Topology of Algebraic Curves (eBook)

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Alex Degtyarev, Bilkent University, Ankara, Turkey.

Preface 7
I Skeletons and dessins 17
1 Graphs 19
1.1 Graphs and trees 19
1.1.1 Graphs 19
1.1.2 Trees 22
1.1.3 Dynkin diagrams 23
1.2 Skeletons 25
1.2.1 Ribbon graphs 25
1.2.2 Regions 28
1.2.3 The fundamental group 32
1.2.4 First applications 38
1.3 Pseudo-trees 42
1.3.1 Admissible trees 42
1.3.2 The counts 47
1.3.3 The associated lattice 52
2 The groups G and B3 57
2.1 The modular group G := PSL(2, Z) 57
2.1.1 The presentation of G 57
2.1.2 Subgroups 63
2.2 The braid group B3 66
2.2.1 Artin’s braid groups Bn 66
2.2.2 The Burau representation 70
2.2.3 The group B3 73
3 Trigonal curves and elliptic surfaces 79
3.1 Trigonal curves 79
3.1.1 Basic definitions and properties 79
3.1.2 Singular fibers 87
3.1.3 Special geometric structures 92
3.2 Elliptic surfaces 95
3.2.1 The local theory 95
3.2.2 Compact elliptic surfaces 99
3.3 Real structures 106
3.3.1 Real varieties 107
3.3.2 Real trigonal curves and real elliptic surfaces 112
3.3.3 Lefschetz fibrations 117
4 Dessins 125
4.1 Dessins 125
4.1.1 Trichotomic graphs 125
4.1.2 Deformations 131
4.2 Trigonal curves via dessins 134
4.2.1 The correspondence theorems 134
4.2.2 Complex curves 136
4.2.3 Generic real curves 147
4.3 First applications 153
4.3.1 Ribbon curves 153
4.3.2 Elliptic Lefschetz fibrations revisited 158
5 The braid monodromy 162
5.1 The Zariski–van Kampen theorem 162
5.1.1 The monodromy of a proper n-gonal curve 162
5.1.2 The fundamental groups 168
5.1.3 Improper curves: slopes 174
5.2 The case of trigonal curves 180
5.2.1 Monodromy via skeletons 180
5.2.2 Slopes 186
5.2.3 The strategy 189
5.3 Universal curves 193
5.3.1 Universal curves 193
5.3.2 The irreducibility criteria 195
II Applications 197
6 The metabelian invariants 199
6.1 Dihedral quotients 199
6.1.1 Uniform dihedral quotients 199
6.1.2 Geometric implications 203
6.2 The Alexander module 206
6.2.1 Statements 206
6.2.2 Proof of Theorem 6.16: the case N . 7 209
6.2.3 Congruence subgroups (the case N . 5) 212
6.2.4 The parabolic case N = 6 215
7 A few simple computations 219
7.1 Trigonal curves in .2 219
7.1.1 Proper curves in .2 219
7.1.2 Perturbations of simple singularities 223
7.2 Sextics with a non-simple triple point 229
7.2.1 A gentle introduction to plane sextics 229
7.2.2 Classification and fundamental groups 236
7.2.3 A summary of further results 237
7.3 Plane quintics 240
8 Fundamental groups of plane sextics 243
8.1 Statements 243
8.1.1 Principal results 243
8.1.2 Beginning of the proof 244
8.2 A distinguished point of type E 247
8.2.1 A point of type E8 248
8.2.2 A point of type E7 254
8.2.3 A point of type E6 260
8.3 A distinguished point of type D 275
8.3.1 A point of type Dp, p . 6 275
8.3.2 A point of type D5 279
8.3.3 A point of type D4 285
9 The transcendental lattice 291
9.1 Extremal elliptic surfaces without exceptional fibers 291
9.1.1 The tripod calculus 291
9.1.2 Proofs and further observations 293
9.2 Generalizations and examples 297
9.2.1 A computation via the homological invariant 297
9.2.2 An example 300
10 Monodromy factorizations 304
10.1 Hurwitz equivalence 304
10.1.1 Statement of the problem 304
10.1.2 Fn-valued factorizations 307
10.1.3 Sn-valued factorizations 308
10.2 Factorizations in G 313
10.2.1 Exponential examples 313
10.2.2 2-factorizations 317
10.2.3 The transcendental lattice 323
10.2.4 2-factorizations via matrices 329
10.3 Geometric applications 332
10.3.1 Extremal elliptic surfaces 332
10.3.2 Ribbon curves via skeletons 334
10.3.3 Maximal Lefschetz fibrations are algebraic 339
Appendices 343
A An algebraic complement 345
A.1 Integral lattices 345
A.1.1 Nikulin’s theory of discriminant forms 345
A.1.2 Definite lattices 347
A.2 Quotient groups 351
A.2.1 Zariski quotients 351
A.2.2 Auxiliary lemmas 352
A.2.3 Alexander module and dihedral quotients 353
B Bigonal curves in .d 356
B.1 Bigonal curves in .d 356
B.2 Plane quartics, quintics, and sextics 360
C Computer implementations 362
C.1 GAP implementations 362
C.1.1 Manipulating skeletons in GAP 362
C.1.2 Proof of Theorem 6.16 368
D Definitions and notation 375
D.1 Common notation 375
D.1.1 Groups and group actions 375
D.1.2 Topology and homotopy theory 376
D.1.3 Algebraic geometry 378
D.1.4 Miscellaneous notation 380
D.2 Index of notation 381
Bibliography 385
Index of figures 395
Index of tables 398
Index 399

"[...] The book is principally designated to researches and graduate students in topology of (complex and real) algebraic varieties, but the monograph is also very useful for mathematicians working in other fields."
Ilia Itenberg, Zentralblatt für Mathematik