Singular points of plane curves:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2004
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society student texts
63 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 370 S. |
| ISBN: | 0521839041 0521547741 |
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Record in the Search Index
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| adam_text | Contents
Preface page ix
1 Preliminaries 1
1.1 What is a plane curve? 1
1.2 Intersection numbers 6
1.3 Resultants and discriminants 8
1.4 Manifolds and the Implicit Function Theorem 9
1.5 Polar curves and inflections 12
2 Puiseux Theorem 15
2.1 Solution in power series 15
2.2 Convergent power series 19
2.3 Curves, branches, multiplicities and tangents 27
2.4 Factorisation 31
2.5 Notes 34
2.6 Exercises 37
3 Resolutions 39
3.1 Puiseux characteristics 39
3.2 Blowing up 40
3.3 Resolution of singularities 42
3.4 Geometry of the resolution 46
3.5 Infinitely near points 49
3.6 The dual graph 57
3.7 Notes 63
3.8 Exercises 65
4 Contact of two branches 67
4.1 Exponents of contact and intersection numbers 67
4.2 The Eggers tree 75
4.3 The semigroup of a branch 78
v
vi Contents
4.4 Intersections and infinitely near points 88
4.5 Decomposition of transverse polar curves 91
4.6 Notes 94
4.7 Exercises 99
5 Topology of the singularity link 103
5.1 Vector fields 103
5.2 Knots and links 108
5.3 Description of the geometry of the link 111
5.4 Cable knots 116
5.5 The Alexander polynomial 122
5.6 Notes 128
5.7 Exercises 129
6 The Milnor fibration 131
6.1 Fibrations 131
6.2 The Milnor fibration 133
6.3 First properties of the Milnor fibre 139
6.4 Euler characteristics and fibrations 141
6.5 Further formulae for /j, 144
6.6 Notes 151
6.7 Exercises 153
7 Projective curves and their duals 156
7.1 The genus of a singular curve 156
7.2 The degree of the dual curve 159
7.3 Constructible functions and Klein s equation 162
7.4 The singularities of the dual 170
7.5 Singularities of curves of a given degree 175
7.6 Notes 181
7.7 Exercises 184
8 Combinatorics on a resolution tree 187
8.1 The homology of a blow up 187
8.2 The exceptional divisor of a curve 194
8.3 Functions on the tree 198
8.4 The topological zeta function 202
8.5 Calculations for a single branch 209
8.6 Notes 216
8.7 Exercises 217
9 Decomposition of the link complement and the Milnor
fibre 219
9.1 Canonical decomposition theorems 220
Contents vii
9.2 The complement of an algebraic link 224
9.3 Resolution and plumbing 227
9.4 The Eggers tree and the resolution tree 237
9.5 Finiteness of the monodromy 242
9.6 Seifert fibre spaces 243
9.7 The Eisenbud Neumann diagram 247
9.8 Calculation of E N diagrams 251
9.9 The polar discriminant 258
9.10 Notes 261
9.11 Exercises 263
10 The monodromy and the Seifert form 265
10.1 Definition of Seifert forms 266
10.2 Use of the Thurston decomposition 269
10.3 Calculation of the monodromy 272
10.4 Algebraic classification of Seifert forms 283
10.5 Hermitian forms 289
10.6 Signatures 297
10.7 Proof of 10.6.2 and 10.6.3 302
10.8 Notes 311
10.9 Exercises 314
11 Ideals and clusters 317
11.1 Blowing up ideals 317
11.2 The valuative closure of an ideal 323
11.3 Ideals and clusters 328
11.4 Integrally closed ideals 335
11.5 Jets and determinacy 341
11.6 Local rings and differentials 348
11.7 Notes 352
11.8 Exercises 354
References 357
Index 368
|
| adam_txt |
Contents
Preface page ix
1 Preliminaries 1
1.1 What is a plane curve? 1
1.2 Intersection numbers 6
1.3 Resultants and discriminants 8
1.4 Manifolds and the Implicit Function Theorem 9
1.5 Polar curves and inflections 12
2 Puiseux' Theorem 15
2.1 Solution in power series 15
2.2 Convergent power series 19
2.3 Curves, branches, multiplicities and tangents 27
2.4 Factorisation 31
2.5 Notes 34
2.6 Exercises 37
3 Resolutions 39
3.1 Puiseux characteristics 39
3.2 Blowing up 40
3.3 Resolution of singularities 42
3.4 Geometry of the resolution 46
3.5 Infinitely near points 49
3.6 The dual graph 57
3.7 Notes 63
3.8 Exercises 65
4 Contact of two branches 67
4.1 Exponents of contact and intersection numbers 67
4.2 The Eggers tree 75
4.3 The semigroup of a branch 78
v
vi Contents
4.4 Intersections and infinitely near points 88
4.5 Decomposition of transverse polar curves 91
4.6 Notes 94
4.7 Exercises 99
5 Topology of the singularity link 103
5.1 Vector fields 103
5.2 Knots and links 108
5.3 Description of the geometry of the link 111
5.4 Cable knots 116
5.5 The Alexander polynomial 122
5.6 Notes 128
