Splitting Deformations of Degenerations of Complex Curves: Towards the Classification of Atoms of Degenerations, III (Lecture Notes in Mathematics, 1886)

Takamura, Shigeru

ISBN 10: 3540333630 / ISBN 13: 9783540333630

Published by Springer, 2006

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Bibliographic Details

• Title: Splitting Deformations of Degenerations of Complex Curves: Towards the Classification of Atoms of Degenerations, III (Lecture Notes in Mathematics, 1886)
• Publisher: Springer
• Publication Date: 2006
• Language: English
• ISBN 10: 3540333630
• ISBN 13: 9783540333630
• Binding: paperback
• Condition: New
• Weight: 30 ounces

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Synopsis

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.

Review

From the reviews:

"This is a 590 pages book on deformation theory, using mostly topological methods, but also ‘translated’ to algebraic geometry and using algebraic methods. ... It is a nice level and should be possible to read. Most commonly, algebraic geometers translate from differential geometry to solve problems. In this book the concept is vice versa: Algebraic methods are used to solve topological problems. Thus this book may at the first glance look elementary for an algebraist, but it is not." (Arvid Siqveland, Zentralblatt MATH, Vol. 1100 (2), 2007)