Resolution of curve and surface singularities in characteristic zero:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
2004
|
| Schriftenreihe: | Algebras and applications
4 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 483 S. |
| ISBN: | 1402020287 1402020295 |
Internformat
MARC
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| 100 | 1 | |a Kiyek, Karl-Heinz |d 1936- |e Verfasser |0 (DE-588)12085001X |4 aut | |
| 245 | 1 | 0 | |a Resolution of curve and surface singularities in characteristic zero |c by K. Kiyek and J. L. Vicente |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 2004 | |
| 300 | |a XXI, 483 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algebras and applications |v 4 | |
| 650 | 4 | |a Cohen-Macaulay, Anneaux | |
| 650 | 4 | |a Courbes | |
| 650 | 4 | |a Singularités (Mathématiques) | |
| 650 | 4 | |a Surfaces algébriques | |
| 650 | 4 | |a Valuations, Théorie des | |
| 650 | 4 | |a Cohen-Macaulay rings | |
| 650 | 4 | |a Curves | |
| 650 | 4 | |a Singularities (Mathematics) | |
| 650 | 4 | |a Surfaces, Algebraic | |
| 650 | 4 | |a Valuation theory | |
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Datensatz im Suchindex
| _version_ | 1804135373076430848 |
|---|---|
| adam_text | Contents
Preface xi
Note to the Reader xx
Terminology xxi
I Valuation Theory 1
1 Marot Rings 1
1.1 Marot Rings 2
1.2 Large Quotient Ring 3
1.3 Rings With Large Jacobson Radical 4
2 Manis Valuation Rings 5
2.1 Manis Valuation Rings 5
2.2 Manis Valuations 9
2.3 The Approximation Theorem For Discrete Manis Valuations 14
3 Valuation Rings and Valuations 17
3.1 Valuation Rings 17
3.2 Subrings and Overrings of Valuation Rings 18
3.3 Valuations 20
3.4 Composite Valuations 23
3.5 Discrete Valuations 24
3.6 Existence of Valuations of the Second Kind 26
4 The Approximation Theorem For Independent Valuations 27
5 Extensions of Valuations 29
5.1 Existence of Extensions 29
5.2 Reduced Ramification Index and Residue Degree 30
5.3 Extension of Composite Valuations 30
6 Extending Valuations to Algebraic Overfields 31
6.1 Some General Results 31
6.2 The Formula ef n 33
6.3 The Formula J2eifi n 36
6.4 The Formula J2eifi = n 38
7 Extensions of Discrete Valuations 40
v
vi
7.1 Intersections of Discrete Valuation Rings 40
7.2 Extensions of Discrete Valuations 40
7.3 Some Classes of Extensions 41
7.4 Quadratic Number Fields 44
8 Ramification Theory of Valuations 46
8.1 Generalities 46
8.2 The Value Groups T, Fz and TT 47
8.3 The Ramification Group 49
9 Extending Valuations to Non Algebraic Overfields 53
10 Valuations of Algebraic Function Fields 56
11 Valuations Dominating a Local Domain 59
II One Dimensional Semilocal Cohen Macaulay Rings 67
1 Transversal Elements 67
1.1 Adic topologies 67
1.2 The Hilbert Polynomial 69
1.3 Transversal Elements 70
2 Integral Closure of One Dimensional Semilocal Cohen Macaulay
Rings 74
2.1 Invertible Modules 74
2.2 The Integral Closure 75
2.3 Integral Closure and Manis Valuation Rings 77
3 One Dimensional Analytically Unramified and Analytically Irre¬
ducible CM Rings 79
3.1 Two Length Formulae 79
3.2 Divisible Modules 80
3.3 Compatible Extensions 81
3.4 Criteria for One Dimensional Analytically Unramified and
Analytically Irreducible CM Rings 84
4 Blowing up Ideals 88
4.1 The Blow up Ring Ra 88
4.2 Integral Closure 91
4.3 Stable Ideals 92
5 Infinitely Near Rings 96
III Differential Modules and Ramification 101
1 Introduction 101
2 Norms and Traces 104
2.1 Some Linear Algebra 105
