Advanced Studies in Pure Mathematics
- Adv. Stud. Pure Math.
- Singularities in Geometry and Topology 2011, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 203 - 256
Singular fibers in barking families of degenerations of elliptic curves
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.
Received: 23 May 2012
Revised: 10 December 2013
First available in Project Euclid: 19 October 2018
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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