Singularities of plane curves:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2000
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society lecture notes series
276 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 345 S. graph. Darst. |
| ISBN: | 0521789591 9780521789592 |
Staff View
MARC
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| 245 | 1 | 0 | |a Singularities of plane curves |c Eduardo Casas-Alvero |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2000 | |
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Record in the Search Index
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| adam_text |
Contents
Preface xi
0 Preliminaries 1
0.1 Projective spaces 1
0.2 Power series 2
0.3 Surfaces, local coordinates 3
0.4 Morphisms 4
0.5 Local rings 4
0.6 Tangent and cotangent spaces • 5
0.7 Curves 7
0.8 Germs of curves 8
0.9 Multiplicity and tangent cone 10
0.10 Smooth germs 11
0.11 Examples of singular germs 12
1 Newton Puiseux algorithm 15
1.1 Newton polygon 15
1.2 Fractionary power series 17
1.3 Search for j/ roots of f(x,y) 20
1.4 The Newton Puiseux algorithm 22
1.5 Puiseux theorem 24
1.6 Separation of y roots 28
1.7 The case of convergent series 30
1.8 Algebraic properties of C{x, y} 32
1.9 Exercises 35
2 First local properties of plane curves 39
2.1 The branches of a germ 39
2.2 The Puiseux series of a germ 40
2.3 Points on curves around O 44
2.4 Local rings of germs 46
2.5 Parameterizing branches 49
2.6 Intersection multiplicity 53
2.7 Pencils and linear systems 57
vii
viii CONTENTS
2.8 Exercises 61
3 Infinitely near points 67
3.1 Blowing up 67
3.2 Transforming curves and germs 70
3.3 Infinitely near points 76
3.4 Enriques' definition of infinitely near points 81
3.5 Proximity 82
3.6 Free and satellite points 86
3.7 Resolution of singularities 88
3.8 Equisingularity 93
3.9 Enriques diagrams 98
3.10 The ring in the first neighbourhood 102
3.11 The rings in the successive neighbourhoods 106
3.12 Artin theorem for plane curves 113
3.13 Exercises 115
4 Virtual multiplicities 119
4.1 Curves through a weighted cluster 119
4.2 When virtual multiplicities are effective 123
4.3 Blowing up all points in a cluster 131
4.4 Exceptional divisors and dual graphs 134
4.5 The total transform of a curve 138
4.6 Unloading 144
4.7 The number of conditions 149
4.8 Adjoint germs and curves 152
4.9 Noether's Af + B p theorem 154
4.10 Exercises 156
5 Analysis of branches 159
5.1 Characteristic exponents 159
5.2 The first characteristic exponent 161
5.3 Beyond the first cluster of satellite points 166
5.4 Composing blowing ups 170
5.5 Enriques' theorem 172
5.6 The inversion formula 178
5.7 The position of points 179
5.8 The semigroup of a branch 184
5.9 Approximate roots of polynomials in C{x}[y] 190
5.10 Exercises 193
6 Polar germs and related invariants 197
6.1 Definition and first properties of polar germs 197
6.2 Transforming polar germs 200
6.3 The Pliicker formula 203
6.4 The Milnor number 206
CONTENTS ix
6.5 Virtual behaviour of polar germs 208
6.6 On the effective behaviour of polar germs 210
6.7 Base points of polars and analytic invariants 217
6.8 Polar invariants of irreducible germs 221
6.9 More on infinitely near points 225
6.10 Components of the polar germs 230
6.11 Polar invariants: the general case 237
6.12 Exercises 243
7 Linear families of germs 249
7.1 Linear series on a projective line 249
7.2 Base points of pencils and linear systems 252
7.3 /i constant equisingularity criteria 259
7.4 Special germs and jacobian determinants 265
7.5 E sufficiency degree 267
7.6 E sufficiency degree and polar invariants 272
7.7 A sufficiency 277
7.8 Exercises 282
8 Valuations and complete ideals 285
8.1 Valuations of the ring Os,o 285
8.2 Classification of valuations of Os,o 295
8.3 Complete ideals and linear systems 304
8.4 Zariski's theory of complete ideals 308
8.5 Generic polar germs 314
8.6 Base points of polar germs 318
8.7 Exercises 323
A Applications to affine Geometry 325
A.I Abhyankar Moh theorem 325
A.2 Jung's theorem 330
Bibliography 333
Index 339 |
| any_adam_object | 1 |
| author | Casas-Alvero, E. 1948- |
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| dewey-full | 516.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.35 |
| dewey-search | 516.35 |
| dewey-sort | 3516.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV013333131 |
| illustrated | Illustrated |
| indexdate | 2024-07-31T00:13:31Z |
| institution | BVB |
| isbn | 0521789591 9780521789592 |
| language | English |
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| physical | XV, 345 S. graph. Darst. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | London Mathematical Society lecture notes series |
| series2 | London Mathematical Society lecture notes series |
| spelling | Casas-Alvero, E. 1948- Verfasser (DE-588)124122876 aut Singularities of plane curves Eduardo Casas-Alvero 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2000 XV, 345 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture notes series 276 Singularität <Mathematik> - Ebene Kurve Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Ebene Kurve (DE-588)4150970-5 gnd rswk-swf Ebene Kurve (DE-588)4150970-5 s Singularität Mathematik (DE-588)4077459-4 s DE-604 London Mathematical Society lecture notes series 276 (DE-604)BV000000130 276 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009091673&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Casas-Alvero, E. 1948- Singularities of plane curves London Mathematical Society lecture notes series Singularität <Mathematik> - Ebene Kurve Singularität Mathematik (DE-588)4077459-4 gnd Ebene Kurve (DE-588)4150970-5 gnd |
| subject_GND | (DE-588)4077459-4 (DE-588)4150970-5 |
| title | Singularities of plane curves |
| title_auth | Singularities of plane curves |
| title_exact_search | Singularities of plane curves |
| title_full | Singularities of plane curves Eduardo Casas-Alvero |
| title_fullStr | Singularities of plane curves Eduardo Casas-Alvero |
| title_full_unstemmed | Singularities of plane curves Eduardo Casas-Alvero |
| title_short | Singularities of plane curves |
| title_sort | singularities of plane curves |
| topic | Singularität <Mathematik> - Ebene Kurve Singularität Mathematik (DE-588)4077459-4 gnd Ebene Kurve (DE-588)4150970-5 gnd |
| topic_facet | Singularität <Mathematik> - Ebene Kurve Singularität Mathematik Ebene Kurve |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009091673&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000130 |
| work_keys_str_mv | AT casasalveroe singularitiesofplanecurves |