Shigeru Takamura (eds.)9783540333630, 3-540-33363-0
The author develops a deformation theory for degenerations of complex curves; specifically, he treats deformations which induce splittings of the singular fiber of a degeneration. He constructs a deformation of the degeneration in such a way that a subdivisor is “barked” (peeled) off from the singular fiber. These “barking deformations” are related to deformations of surface singularities (in particular, cyclic quotient singularities) as well as the mapping class groups of Riemann surfaces (complex curves) via monodromies. Important applications, such as the classification of atomic degenerations, are also explained.
Table of contents :
Front Matter….Pages i-xxix
Front Matter….Pages 22-22
Splitting Deformations of Degenerations….Pages 23-31
What is a barking?….Pages 33-39
Semi-Local Barking Deformations: Ideas and Examples….Pages 41-56
Global Barking Deformations: Ideas and Examples….Pages 57-81
Front Matter….Pages 84-84
Deformations of Tubular Neighborhoods of Branches (Preparation)….Pages 85-98
Construction of Deformations by Tame Subbranches….Pages 99-117
Construction of Deformations of type A l ….Pages 119-141
Construction of Deformations by Wild Subbranches….Pages 143-152
Subbranches of Types A l , B l , C l ….Pages 153-176
Construction of Deformations of Type B l ….Pages 177-181
Construction of Deformations of Type C l ….Pages 183-207
Recursive Construction of Deformations of Type C l ….Pages 209-234
Types A l , B l , and C l Exhaust all Cases….Pages 235-251
Construction of Deformations by Bunches of Subbranches….Pages 253-262
Front Matter….Pages 264-264
Construction of Barking Deformations (Stellar Case)….Pages 265-278
Simple Crusts (Stellar Case)….Pages 279-302
Compound barking (Stellar Case)….Pages 303-307
Deformations of Tubular Neighborhoods of Trunks….Pages 309-326
Construction of Barking Deformations (Constellar Case)….Pages 327-347
Further Examples….Pages 349-379
Front Matter….Pages 382-382
Singularities of Fibers around Cores….Pages 383-419
Arrangement Functions and Singularities, I….Pages 421-438
Arrangement Functions and Singularities, II….Pages 439-459
Supplement….Pages 461-479
Front Matter….Pages 482-482
Classification Theorem….Pages 483-485
List of Weighted Crustal Sets for Singular Fibers of Genus ≤ 5….Pages 487-580
Back Matter….Pages 581-594
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