2.2 Determinant and Characteristic Polynomial 106
2.3 The Trace Form 109
3 Formally Unramified and Unramified Extensions 113
3.1 The Branch Locus 113
3.2 Some Ramification Criteria 115
vii
3.3 Ramification for Local Rings and Applications 120
3.4 Discrete Valuation Rings and Ramification 124
4 Unramified Extensions and Discriminants 125
5 Ramification For Quasilocal Rings 131
6 Integral Closure and Completion 134
IV Formal and Convergent Power Series Rings 143
1 Formal Power Series Rings 143
2 Convergent Power Series Rings 146
3 Weierstrafi Preparation Theorem 147
3.1 Weierstrafi Division Theorem 147
3.2 Weierstrafi Preparation Theorem and Applications 152
4 The Category of Formal and Analytic Algebras 157
4.1 Local fc algebras 157
4.2 Morphisms of Formal and Analytic Algebras 157
4.3 Integral Extensions 162
4.4 Noether Normalization 163
5 Extensions of Formal and Analytic Algebras 166
V Quasiordinary Singularities 169
1 Fractionary Power Series 169
1.1 Generalities 169
1.2 Intermediate Fields 171
1.3 Intermediate Fields Generated by a Fractionary Power Series 175
2 The Jung Abhyankar Theorem: Formal Case 177
3 The Jung Abhyankar Theorem: Analytic Case 181
4 Quasiordinary Power Series 182
5 A Generalized Newton Algorithm 190
5.1 The Algorithm 190
5.2 An Example 195
6 Strictly Generated Semigroups 198
6.1 Generalities 198
6.2 Strictly Generated Semigroups 202
VI The Singularity Z« = XY? 205
1 Hirzebruch Jung Singularities 205
2 Semigroups and Semigroup Rings 210
2.1 Generalities 210
2.2 Integral Closure of Semigroup Rings 213
3 Continued Fractions 215
3.1 Continued Fractions 216
3.2 Hirzebruch Jung Continued Fractions 218
4 Two Dimensional Cones 222
4.1 Two dimensional Cones and Semigroups 222
4.2 The Boundary Polygon of a and the Ideal of Xa 225
viii
5 Resolution of Singularities 235
5.1 Some Useful Formulae 235
5.2 The Case p = 1 237
5.3 The General Case 238
5.4 Counting Singularities of the Blow up 244
VII Two Dimensional Regular Local Rings 247
1 Ideal Transform 247
1.1 Generalities 247
1.2 Ideal Transforms 249
2 Quadratic Transforms and Ideal Transforms 252
2.1 Generalities 252
2.2 Quadratic Transforms and the First Neighborhood 255
2.3 Ideal Transforms 257
2.4 Valuations Dominating R 260
3 Complete Ideals 261
3.1 Generalities 261
3.2 Complete Ideals as Intersections 263
3.3 When Does m Divide a Complete Ideal? 268
3.4 An Existence Theorem 271
4 Factorization of Complete Ideals 273
4.1 Preliminary Results 273
4.2 Contracted Ideals 278
4.3 Unique Factorization 281
5 The Predecessors of a Simple Ideal 282
6 The Quadratic Sequence 287
7 Proximity 292
8 Resolution of Embedded Curves 296
VIII Resolution of Singularities 303
1 Blowing up Curve Singularities 303
2 Resolution of Surface Singularities I: Jung s Method 310
3 Quadratic Dilatations 313
3.1 Quadratic Dilatations 313
3.2 Quadratic Dilatations and Algebraic Varieties 314
4 Quadratic Dilatations of Two Dimensional Regular Local Rings . . 316
5 Valuations of Algebraic Function Fields in Two Variables 320
6 Uniformization 324
6.1 Classification of Valuations and Local Uniformization . . . 324
6.2 Existence of Subrings Lying Under a Local Ring 328
6.3 Uniformization 330
7 Resolution of Surface Singularities II: Blowing up and Normalizing 334
7.1 Principalization 334
7.2 Tangential Ideals 339
ix
7.3 The Main Result 343
Appendices 345
A Results from Classical Algebraic Geometry 345
1 Generalities 345
1.1 Ideals and Varieties 345
1.2 Rational Functions and Maps 347
