Topology of algebraic curves: an approach via dessins d'enfants
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2012
|
| Series: | De Gruyter Studies in Mathematics
44 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVI, 393 S. graph. Darst. |
| ISBN: | 311025591X 9783110255911 |
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Record in the Search Index
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IMAGE 1
CONTENTS
PREFACE VII
I SKELETONS AND DESSINS
1 GRAPHS 3
1.1 GRAPHS AND TREES 3
1.1.1 GRAPHS 3
1.1.2 TREES 6
1.1.3 DYNKIN DIAGRAMS 7
1.2 SKELETONS 9
1.2.1 RIBBON GRAPHS 9
1.2.2 REGIONS 12
1.2.3 THE FUNDAMENTAL GROUP 16
1.2.4 FIRST APPLICATIONS 22
1.3 PSEUDO-TREES 26
1.3.1 ADMISSIBLE TREES 26
1.3.2 THE COUNTS 31
1.3.3 THE ASSOCIATED LATTICE 36
2 THE GROUPS T AND B3 41
2.1 THE MODULAR GROUP T := PSL{2, Z) 41
2.1.1 THE PRESENTATION OF T 41
2.1.2 SUBGROUPS 47
2.2 THE BRAID GROUP B3 50
2.2.1 ARTIN'S BRAID GROUPS B N 50
2.2.2 - THE BURAU REPRESENTATION 54
2.2.3 THE GROUP B3 57
3 TRIGONAL CURVES AND ELLIPTIC SURFACES 63
3.1 TRIGONAL CURVES 63
3.1.1 BASIC DEFINITIONS AND PROPERTIES 63
3.1.2 SINGULAR FIBERS 71
3.1.3 SPECIAL GEOMETRIC STRUCTURES 76
HTTP://D-NB.INFO/1018440798
IMAGE 2
XIV CONTENTS
3.2 ELLIPTIC SURFACES 79
3.2.1 THE LOCAL THEORY 79
3.2.2 COMPACT ELLIPTIC SURFACES 83
3.3 REAL STRUCTURES 90
3.3.1 REAL VARIETIES 91
3.3.2 REAL TRIGONAL CURVES AND REAL ELLIPTIC SURFACES 96
3.3.3 LEFSCHETZ FIBRATIONS 101
4 DESSINS 109
4.1 DESSINS 109
4.1.1 TRICHOTOMIC GRAPHS 109
4.1.2 DEFORMATIONS 115
4.2 TRIGONAL CURVES VIA DESSINS 118
4.2.1 THE CORRESPONDENCE THEOREMS 118
4.2.2 COMPLEX CURVES 120
4.2.3 GENERIC REAL CURVES 131
4.3 FIRST APPLICATIONS 137
4.3.1 RIBBON CURVES 137
4.3.2 ELLIPTIC LEFSCHETZ FIBRATIONS REVISITED 142
5 THE BRAID MONODROMY 146
5.1 THE ZARISKI-VAN KAMPEN THEOREM 146
5.1.1 THE MONODROMY OF A PROPER N-GONAL CURVE 146
5.1.2 THE FUNDAMENTAL GROUPS 152
5.1.3 IMPROPER CURVES: SLOPES 158
5.2 THE CASE OF TRIGONAL CURVES 164
5.2.1 MONODROMY VIA SKELETONS 164
5.2.2 SLOPES 170
5.2.3 THE STRATEGY 173
5.3 UNIVERSAL CURVES 177
5.3.1 UNIVERSAL CURVES 177
5.3.2 THE IRREDUCIBILITY CRITERIA 179
II APPLICATIONS
6 THE METABELIAN INVARIANTS 183
6.1 DIHEDRAL QUOTIENTS 183
6.1.1 UNIFORM DIHEDRAL QUOTIENTS 183
6.1.2 GEOMETRIC IMPLICATIONS 187
IMAGE 3
CONTENTS X V
6.2 THE ALEXANDER MODULE 190
6.2.1 STATEMENTS 190
6.2.2 PROOF OF THEOREM 6.16: THE CASE N ^ 7 193
6.2.3 CONGRUENCE SUBGROUPS (THE CASE N ^ 5) 196
6.2.4 THE PARABOLIC CASE N - 6 199
7 A FEW SIMPLE COMPUTATIONS 203
7.1 TRIGONAL CURVES IN Z2 203
7.1.1 PROPER CURVES IN 2 203
7.1.2 PERTURBATIONS OF SIMPLE SINGULARITIES 207
7.2 SEXTICS WITH A NON-SIMPLE TRIPLE POINT 213
7.2.1 A GENTLE INTRODUCTION TO PLANE SEXTICS 213
7.2.2 CLASSIFICATION AND FUNDAMENTAL GROUPS 220
7.2.3 A SUMMARY OF FURTHER RESULTS 221
7.3 PLANE QUINTICS 224
8 FUNDAMENTAL GROUPS OF PLANE SEXTICS 227
8.1 STATEMENTS 227
8.1.1 PRINCIPAL RESULTS 227
8.1.2 BEGINNING OF THE PROOF 228
8.2 A DISTINGUISHED POINT OF TYPE E 231
8.2.1 A POINT OF TYPE EG 232
8.2.2 A POINT OF TYPE E7 238
8.2.3 A POINT OF TYPE EG 244
8.3 A DISTINGUISHED POINT OF TYPE D 259
8.3.1 A POINT OF TYPE D P , P ^ 6 259
8.3.2 A POINT OF TYPE D5 263
8.3.3 A POINT OF TYPE D4 269
9 THE TRANSCENDENTAL LATTICE 275
9.1 EXTREMAL ELLIPTIC SURFACES WITHOUT EXCEPTIONAL FIBERS 275
9.1.1 THE TRIPOD CALCULUS 275
9.1.2 PROOFS AND FURTHER OBSERVATIONS 277
9.2 GENERALIZATIONS AND EXAMPLES 281
9.2.1 A COMPUTATION VIA THE HOMOLOGICAL INVARIANT 281
9.2.2 AN EXAMPLE 284
10 MONODROMY FACTORIZATIONS 288
10.1 HURWITZ EQUIVALENCE 288
10.1.1 STATEMENT OF THE PROBLEM 288
10.1.2 F N -VALUED FACTORIZATIONS 291
10.1.3 § N -VALUED FACTORIZATIONS 292
IMAGE 4
XVI CONTENTS
10.2 FACTORIZATIONS IN T 297
10.2.1 EXPONENTIAL EXAMPLES 297
10.2.2 2-FACTORIZATIONS 301
10.2.3 THE TRANSCENDENTAL LATTICE 307
