Splitting Deformations of Degenerations of Complex Curves Towards th
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 s
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Shigeru Takamura
Splitting Deformations of Degenerations of Complex Curves Towards the Classification of Atoms of Degenerations, III
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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
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Part I Basic Notions and Ideas 1
Splitting Deformations of Degenerations . . . . . . . . . . . . . . . . . . . 23 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2 Splitting criteria via configuration of singular fibers . . . . . . . . . . 30
2
What is a barking? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 Barking, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Barking, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3
Semi-Local Barking Deformations: Ideas and Examples . . . . 3.1 Semi-local example, I (Reduced barking) . . . . . . . . . . . . . . . . . . . 3.2 Semi-local example, II (Multiple barking) . . . . . . . . . . . . .
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