1.3 Coordinate Ring and Local Rings 348
1.4 Dominant Morphisms and Closed Embeddings 349
1.5 Elementary Open Sets 350
1.6 Varieties as Topological Spaces 351
1.7 Local Ring on a Subvariety 353
2 Affine and Finite Morphisms 355
3 Products 357
4 Proper Morphisms 361
4.1 Space of Irreducible Closed Subsets 362
4.2 Varieties and the Functor t 365
4.3 Proper Morphisms 367
5 Algebraic Cones and Projective Varieties 370
6 Regular and Singular Points 373
7 Normalization of a Variety 378
8 Desingularization of a Variety 384
9 Dimension of Fibres 385
10 Quasifinite Morphisms and Ramification 387
10.1 Quasifinite Morphisms 387
10.2 Ramification 389
11 Divisors 392
12 Some Results on Projections 395
13 Blowing up 398
14 Blowing up: The Local Rings 403
B Miscellaneous Results 409
1 Ordered Abelian Groups 409
1.1 Isolated Subgroups 409
1.2 Initial Index 411
1.3 Archimedean Ordered Groups 412
1.4 The Rational Rank of an Abelian Group 415
2 Localization 419
3 Integral Extensions 421
4 Some Results on Graded Rings and Modules 423
4.1 Generalities 423
4.2 M Graded Rings and M Graded Modules 424
4.3 Homogeneous Localization 426
4.4 Integral Closure of Graded Rings 430
X
5 Properties of the Rees Ring 433
6 Integral Closure of Ideals 437
6.1 Generalities 437
6.2 Integral Closure of Ideals 437
6.3 Integral Closure of Ideals and Valuation Theory 440
7 Decomposition Group and Inertia Group 442
8 Decomposable Rings 448
9 The Dimension Formula 449
10 Miscellaneous Results 452
10.1 The Chinese Remainder Theorem 452
10.2 Separable Noether Normalization 453
10.3 The Segre Ideal 454
10.4 Adjoining an Indeterminate 456
10.5 Divisor Group and Class Group 457
10.6 Calculating a Multiplicity 457
10.7 A Length Formula 458
10.8 Quasifmite Modules 459
10.9 Maximal Primary Ideals 460
10.10 Primary Decomposition in Non Noetherian Rings 461
10.11 Discriminant of a Polynomial 462
Bibliography 463
Index of Symbols 475
Index 478
|
| adam_txt |
Contents
Preface xi
Note to the Reader xx
Terminology xxi
I Valuation Theory 1
1 Marot Rings 1
1.1 Marot Rings 2
1.2 Large Quotient Ring 3
1.3 Rings With Large Jacobson Radical 4
2 Manis Valuation Rings 5
2.1 Manis Valuation Rings 5
2.2 Manis Valuations 9
2.3 The Approximation Theorem For Discrete Manis Valuations 14
3 Valuation Rings and Valuations 17
3.1 Valuation Rings 17
3.2 Subrings and Overrings of Valuation Rings 18
3.3 Valuations 20
3.4 Composite Valuations 23
3.5 Discrete Valuations 24
3.6 Existence of Valuations of the Second Kind 26
4 The Approximation Theorem For Independent Valuations 27
5 Extensions of Valuations 29
5.1 Existence of Extensions 29
5.2 Reduced Ramification Index and Residue Degree 30
5.3 Extension of Composite Valuations 30
6 Extending Valuations to Algebraic Overfields 31
6.1 Some General Results 31
6.2 The Formula ef n 33
6.3 The Formula J2eifi n 36
6.4 The Formula J2eifi = n 38
7 Extensions of Discrete Valuations 40
v
vi
7.1 Intersections of Discrete Valuation Rings 40
7.2 Extensions of Discrete Valuations 40
7.3 Some Classes of Extensions 41
7.4 Quadratic Number Fields 44
8 Ramification Theory of Valuations 46
8.1 Generalities 46
8.2 The Value Groups T, Fz and TT 47
8.3 The Ramification Group 49
9 Extending Valuations to Non Algebraic Overfields 53
10 Valuations of Algebraic Function Fields 56
11 Valuations Dominating a Local Domain 59
II One Dimensional Semilocal Cohen Macaulay Rings 67
1 Transversal Elements 67
1.1 Adic topologies 67
1.2 The Hilbert Polynomial 69
1.3 Transversal Elements 70