10.2.4 2-FACTORIZATIONS VJ'A MATRICES 313
10.3 GEOMETRIC APPLICATIONS 316
10.3.1 EXTREMAL ELLIPTIC SURFACES 316
10.3.2 RIBBON CURVES VIA SKELETONS 318
10.3.3 MAXIMAL LEFSCHETZ FIBRATIONS ARE ALGEBRAIC 323
APPENDICES
A AN ALGEBRAIC COMPLEMENT 329
A. 1 INTEGRAL LATTICES 329
A . L . L NIKULIN'S THEORY OF DISCRIMINANT FORMS 329
A. 1.2 DEFINITE LATTICES 331
A.2 QUOTIENT GROUPS 335
A.2.1 ZARISKI QUOTIENTS 335
A.2.2 AUXILIARY LEMMAS 336
A.2.3 ALEXANDER MODULE AND DIHEDRAL QUOTIENTS 337
B BIGONAL CURVES IN 340
B.L BIGONAL CURVES IN ZRF 340
B.2 PLANE QUARTICS, QUINTICS, AND SEXTICS 344
C COMPUTER IMPLEMENTATIONS 346
C.L GAP IMPLEMENTATIONS 346
C.L.L MANIPULATING SKELETONS IN GAP 346
C.1.2 PROOF OF THEOREM 6.16 352
D DEFINITIONS AND NOTATION 359
D.L COMMON NOTATION 359
D . L . L GROUPS AND GROUP ACTIONS 359
D.L.2 TOPOLOGY AND HOMOTOPY THEORY 360
D. 1.3 ALGEBRAIC GEOMETRY 362
D.L.4 MISCELLANEOUS NOTATION 364
D.2 INDEX OF NOTATION 365
BIBLIOGRAPHY 369
INDEX OF FIGURES 379
INDEX OF TABLES 382
INDEX 383 |
| any_adam_object | 1 |
| author | Degtjarev, Aleksandr 1962- |
| author_GND | (DE-588)122288688 |
| author_facet | Degtjarev, Aleksandr 1962- |
| author_role | aut |
| author_sort | Degtjarev, Aleksandr 1962- |
| author_variant | a d ad |
| building | Verbundindex |
| bvnumber | BV040267976 |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)799887586 (DE-599)DNB1018440798 |
| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| isbn | 311025591X 9783110255911 |
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| spelling | Degtjarev, Aleksandr 1962- Verfasser (DE-588)122288688 aut Topology of algebraic curves an approach via dessins d'enfants Alex Degtyarev Berlin [u.a.] De Gruyter 2012 XVI, 393 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematics 44 Verbraucherverhalten (DE-588)4062644-1 gnd rswk-swf Topologische Graphentheorie (DE-588)4341226-9 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 s Topologische Graphentheorie (DE-588)4341226-9 s DE-604 Verbraucherverhalten (DE-588)4062644-1 s Erscheint auch als Online-Ausgabe 978-3-11-025842-4 De Gruyter Studies in Mathematics 44 (DE-604)BV000005407 44 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3950669&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025123606&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Degtjarev, Aleksandr 1962- Topology of algebraic curves an approach via dessins d'enfants De Gruyter Studies in Mathematics Verbraucherverhalten (DE-588)4062644-1 gnd Topologische Graphentheorie (DE-588)4341226-9 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| subject_GND | (DE-588)4062644-1 (DE-588)4341226-9 (DE-588)4001165-3 |
| title | Topology of algebraic curves an approach via dessins d'enfants |
| title_auth | Topology of algebraic curves an approach via dessins d'enfants |
| title_exact_search | Topology of algebraic curves an approach via dessins d'enfants |
| title_full | Topology of algebraic curves an approach via dessins d'enfants Alex Degtyarev |
| title_fullStr | Topology of algebraic curves an approach via dessins d'enfants Alex Degtyarev |
| title_full_unstemmed | Topology of algebraic curves an approach via dessins d'enfants Alex Degtyarev |
| title_short | Topology of algebraic curves |
| title_sort | topology of algebraic curves an approach via dessins d enfants |
| title_sub | an approach via dessins d'enfants |
| topic | Verbraucherverhalten (DE-588)4062644-1 gnd Topologische Graphentheorie (DE-588)4341226-9 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| topic_facet | Verbraucherverhalten Topologische Graphentheorie Algebraische Kurve |
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| volume_link | (DE-604)BV000005407 |
| work_keys_str_mv | AT degtjarevaleksandr topologyofalgebraiccurvesanapproachviadessinsdenfants |