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.
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1886 Shigeru Takamura Splitting Deformations of Degenerations of Complex Curves Towards the Classification of Atoms of Degenerations, III ABC Author Shigeru Takamura Department of Mathematics Graduate School of Science Kyoto University Oiwakecho, Kitashirakawa Sakyo-Ku, Kyoto 606-8502 Japan e-mail:
[email protected] Library of Congress Control Number: 2006923235 Mathematics Subject Classification (2000): 14D05, 14J15, 14H15, 32S30 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-33363-0 Springer Berlin Heidelberg New York ISBN-13 978-3-33363-0 Springer Berlin Heidelberg New York DOI 10.1007/b138136 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this pub