2 Integral Closure of One Dimensional Semilocal Cohen Macaulay
Rings 74
2.1 Invertible Modules 74
2.2 The Integral Closure 75
2.3 Integral Closure and Manis Valuation Rings 77
3 One Dimensional Analytically Unramified and Analytically Irre¬
ducible CM Rings 79
3.1 Two Length Formulae 79
3.2 Divisible Modules 80
3.3 Compatible Extensions 81
3.4 Criteria for One Dimensional Analytically Unramified and
Analytically Irreducible CM Rings 84
4 Blowing up Ideals 88
4.1 The Blow up Ring Ra 88
4.2 Integral Closure 91
4.3 Stable Ideals 92
5 Infinitely Near Rings 96
III Differential Modules and Ramification 101
1 Introduction 101
2 Norms and Traces 104
2.1 Some Linear Algebra 105
2.2 Determinant and Characteristic Polynomial 106
2.3 The Trace Form 109
3 Formally Unramified and Unramified Extensions 113
3.1 The Branch Locus 113
3.2 Some Ramification Criteria 115
vii
3.3 Ramification for Local Rings and Applications 120
3.4 Discrete Valuation Rings and Ramification 124
4 Unramified Extensions and Discriminants 125
5 Ramification For Quasilocal Rings 131
6 Integral Closure and Completion 134
IV Formal and Convergent Power Series Rings 143
1 Formal Power Series Rings 143
2 Convergent Power Series Rings 146
3 Weierstrafi Preparation Theorem 147
3.1 Weierstrafi Division Theorem 147
3.2 Weierstrafi Preparation Theorem and Applications 152
4 The Category of Formal and Analytic Algebras 157
4.1 Local fc algebras 157
4.2 Morphisms of Formal and Analytic Algebras 157
4.3 Integral Extensions 162
4.4 Noether Normalization 163
5 Extensions of Formal and Analytic Algebras 166
V Quasiordinary Singularities 169
1 Fractionary Power Series 169
1.1 Generalities 169
1.2 Intermediate Fields 171
1.3 Intermediate Fields Generated by a Fractionary Power Series 175
2 The Jung Abhyankar Theorem: Formal Case 177
3 The Jung Abhyankar Theorem: Analytic Case 181
4 Quasiordinary Power Series 182
5 A Generalized Newton Algorithm 190
5.1 The Algorithm 190
5.2 An Example 195
6 Strictly Generated Semigroups 198
6.1 Generalities 198
6.2 Strictly Generated Semigroups 202
VI The Singularity Z« = XY? 205
1 Hirzebruch Jung Singularities 205
2 Semigroups and Semigroup Rings 210
2.1 Generalities 210
2.2 Integral Closure of Semigroup Rings 213
3 Continued Fractions 215
3.1 Continued Fractions 216
3.2 Hirzebruch Jung Continued Fractions 218
4 Two Dimensional Cones 222
4.1 Two dimensional Cones and Semigroups 222
4.2 The Boundary Polygon of a and the Ideal of Xa 225
viii
5 Resolution of Singularities 235
5.1 Some Useful Formulae 235
5.2 The Case p = 1 237
5.3 The General Case 238
5.4 Counting Singularities of the Blow up 244
VII Two Dimensional Regular Local Rings 247
1 Ideal Transform 247
1.1 Generalities 247
1.2 Ideal Transforms 249
2 Quadratic Transforms and Ideal Transforms 252
2.1 Generalities 252
2.2 Quadratic Transforms and the First Neighborhood 255
2.3 Ideal Transforms 257
2.4 Valuations Dominating R 260
3 Complete Ideals 261
3.1 Generalities 261
3.2 Complete Ideals as Intersections 263
3.3 When Does m Divide a Complete Ideal? 268
3.4 An Existence Theorem 271
4 Factorization of Complete Ideals 273
4.1 Preliminary Results 273
4.2 Contracted Ideals 278
4.3 Unique Factorization 281
5 The Predecessors of a Simple Ideal 282
6 The Quadratic Sequence 287
7 Proximity 292
8 Resolution of Embedded Curves 296
VIII Resolution of Singularities 303
1 Blowing up Curve Singularities 303
2 Resolution of Surface Singularities I: Jung's Method 310
3 Quadratic Dilatations 313
