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El. knyga: Topology of Algebraic Curves: An Approach Via Dessins D'Enfants [De Gruyter E-books]

  • Formatas: 409 pages, 25 Tables, black and white; 75 Illustrations, black and white
  • Serija: De Gruyter Studies in Mathematics
  • Išleidimo metai: 14-Jun-2012
  • Leidėjas: De Gruyter
  • ISBN-13: 9783110258424
Kitos knygos pagal šią temą:
  • De Gruyter E-books
  • Kaina: 167,94 €*
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Topology of Algebraic Curves: An Approach Via Dessins D'EnfantsDidesnis paveikslėlis
  • Formatas: 409 pages, 25 Tables, black and white; 75 Illustrations, black and white
  • Serija: De Gruyter Studies in Mathematics
  • Išleidimo metai: 14-Jun-2012
  • Leidėjas: De Gruyter
  • ISBN-13: 9783110258424
Kitos knygos pagal šią temą:
Wiley OnlineMore info about De Gruyter e-books
Partnerių interneto svetainė: http://dx.doi.org/10.1515/9783110258424
Degtyarev (Bilkent U., Ankara, Turkey) explores ramifications of the close relation between elliptic surfaces and trigonal curves in ruled surfaces, skeletons (also called dessins d'enfants and quilts among other things), and subgroups of the modular group gamma : = PSL(2,Z). At a level suitable to advanced students and researchers in mathematics and theoretical physics, he summarizes, unifies, and extends a number of results in articles that he published or submitted during the past five years in collaboration with several colleagues. The work remains in progress, he warns, so he presents some of the older results in a more complete and general form as well as introducing new results. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)
Preface vii
I Skeletons and dessins
1 Graphs
3(38)
1.1 Graphs and trees
3(6)
1.1.1 Graphs
3(3)
1.1.2 Trees
6(1)
1.1.3 Dynkin diagrams
7(2)
1.2 Skeletons
9(17)
1.2.1 Ribbon graphs
9(3)
1.2.2 Regions
12(4)
1.2.3 The fundamental group
16(6)
1.2.4 First applications
22(4)
1.3 Pseudo-trees
26(15)
1.3.1 Admissible trees
26(5)
1.3.2 The counts
31(5)
1.3.3 The associated lattice
36(5)
2 The groups Γ and B3
41(22)
2.1 The modular group Γ:=PSL(2,Z)
41(9)
2.1.1 The presentation of Γ
41(6)
2.1.2 Subgroups
47(3)
2.2 The braid group B3
50(13)
2.2.1 Artin's braid groups Bn
50(4)
2.2.2 The Burau representation
54(3)
2.2.3 The group B3
57(6)
3 Trigonal curves and elliptic surfaces
63(46)
3.1 Trigonal curves
63(16)
3.1.1 Basic definitions and properties
63(8)
3.1.2 Singular fibers
71(5)
3.1.3 Special geometric structures
76(3)
3.2 Elliptic surfaces
79(11)
3.2.1 The local theory
79(4)
3.2.2 Compact elliptic surfaces
83(7)
3.3 Real structures
90(19)
3.3.1 Real varieties
91(5)
3.3.2 Real trigonal curves and real elliptic surfaces
96(5)
3.3.3 Lefschetz fibrations
101(8)
4 Dessins
109(37)
4.1 Dessins
109(9)
4.1.1 Trichotomic graphs
109(6)
4.1.2 Deformations
115(3)
4.2 Trigonal curves via dessins
118(19)
4.2.1 The correspondence theorems
118(2)
4.2.2 Complex curves
120(11)
4.2.3 Generic real curves
131(6)
4.3 First applications
137(9)
4.3.1 Ribbon curves
137(5)
4.3.2 Elliptic Lefschetz fibrations revisited
142(4)
5 The braid monodromy
146(37)
5.1 The Zariski-van Kampen theorem
146(18)
5.1.1 The monodromy of a proper n-gonal curve
146(6)
5.1.2 The fundamental groups
