Journal of the Mathematical Society of Japan

${\Bbb C}^*$-equivariant degenerations of curves and normal surface singularities with ${\Bbb C}^*$-action

Tadashi TOMARU

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Abstract

This paper presents a definition of ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb C}$. Those families are called ${\Bbb C}^*$-pencils of curves. We give the canonical method to construct them and prove some results on relations between them and normal surface singularities with ${\Bbb C}^*$-action. We also define ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb P}^1$. From this, it is possible to introduce a notion of dual ${\Bbb C}^*$-pencils of curves naturally. Associating it, we prove a duality for cyclic covers of normal surface singularities with ${\Bbb C}^*$-action.

Article information

Source
J. Math. Soc. Japan, Volume 65, Number 3 (2013), 829-885.

Dates
First available in Project Euclid: 23 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1374586628

Digital Object Identifier
doi:10.2969/jmsj/06530829

Mathematical Reviews number (MathSciNet)
MR3084983

Zentralblatt MATH identifier
1282.14017

Subjects
Primary: 14D06: Fibrations, degenerations
Secondary: 32S25: Surface and hypersurface singularities [See also 14J17]

Keywords
${\Bbb C}^*$-pencils of curves normal surface singularities with ${\Bbb C}^*$-action

Citation

TOMARU, Tadashi. ${\Bbb C}^*$-equivariant degenerations of curves and normal surface singularities with ${\Bbb C}^*$-action. J. Math. Soc. Japan 65 (2013), no. 3, 829--885. doi:10.2969/jmsj/06530829. https://projecteuclid.org/euclid.jmsj/1374586628


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