Preface
Ⅰ Skeletons and dessins
1 Graphs
1.1 Graphs and trees
1.1.1 Graphs
1.1.2 Trees
1.1.3 Dynkin diagrams
1.2 Skeletons
1.2.1 Ribbon graphs
1.2.2 Regions
1.2.3 The fundamental group
1.2.4 First applications
1.3 Pseudo-trees
1.3.1 Admissible trees
1.3.2 The counts
1.3.3 The associated lattice
2 The groups г and в3
2.1 The modular group г := PSL(2, Z)
2.1.1 The presentation of г
2.1.2 Subgroups
2.2 The braid group в3
2.2.1 Artin's braid groups вn
2.2.2 The Burau representation
2.2.3 The group в3
3 Trigonai curves and elliptic surfaces
3.1 Trigonal curves
3.1.1 Basic definitions and properties
3.1.2 Singular fibers
3.1.3 Special geometric structures
3.2 Elliptic surfaces
3.2.1 The local theory
3.2.2 Compact elliptic surfaces
3.3 Real structures
3.3.1 Real varieties
3.3.2 Real trigonal curves and real elliptic surfaces
3.3.3 Lefschetz fibrations
Dessins
4.1 Dessins
4.1.1 Trichotomic graphs
4.1.2 Deformations
4.2 Trigonal curves via dessins
4.2.1 The correspondence theorems
4.2.2 Complex curves
4.2.3 Generic real curves
4.3 First applications
4.3.1 Ribbon curves
4.3.2 Elliptic Lefschetz fibrations revisited
5 The braid monodromy
5.1 The Zariski-van Kampen theorem
5.1.1 The monodromy of a proper n-gonal curve
5.1.2 The fundamental groups
5.1.3 Improper curves: slopes
5.2 The case of trigonal curves
5.2.1 Monodromy via skeletons
5.2.2 Slopes
5.2.3 The strategy
5.3 Universal curves
5.3.1 Universal Curves
5.3.2 The irreducibility criteria
Ⅱ Applications
6 The metabelian invariants
6.1 Dihedral quotients
6.1.1 Uniform dihedral quotients
6.1.2 Geometric implications
6.2 The Alexander module
6.2.1 Statements
6.2.2 Proof of Theorem 6.16: the case N ≥ 7
6.2.3 Congruence subgroups (the case N ≤ 5)
6.2.4 The parabolic case N = 6
A few simple computations
7.1 Trigonal curves in ∑2
7.1.1 Proper curves in ∑2
7.1.2 Perturbations of simple singularities
7.2 Sextics with a non-simple triple point
7.2.1 A gentle introduction to plane sextics
7.2.2 Classification and fundamental groups
7.2.3 A summary of further results
7.3 Plane quintics
8 Fundamental groups of plane sextics
8.1 Statements
8.1.1 Principal results
8.1.2 Beginning of the proof
8.2 A distinguished point of type E
8.2.1 A point of type E8
8.2.2 A point of type E7
8.2.3 A point of type E6
8.3 A distinguished point of type D
8.3.1 A point of type Dp, p ≥ 6
8.3.2 A point of type D5
8.3.3 A point of type D4
9 The transcendental lattice
9.1 Extremal elliptic surfaces without exceptional fibers
9.1.1 The tripod calculus
9.1.2 Proofs and further observations
9.2 Generalizations and examples
9.2.1 A computation via the homological invariant
9.2.2 An example
10 Monodromy factorizations
10.1 Hurwitz equivalence
10.1.1 Statement of the problem
10.1.2 En-valued factorizations
10.1.3 Sn-valued factorizations
10.2 Factorizations in Г
10.2.1 Exponential examples
10.2.2 2-factorizations
10.2.3 The transcendental lattice
10.2.4 2-factorizations via matrices
10.3 Geometric applications
10.3.1 Extremal elliptic surfaces
10.3.2 Ribbon curves via skeletons
10.3.3 Maximal Lefschetz fibrations are algebraic
Appendices
A An algebraic complement
A.1 Integral lattices
A.1.1 Nikulin's theory of discriminant forms
A.I.2 Definite lattices
A.2 Quotient groups
A.2.1 Zariski quotients
A.2.2 Auxiliary lemmas
A.2.3 Alexander module and dihedral quotients
B Bigonal curves in ∑d
B. 1 Bigonal curves in ∑d
B.2 Plane quartics, quintics, and sextics
C Computer implementations
C.1 GAP implementations
C.I.1 Manipulating skeletons in GAP
C.1.2 Proof of Theorem 6.16
D Definitions and notation
D.1 Common notation
D.I.1 Groups and group actions
D.1.2 Topology and homotopy theory
D.1.3 Algebraic geometry
D.1.4 Miscellaneous notation
D.2 Index of notation
Bibliography
Index of figures
Index of tables
Index
