Umfang:
1 Online-Ressource (XVI, 393 Seiten)
ISBN:
9783110258424
,
1283627736
,
9781283627733
Serie:
De Gruyter Studies in Mathematics 44
Inhalt:
Biographical note: Alex Degtyarev, Bilkent University, Ankara, Turkey.
Inhalt:
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.
Inhalt:
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
Anmerkung:
Description based upon print version of record
,
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
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3.1.1 Basic definitions and properties3.1.2 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
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5 The braid monodromy5.1 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
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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
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8.2.3 A point of type E68.3 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
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10.2.1 Exponential examples
,
In English
Weitere Ausg.:
ISBN 9783110255911
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Degtjarev, Aleksandr, 1962 - Topology of algebraic curves Berlin [u.a.] : de Gruyter, 2012 ISBN 311025591X
Weitere Ausg.:
ISBN 9783110255911
Weitere Ausg.:
ISBN 9783110255911
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Algebraische Kurve
;
Topologische Graphentheorie
;
Algebraische Kurve
;
Topologische Graphentheorie
DOI:
10.1515/9783110258424
URL:
Volltext
(lizenzpflichtig)
Mehr zum Autor:
Degtjarev, Aleksandr 1962-
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