Unramified Extensions
3.1 The Branch Locus
3.2 Some Ramification Criteria
3.3 Ramification for Local Rings and Applications
3.4 Discrete Valuation Rings and Ramification
4. Unramified Extensions and Discriminants
5. Ramification For Quasilocal Rings
6. Integral Closure and Completion
Chapter IV: Formal and Convergent Power Series Rings
1. Formal Power Series Rings
2. Convergent Power Series Rings
3. Weierstraß Preparation Theorem
3.1 Weierstraß Division Theorem
3.2 Weierstraß Preparation Theorem and Applications
4. The Category of Formal and Analytic Algebras
4.1 Local k-algebras
4.2 Morphisms of Formal and Analytic Algebras
4.3 Integral Extensions
4.4 Noether Normalization
5. Extensions of Formal and Analytic Algebras
Chapter V: Quasiordinary Singularities
1. Fractionary Power Series
1.1 Generalities
1.2 Intermediate Fields
1.3 Intermediate Fields Generated by a 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
5.1 The Algorithm
5.2 An Example
6. Strictly Generated Semigroups
6.1 Generalities
6.2 Strictly Generated Semigroups
Chapter VI: The Singularity Z^q = XY^p
1. Hirzebruch-Jung Singularities
2. Semigroups and Semigroup Rings
2.1 Generalities
2.2 Integral Closure of Semigroup Rings
3. Continued Fractions
3.1 Continued Fractions
3.2 Hirzebruch-Jung Continued Fractions
4. Two-Dimensional Cones
4.1 Two-dimensional Cones and Semigroups
4.2 The Boundary Polygon of σ and the Ideal of X_σ
5. Resolution of Singularities
5.1 Some Useful Formulae
5.2 The Case p=1
5.3 The General Case
5.4 Counting Singularities of the Blow-up
Chapter VII: Two-Dimensional Regular Local Rings
1. Ideal Transform
1.1 Generalities
1.2 Ideal Transforms
2. Quadratic Transforms and Ideal Transforms
2.1 Generalities
2.2 Quadratic Transforms and the First Neighborhood
2.3 Ideal Transforms
2.4 Valuations Dominating R
3. Complete Ideals
3.1 Generalities
3.2 Complete Ideals as Intersections
3.3 When Does m Divide a Complete Ideal?
3.4 An Existence Theorem
4. Factorization of Complete Ideals
4.1 Preliminary Results
4.2 Contracted Ideals
4.3 Unique Factorization
5. The Predecessors of a Simple Ideal
6. The Quadratic Sequence
7. Proximity
8. Resolution of Embedded Curves
Chapter VIII: Resolution of Singularities
1. Blowing up Curve Singularities
2. Resolution of Surface Singularities I: Jung's Method
3. Quadratic Dilatations
3.1 Quadratic Dilatations
3.2 Quadratic Dilatations and Algebraic Varieties
4. Quadratic Dilatations of Two-Dimensional Regular Local Rings
5. Valuations of Algebraic Function Fields in Two Variables
6. Uniformization
6.1 Classification of Valuations and Local Uniformization
6.2 Existence of Subrings Lying Under a Local Ring
6.3 Uniformization
7. Resolution of Surface Singularities II: Blowing up and Normalizing
7.1 Principalization
7.2 Tangential Ideals
7.3 The Main Result
Appendix A: Results from Classical Algebraic Geometry
1. Generalities
1.1 Ideals and Varieties
1.2 Rational Functions and Maps
1.3 Coordinate Ring and Local Rings
1.4 Dominant Morphisms and Closed Embeddings
1.5 Elementary Open Sets
1.6 Varieties as Topological Spaces
1.7 Local Ring on a Subvariety
2. Affine and Finite Morphisms
3. Products
4. Proper Morphisms
4.1 Space of Irreducible Closed Subsets
4.2 Varieties and the Functor t
4.3 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
10.1 Quasifinite Morphisms
10.2 Ramification
11. Divisors
12. Some Results on Projections
13. Blowing up
14. Blowing up: The Local Rings
Appendix B.
Miscellaneous Results
1. Ordered Abelian Groups
1.1 Isolated Subgroups
1.2 Initial Index
1.3 Archimedean Ordered Groups
1.4 The Rational Rank of an Abelian Group
2. Localization
3. Integral Extensions
4. Some Results on Graded Rings and Modules
4.1 Generalities
4.2 M-Graded Rings and M-Graded Modules
4.3 Homogeneous Localization
4.4 Integral Closure of Graded Rings
5. Properties of the Rees Ring
6. Integral Closure of Ideals
6.1 Generalities
6.2 Integral Closure of Ideals
6.3 Integral Closure of Ideals and Valuation Theory
7. Decomposition Group and Inertia Group
8. Decomposable Rings
9. The Dimension Formula
10. Miscellaneous Results
10.1 The Chinese Remainder Theorem
10.2 Separable Noether Normalization
10.3 The Segre Ideal
10.4 Adjoining an Indeterminate
10.5 Divisor Group and Class Group
10.6 Calculating a Multiplicity
10.7 A Length Formula
10.8 Quasifinite Modules
10.9 Maximal Primary Ideals
10.10 Primary Decomposition in Non-Noetherian Rings
10.11 Discriminant of a Polynomial
Bibliography
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[52]
[69]
[87]
[105]
[124]
[143]
[161]
[180]
[198]
Index of Symbols
Index
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