Degtyarev Topology of Algebraic Curves

An Approach via Dessins d'Enfants
1. Auflage 2012
ISBN: 978-3-11-025842-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 44, 409 Seiten

Reihe: De Gruyter Studies in MathematicsISSN

This monograph summarizes and extends a number of results on the topology of trigonal curves in geometrically ruled surfaces. An emphasis is given to various applications of the theory to a few related areas, most notably singular plane curves of small degree, elliptic surfaces, and Lefschetz fibrations (both complex and real), and Hurwitz equivalence of braid monodromy factorizations. The approach relies on a close relation between trigonal curves/elliptic surfaces, a certain class of ribbon graphs, and subgroups of the modular group, which provides a combinatorial framework for the study of geometric objects. A brief summary of the necessary auxiliary results and techniques used and a background of the principal problems dealt with are included in the text. The book is intended to researchers and graduate students in the field of topology of complex and real algebraic varieties.

Zielgruppe

Researchers, Lecturers, and Graduate Students in Mathematics; Academic Libraries

Autoren/Hrsg.

Degtyarev, Alex

Fachgebiete

• Mathematik | Informatik Mathematik Topologie Algebraische Topologie
• Mathematik | Informatik Mathematik Operations Research Graphentheorie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

1;Preface;7
2;I Skeletons and dessins;17
2.1;1 Graphs;19
2.1.1;1.1 Graphs and trees;19
2.1.1.1;1.1.1 Graphs;19
2.1.1.2;1.1.2 Trees;22
2.1.1.3;1.1.3 Dynkin diagrams;23
2.1.2;1.2 Skeletons;25
2.1.2.1;1.2.1 Ribbon graphs;25
2.1.2.2;1.2.2 Regions;28
2.1.2.3;1.2.3 The fundamental group;32
2.1.2.4;1.2.4 First applications;38
2.1.3;1.3 Pseudo-trees;42
2.1.3.1;1.3.1 Admissible trees;42
2.1.3.2;1.3.2 The counts;47
2.1.3.3;1.3.3 The associated lattice;52
2.2;2 The groups G and B3;57
2.2.1;2.1 The modular group G := PSL(2, Z);57
2.2.1.1;2.1.1 The presentation of G;57
2.2.1.2;2.1.2 Subgroups;63
2.2.2;2.2 The braid group B3;66
2.2.2.1;2.2.1 Artin’s braid groups Bn;66
2.2.2.2;2.2.2 The Burau representation;70
2.2.2.3;2.2.3 The group B3;73
2.3;3 Trigonal curves and elliptic surfaces;79
2.3.1;3.1 Trigonal curves;79
2.3.1.1;3.1.1 Basic definitions and properties;79
2.3.1.2;3.1.2 Singular fibers;87
2.3.1.3;3.1.3 Special geometric structures;92
2.3.2;3.2 Elliptic surfaces;95
2.3.2.1;3.2.1 The local theory;95
2.3.2.2;3.2.2 Compact elliptic surfaces;99
2.3.3;3.3 Real structures;106
2.3.3.1;3.3.1 Real varieties;107
2.3.3.2;3.3.2 Real trigonal curves and real elliptic surfaces;112
2.3.3.3;3.3.3 Lefschetz fibrations;117
2.4;4 Dessins;125
2.4.1;4.1 Dessins;125
2.4.1.1;4.1.1 Trichotomic graphs;125
2.4.1.2;4.1.2 Deformations;131
2.4.2;4.2 Trigonal curves via dessins;134
2.4.2.1;4.2.1 The correspondence theorems;134
2.4.2.2;4.2.2 Complex curves;136
2.4.2.3;4.2.3 Generic real curves;147
2.4.3;4.3 First applications;153
2.4.3.1;4.3.1 Ribbon curves;153
2.4.3.2;4.3.2 Elliptic Lefschetz fibrations revisited;158
2.5;5 The braid monodromy;162
2.5.1;5.1 The Zariski–van Kampen theorem;162
2.5.1.1;5.1.1 The monodromy of a proper n-gonal curve;162
2.5.1.2;5.1.2 The fundamental groups;168
2.5.1.3;5.1.3 Improper curves: slopes;174
2.5.2;5.2 The case of trigonal curves;180
2.5.2.1;5.2.1 Monodromy via skeletons;180
2.5.2.2;5.2.2 Slopes;186
2.5.2.3;5.2.3 The strategy;189