5.7 Exercises 129
6 The Milnor fibration 131
6.1 Fibrations 131
6.2 The Milnor fibration 133
6.3 First properties of the Milnor fibre 139
6.4 Euler characteristics and fibrations 141
6.5 Further formulae for /j, 144
6.6 Notes 151
6.7 Exercises 153
7 Projective curves and their duals 156
7.1 The genus of a singular curve 156
7.2 The degree of the dual curve 159
7.3 Constructible functions and Klein's equation 162
7.4 The singularities of the dual 170
7.5 Singularities of curves of a given degree 175
7.6 Notes 181
7.7 Exercises 184
8 Combinatorics on a resolution tree 187
8.1 The homology of a blow up 187
8.2 The exceptional divisor of a curve 194
8.3 Functions on the tree 198
8.4 The topological zeta function 202
8.5 Calculations for a single branch 209
8.6 Notes 216
8.7 Exercises 217
9 Decomposition of the link complement and the Milnor
fibre 219
9.1 Canonical decomposition theorems 220
Contents vii
9.2 The complement of an algebraic link 224
9.3 Resolution and plumbing 227
9.4 The Eggers tree and the resolution tree 237
9.5 Finiteness of the monodromy 242
9.6 Seifert fibre spaces 243
9.7 The Eisenbud Neumann diagram 247
9.8 Calculation of E N diagrams 251
9.9 The polar discriminant 258
9.10 Notes 261
9.11 Exercises 263
10 The monodromy and the Seifert form 265
10.1 Definition of Seifert forms 266
10.2 Use of the Thurston decomposition 269
10.3 Calculation of the monodromy 272
10.4 Algebraic classification of Seifert forms 283
10.5 Hermitian forms 289
10.6 Signatures 297
10.7 Proof of 10.6.2 and 10.6.3 302
10.8 Notes 311
10.9 Exercises 314
11 Ideals and clusters 317
11.1 Blowing up ideals 317
11.2 The valuative closure of an ideal 323
11.3 Ideals and clusters 328
11.4 Integrally closed ideals 335
11.5 Jets and determinacy 341
11.6 Local rings and differentials 348
11.7 Notes 352
11.8 Exercises 354
References 357
Index 368 |
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| spelling | Wall, C. T. C. 1936- Verfasser (DE-588)107575256 aut Singular points of plane curves C. T. C. Wall 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2004 XI, 370 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society student texts 63 Courbes planes - Congrès Curvas planas larpcal Ebene algebraische Kurve swd Singularidades (geometria algébrica) larpcal Singularités (Mathématiques) - Congrès Curves, Plane Congresses Singularities (Mathematics) Congresses Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Ebene Kurve (DE-588)4150970-5 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Singularität Mathematik (DE-588)4077459-4 s DE-604 Ebene Kurve (DE-588)4150970-5 s London Mathematical Society student texts 63 (DE-604)BV000841726 63 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014278843&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wall, C. T. C. 1936- Singular points of plane curves London Mathematical Society student texts Courbes planes - Congrès Curvas planas larpcal Ebene algebraische Kurve swd Singularidades (geometria algébrica) larpcal Singularités (Mathématiques) - Congrès Curves, Plane Congresses Singularities (Mathematics) Congresses Singularität Mathematik (DE-588)4077459-4 gnd Ebene Kurve (DE-588)4150970-5 gnd |
| subject_GND | (DE-588)4077459-4 (DE-588)4150970-5 (DE-588)1071861417 |
| title | Singular points of plane curves |
| title_auth | Singular points of plane curves |
| title_exact_search | Singular points of plane curves |
| title_exact_search_txtP | Singular points of plane curves |
| title_full | Singular points of plane curves C. T. C. Wall |
| title_fullStr | Singular points of plane curves C. T. C. Wall |
| title_full_unstemmed | Singular points of plane curves C. T. C. Wall |
| title_short | Singular points of plane curves |
| title_sort | singular points of plane curves |
| topic | Courbes planes - Congrès Curvas planas larpcal Ebene algebraische Kurve swd Singularidades (geometria algébrica) larpcal Singularités (Mathématiques) - Congrès Curves, Plane Congresses Singularities (Mathematics) Congresses Singularität Mathematik (DE-588)4077459-4 gnd Ebene Kurve (DE-588)4150970-5 gnd |
| topic_facet | Courbes planes - Congrès Curvas planas Ebene algebraische Kurve Singularidades (geometria algébrica) Singularités (Mathématiques) - Congrès Curves, Plane Congresses Singularities (Mathematics) Congresses Singularität Mathematik Ebene Kurve Konferenzschrift |
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| work_keys_str_mv | AT wallctc singularpointsofplanecurves |