3.1 Quadratic Dilatations 313
3.2 Quadratic Dilatations and Algebraic Varieties 314
4 Quadratic Dilatations of Two Dimensional Regular Local Rings . . 316
5 Valuations of Algebraic Function Fields in Two Variables 320
6 Uniformization 324
6.1 Classification of Valuations and Local Uniformization . . . 324
6.2 Existence of Subrings Lying Under a Local Ring 328
6.3 Uniformization 330
7 Resolution of Surface Singularities II: Blowing up and Normalizing 334
7.1 Principalization 334
7.2 Tangential Ideals 339
ix
7.3 The Main Result 343
Appendices 345
A Results from Classical Algebraic Geometry 345
1 Generalities 345
1.1 Ideals and Varieties 345
1.2 Rational Functions and Maps 347
1.3 Coordinate Ring and Local Rings 348
1.4 Dominant Morphisms and Closed Embeddings 349
1.5 Elementary Open Sets 350
1.6 Varieties as Topological Spaces 351
1.7 Local Ring on a Subvariety 353
2 Affine and Finite Morphisms 355
3 Products 357
4 Proper Morphisms 361
4.1 Space of Irreducible Closed Subsets 362
4.2 Varieties and the Functor t 365
4.3 Proper Morphisms 367
5 Algebraic Cones and Projective Varieties 370
6 Regular and Singular Points 373
7 Normalization of a Variety 378
8 Desingularization of a Variety 384
9 Dimension of Fibres 385
10 Quasifinite Morphisms and Ramification 387
10.1 Quasifinite Morphisms 387
10.2 Ramification 389
11 Divisors 392
12 Some Results on Projections 395
13 Blowing up 398
14 Blowing up: The Local Rings 403
B Miscellaneous Results 409
1 Ordered Abelian Groups 409
1.1 Isolated Subgroups 409
1.2 Initial Index 411
1.3 Archimedean Ordered Groups 412
1.4 The Rational Rank of an Abelian Group 415
2 Localization 419
3 Integral Extensions 421
4 Some Results on Graded Rings and Modules 423
4.1 Generalities 423
4.2 M Graded Rings and M Graded Modules 424
4.3 Homogeneous Localization 426
4.4 Integral Closure of Graded Rings 430
X
5 Properties of the Rees Ring 433
6 Integral Closure of Ideals 437
6.1 Generalities 437
6.2 Integral Closure of Ideals 437
6.3 Integral Closure of Ideals and Valuation Theory 440
7 Decomposition Group and Inertia Group 442
8 Decomposable Rings 448
9 The Dimension Formula 449
10 Miscellaneous Results 452
10.1 The Chinese Remainder Theorem 452
10.2 Separable Noether Normalization 453
10.3 The Segre Ideal 454
10.4 Adjoining an Indeterminate 456
10.5 Divisor Group and Class Group 457
10.6 Calculating a Multiplicity 457
10.7 A Length Formula 458
10.8 Quasifmite Modules 459
10.9 Maximal Primary Ideals 460
10.10 Primary Decomposition in Non Noetherian Rings 461
10.11 Discriminant of a Polynomial 462
Bibliography 463
Index of Symbols 475
Index 478 |
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| author | Kiyek, Karl-Heinz 1936- Vicente, José Luis |
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| ctrlnum | (OCoLC)55535141 (DE-599)BVBBV021594561 |
| dewey-full | 516.3/5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| genre | (DE-588)4006432-3 Bibliografie gnd-content |
| genre_facet | Bibliografie |
| id | DE-604.BV021594561 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:45:35Z |
| indexdate | 2024-07-09T20:39:28Z |
| institution | BVB |
| isbn | 1402020287 1402020295 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014809986 |
| oclc_num | 55535141 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | XXI, 483 S. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Kluwer Acad. Publ. |
| record_format | marc |
| series | Algebras and applications |
| series2 | Algebras and applications |