2.5.3;5.3 Universal curves;193
2.5.3.1;5.3.1 Universal curves;193
2.5.3.2;5.3.2 The irreducibility criteria;195
3;II Applications;197
3.1;6 The metabelian invariants;199
3.1.1;6.1 Dihedral quotients;199
3.1.1.1;6.1.1 Uniform dihedral quotients;199
3.1.1.2;6.1.2 Geometric implications;203
3.1.2;6.2 The Alexander module;206
3.1.2.1;6.2.1 Statements;206
3.1.2.2;6.2.2 Proof of Theorem 6.16: the case N . 7;209
3.1.2.3;6.2.3 Congruence subgroups (the case N . 5);212
3.1.2.4;6.2.4 The parabolic case N = 6;215
3.2;7 A few simple computations;219
3.2.1;7.1 Trigonal curves in .2;219
3.2.1.1;7.1.1 Proper curves in .2;219
3.2.1.2;7.1.2 Perturbations of simple singularities;223
3.2.2;7.2 Sextics with a non-simple triple point;229
3.2.2.1;7.2.1 A gentle introduction to plane sextics;229
3.2.2.2;7.2.2 Classification and fundamental groups;236
3.2.2.3;7.2.3 A summary of further results;237
3.2.3;7.3 Plane quintics;240
3.3;8 Fundamental groups of plane sextics;243
3.3.1;8.1 Statements;243
3.3.1.1;8.1.1 Principal results;243
3.3.1.2;8.1.2 Beginning of the proof;244
3.3.2;8.2 A distinguished point of type E;247
3.3.2.1;8.2.1 A point of type E8;248
3.3.2.2;8.2.2 A point of type E7;254
3.3.2.3;8.2.3 A point of type E6;260
3.3.3;8.3 A distinguished point of type D;275
3.3.3.1;8.3.1 A point of type Dp, p . 6;275
3.3.3.2;8.3.2 A point of type D5;279
3.3.3.3;8.3.3 A point of type D4;285
3.4;9 The transcendental lattice;291
3.4.1;9.1 Extremal elliptic surfaces without exceptional fibers;291
3.4.1.1;9.1.1 The tripod calculus;291
3.4.1.2;9.1.2 Proofs and further observations;293
3.4.2;9.2 Generalizations and examples;297
3.4.2.1;9.2.1 A computation via the homological invariant;297
3.4.2.2;9.2.2 An example;300
3.5;10 Monodromy factorizations;304
3.5.1;10.1 Hurwitz equivalence;304
3.5.1.1;10.1.1 Statement of the problem;304
3.5.1.2;10.1.2 Fn-valued factorizations;307
3.5.1.3;10.1.3 Sn-valued factorizations;308
3.5.2;10.2 Factorizations in G;313
3.5.2.1;10.2.1 Exponential examples;313
3.5.2.2;10.2.2 2-factorizations;317
3.5.2.3;10.2.3 The transcendental lattice;323
3.5.2.4;10.2.4 2-factorizations via matrices;329
3.5.3;10.3 Geometric applications;332
3.5.3.1;10.3.1 Extremal elliptic surfaces;332
3.5.3.2;10.3.2 Ribbon curves via skeletons;334
3.5.3.3;10.3.3 Maximal Lefschetz fibrations are algebraic;339
4;Appendices;343
4.1;A An algebraic complement;345
4.1.1;A.1 Integral lattices;345
4.1.1.1;A.1.1 Nikulin’s theory of discriminant forms;345
4.1.1.2;A.1.2 Definite lattices;347
4.1.2;A.2 Quotient groups;351
4.1.2.1;A.2.1 Zariski quotients;351
4.1.2.2;A.2.2 Auxiliary lemmas;352
4.1.2.3;A.2.3 Alexander module and dihedral quotients;353
4.2;B Bigonal curves in .d;356
4.2.1;B.1 Bigonal curves in .d;356
4.2.2;B.2 Plane quartics, quintics, and sextics;360
4.3;C Computer implementations;362
4.3.1;C.1 GAP implementations;362
4.3.1.1;C.1.1 Manipulating skeletons in GAP;362
4.3.1.2;C.1.2 Proof of Theorem 6.16;368
4.4;D Definitions and notation;375
4.4.1;D.1 Common notation;375
4.4.1.1;D.1.1 Groups and group actions;375
4.4.1.2;D.1.2 Topology and homotopy theory;376
4.4.1.3;D.1.3 Algebraic geometry;378
4.4.1.4;D.1.4 Miscellaneous notation;380
4.4.2;D.2 Index of notation;381
5;Bibliography;385
6;Index of figures;395
7;Index of tables;398
8;Index;399

Über Autor(innen)

Degtyarev, Alex
Alex Degtyarev, Bilkent University, Ankara, Turkey.

Alex Degtyarev, Bilkent University, Ankara, Turkey.