Topology of algebraic curves. An approach via dessins d'enfants (Q435892)

The monograph is devoted to the topology of trigonal curves in geometrically ruled surfaces. The book summarizes and extends many results in the theory and presents several applications, in particular, those concerning singular plane curves of small degrees, elliptic surfaces, (complex and real) Lefschetz fibrations, Hurwitz equivalence of braid monodromy factorizations.NEWLINENEWLINEThe core of the monograph is the important close relation between elliptic surfaces and trigonal curves in ruled surfaces, subgroups of the modular group \(\Gamma = \mathrm{PSL}(2, \mathbb Z)\), and certain bipartite ribbon graphs (a version of \textit{dessins d'enfants}). The principal geometric applications concern the topology of trigonal curves and plane curves with deep singularities.NEWLINENEWLINEThe main concepts (bipartite ribbon graphs and their relation to the subgroups of appropriate quotients of the free group \({\mathbb F}_2\), the modular group \(\Gamma\) and the closely related braid group \({\mathbb B}_3\), trigonal curves, elliptic surfaces, Lefschetz fibrations) are introduced in Part I of the monograph. Part II is devoted to geometric applications. The author computes and studies the fundamental groups of trigonal curves and related plane curves (Chapters 6, 7, and 8), discusses the transcendental lattices of extremal elliptic surfaces (Chapter 9), and makes a few steps towards the understanding of \(\Gamma\)-valued monodromy factorizations and their applications to the topology of trigonal curves, elliptic surfaces, and Lefschetz fibrations (Chapter 10).NEWLINENEWLINEAppendices contain, in particular, the material concerning integral lattices, Zari\-ski quotients, and bigonal (hyperelliptic) curves in rational geometrically ruled surfaces.NEWLINENEWLINEThe 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.