| spelling | Kiyek, Karl-Heinz 1936- Verfasser (DE-588)12085001X aut Resolution of curve and surface singularities in characteristic zero by K. Kiyek and J. L. Vicente Dordrecht [u.a.] Kluwer Acad. Publ. 2004 XXI, 483 S. txt rdacontent n rdamedia nc rdacarrier Algebras and applications 4 Cohen-Macaulay, Anneaux Courbes Singularités (Mathématiques) Surfaces algébriques Valuations, Théorie des Cohen-Macaulay rings Curves Singularities (Mathematics) Surfaces, Algebraic Valuation theory Fläche (DE-588)4129864-0 gnd rswk-swf Charakteristik Null (DE-588)4472918-2 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Kurve (DE-588)4033824-1 gnd rswk-swf (DE-588)4006432-3 Bibliografie gnd-content Singularität Mathematik (DE-588)4077459-4 s Kurve (DE-588)4033824-1 s Fläche (DE-588)4129864-0 s Charakteristik Null (DE-588)4472918-2 s b DE-604 Vicente, José Luis Verfasser aut Algebras and applications 4 (DE-604)BV035420975 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014809986&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kiyek, Karl-Heinz 1936- Vicente, José Luis Resolution of curve and surface singularities in characteristic zero Algebras and applications Cohen-Macaulay, Anneaux Courbes Singularités (Mathématiques) Surfaces algébriques Valuations, Théorie des Cohen-Macaulay rings Curves Singularities (Mathematics) Surfaces, Algebraic Valuation theory Fläche (DE-588)4129864-0 gnd Charakteristik Null (DE-588)4472918-2 gnd Singularität Mathematik (DE-588)4077459-4 gnd Kurve (DE-588)4033824-1 gnd |
| subject_GND | (DE-588)4129864-0 (DE-588)4472918-2 (DE-588)4077459-4 (DE-588)4033824-1 (DE-588)4006432-3 |
| title | Resolution of curve and surface singularities in characteristic zero |
| title_auth | Resolution of curve and surface singularities in characteristic zero |
| title_exact_search | Resolution of curve and surface singularities in characteristic zero |
| title_exact_search_txtP | Resolution of curve and surface singularities in characteristic zero |
| title_full | Resolution of curve and surface singularities in characteristic zero by K. Kiyek and J. L. Vicente |
| title_fullStr | Resolution of curve and surface singularities in characteristic zero by K. Kiyek and J. L. Vicente |
| title_full_unstemmed | Resolution of curve and surface singularities in characteristic zero by K. Kiyek and J. L. Vicente |
| title_short | Resolution of curve and surface singularities in characteristic zero |
| title_sort | resolution of curve and surface singularities in characteristic zero |
| topic | Cohen-Macaulay, Anneaux Courbes Singularités (Mathématiques) Surfaces algébriques Valuations, Théorie des Cohen-Macaulay rings Curves Singularities (Mathematics) Surfaces, Algebraic Valuation theory Fläche (DE-588)4129864-0 gnd Charakteristik Null (DE-588)4472918-2 gnd Singularität Mathematik (DE-588)4077459-4 gnd Kurve (DE-588)4033824-1 gnd |
| topic_facet | Cohen-Macaulay, Anneaux Courbes Singularités (Mathématiques) Surfaces algébriques Valuations, Théorie des Cohen-Macaulay rings Curves Singularities (Mathematics) Surfaces, Algebraic Valuation theory Fläche Charakteristik Null Singularität Mathematik Kurve Bibliografie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014809986&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035420975 |
| work_keys_str_mv | AT kiyekkarlheinz resolutionofcurveandsurfacesingularitiesincharacteristiczero AT vicentejoseluis resolutionofcurveandsurfacesingularitiesincharacteristiczero |