Topology of Algebraic Curves An Approach via Dessins d'Enfants
:
Alex Degtyarev
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Topology of Algebraic Curves An Approach via Dessins d'Enfants
:
Walter de Gruyter GmbH& Co.KG
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9783110258424
:
De Gruyter Studies in MathematicsISSN
:
1
:
CHF 209.10
:
:
Mathematik
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English
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409
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Wasserzeichen
:
PC/MAC/eReader/Tablet
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PDF
< >The book summarizes the state and new results on the topology of trigonal curves in geometrically ruled surfaces. Emphasis is placed upon various applications of the theory to related areas, most notably singularplane curves of small degree, elliptic surfaces, and Lefschetz fibrations (both complex and real), and Hurwitz equivalence of braid monodromy factorizations. The monograph conveys recent knowledge about related objects and is of interest to researchers and graduate students in the fields of topology and of complex and real algebraic varieties.
< >
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
