Journal of the Mathematical Society of Japan

Pencil genus for normal surface singularities

Tadashi TOMARU

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Abstract

Let ( X , o ) be a normal complex surface singularity. We define an invariant p e ( X , o ) for ( X , o ) in terms of pencils of compact complex curves. Similarly, for a pair of ( X , o ) and h 𝔪 X , o (the maximal ideal of 𝒪 X , o ), we define an invariant p e ( X , o , h ) . We call p e ( X , o ) (resp. p e ( X , o , h ) ) the pencil genus of ( X , o ) (resp. a pair of ( X , o ) and h ). In this paper, we give a method to construct pencils of compact complex curves by gluing a resolution space of ( X , o ) and resolution spaces of some cyclic quotient singularities. Using this, we prove some formulae on p e ( X , o , h ) and estimate p e ( X , o ) . We also characterize Kodaira singularities in terms of p e ( X , o , h ) .

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 1 (2007), 35-80.

Dates
First available in Project Euclid: 25 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1180135500

Digital Object Identifier
doi:10.2969/jmsj/1180135500

Mathematical Reviews number (MathSciNet)
MR2302662

Zentralblatt MATH identifier
1140.32016

Subjects
Primary: 32S10: Invariants of analytic local rings 32S25: Surface and hypersurface singularities [See also 14J17]
Secondary: 14D06: Fibrations, degenerations

Keywords
surface singularities pencils of curves Kodaira singularities

Citation

TOMARU, Tadashi. Pencil genus for normal surface singularities. J. Math. Soc. Japan 59 (2007), no. 1, 35--80. doi:10.2969/jmsj/1180135500. https://projecteuclid.org/euclid.jmsj/1180135500


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