Topology of Algebraic Curves: An Approach via Dessins d'Enfants (De Gruyter Studies in Mathematics Book 44)

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. 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 Γ and B3 2.1 The modular group Γ := PSL(2, Z) 2.1.1 The presentation of Γ 2.1.2 Subgroups 2.2 The braid group B3 2.2.1 Artin’s braid groups Bn 2.2.2 The Burau representation 2.2.3 The group B3 3 Trigonal 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 4 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 7 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 Fn-valued factorizations 10.1.3 Sn-valued factorizations 10.2 Factorizations in Γ 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.1.2 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.1.1 Manipulating skeletons in GAP C.1.2 Proof of Theorem 6.16 D Definitions and notation D.1 Common notation D.1.1 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 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 singular plane 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.