152(6)
5.1.3 Improper curves: slopes
158(6)
5.2 The case of trigonal curves
164(13)
5.2.1 Monodromy via skeletons
164(6)
5.2.2 Slopes
170(3)
5.2.3 The strategy
173(4)
5.3 Universal curves
177(6)
5.3.1 Universal curves
177(2)
5.3.2 The irreducibility criteria
179(4)
II Applications
6 The metabelian invariants
183(20)
6.1 Dihedral quotients
183(7)
6.1.1 Uniform dihedral quotients
183(4)
6.1.2 Geometric implications
187(3)
6.2 The Alexander module
190(13)
6.2.1 Statements
190(3)
6.2.2 Proof of Theorem 6.16: the case N ≥ 7
193(3)
6.2.3 Congruence subgroups (the case N ≤ 5)
196(3)
6.2.4 The parabolic case N = 6
199(4)
7 A few simple computations
203(24)
7.1 Trigonal curves in Σ2
203(10)
7.1.1 Proper curves in Σ2
203(4)
7.1.2 Perturbations of simple singularities
207(6)
7.2 Sextics with a non-simple triple point
213(11)
7.2.1 A gentle introduction to plane sextics
213(7)
7.2.2 Classification and fundamental groups
220(1)
7.2.3 A summary of further results
221(3)
7.3 Plane quintics
224(3)
8 Fundamental groups of plane sextics
227(48)
8.1 Statements
227(4)
8.1.1 Principal results
227(1)
8.1.2 Beginning of the proof
228(3)
8.2 A distinguished point of type E
231(28)
8.2.1 A point of type E8
232(6)
8.2.2 A point of type E7
238(6)
8.2.3 A point of type E6
244(15)
8.3 A distinguished point of type D
259(16)
8.3.1 A point of type Dp, p ≥ 6
259(4)
8.3.2 A point of type D5
263(6)
8.3.3 A point of type D4
269(6)
9 The transcendental lattice
275(13)
9.1 Extremal elliptic surfaces without exceptional fibers
275(6)
9.1.1 The tripod calculus
275(2)
9.1.2 Proofs and further observations
277(4)
9.2 Generalizations and examples
281(7)
9.2.1 A computation via the homological invariant
281(3)
9.2.2 An example
284(4)
10 Monodromy factorizations
288(41)
10.1 Hurwitz equivalence
288(9)
10.1.1 Statement of the problem
288(3)
10.1.2 Fn-valued factorizations
291(1)
10.1.3 Sn-valued factorizations
292(5)
10.2 Factorizations in Γ
297(19)
10.2.1 Exponential examples
297(4)
10.2.2 2-factorizations
301(6)
10.2.3 The transcendental lattice
307(6)
10.2.4 2-factorizations via matrices
313(3)
10.3 Geometric applications
316(13)
10.3.1 Extremal elliptic surfaces
316(2)
10.3.2 Ribbon curves via skeletons
318(5)
10.3.3 Maximal Lefschetz fibrations are algebraic
323(6)
Appendices
A An algebraic complement
329(11)
A.1 Integral lattices
329(1)
A.1.1 Nikulin's theory of discriminant forms
329(2)
A.1.2 Definite lattices
331(4)
A.2 Quotient groups
335(1)
A.2.1 Zariski quotients
335(1)
A.2.2 Auxiliary lemmas
336(1)
A.2.3 Alexander module and dihedral quotients
337(3)
B Bigonal curves in Σd
340(6)
B.1 Bigonal curves in Σd
340(4)
B.2 Plane quartics, quintics, and sextics
344(2)
C Computer implementations
346(13)
C.1 GAP implementations
346(1)
C.1.1 Manipulating skeletons in GAP
346(6)
C.1.2 Proof of Theorem 6.16
352(7)
D Definitions and notation
359(10)
D.1 Common notation
359(1)
D.1.1 Groups and group actions
359(1)
D.1.2 Topology and homotopy theory
360(2)
D.1.3 Algebraic geometry
362(2)
D.1.4 Miscellaneous notation
364(1)
D.2 Index of notation
365(4)
Bibliography 369(10)
Index of figures 379(3)
Index of tables 382(1)
Index 383
Alex Degtyarev, Bilkent University, Ankara, Turkey.
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