Advanced Studies in Pure Mathematics

Singular fibers in barking families of degenerations of elliptic curves

Takayuki Okuda

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Abstract

Takamura [Ta3] established a theory of splitting families of degenerations of complex curves of genus $g \ge 1$. He introduced a powerful method for constructing a splitting family, called a barking family, in which the resulting family of complex curves has a singular fiber over the origin (the main fiber) together with other singular fibers (subordinate fibers). He made a list of barking families for genera up to 5 and determined the main fibers appearing in them. This paper determines most of the subordinate fibers of the barking families in Takamura's list for the case $g = 1$. (There remain four undetermined cases.) Also, we show that some splittings never occur in a splitting family.

Article information

Source
Singularities in Geometry and Topology 2011, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 203-256

Dates
Received: 23 May 2012
Revised: 10 December 2013
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1539916288

Digital Object Identifier
doi:10.2969/aspm/06610203

Mathematical Reviews number (MathSciNet)
MR3382051

Zentralblatt MATH identifier
1360.14032

Subjects
Primary: 14D06: Fibrations, degenerations
Secondary: 14H15: Families, moduli (analytic) [See also 30F10, 32G15] 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 32S50: Topological aspects: Lefschetz theorems, topological classification, invariants

Keywords
Degeneration of complex curves splitting family elliptic curve singular fiber monodromy

Citation

Okuda, Takayuki. Singular fibers in barking families of degenerations of elliptic curves. Singularities in Geometry and Topology 2011, 203--256, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06610203. https://projecteuclid.org/euclid.aspm/1539916288


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