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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Poincaré series of resolutions of surface singularities


Authors: Steven Dale Cutkosky, Jürgen Herzog and Ana Reguera
Journal: Trans. Amer. Math. Soc. 356 (2004), 1833-1874
MSC (2000): Primary 14B05, 14F05, 13A30
Published electronically: August 26, 2003
MathSciNet review: 2031043
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Abstract: Let $X\rightarrow\mathrm{spec}(R)$ be a resolution of singularities of a normal surface singularity $\mathrm{spec}(R)$, with integral exceptional divisors $E_1,\dotsc,E_r$. We consider the Poincaré series

\begin{displaymath}g= \sum_{\underline{n}\in\mathbf{N}^r} h(\underline{n})t^{\underline{n}}, \end{displaymath}

where

\begin{displaymath}h(\underline{n})=\ell(R/\Gamma(X,\mathcal{O}_X(-n_1E-1-\cdots-n_rE_r)). \end{displaymath}

We show that if $R/m$ has characteristic zero and $\mathrm{Pic}^0(X)$ is a semi-abelian variety, then the Poincaré series $g$ is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.


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Additional Information

Steven Dale Cutkosky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: cutkoskys@missouri.edu

Jürgen Herzog
Affiliation: FB 6 Mathematik und Informatik, Universität-GHS-Essen, Postfach 103764, D-45117 Essen, Germany
Email: mat300@uni-essen.de

Ana Reguera
Affiliation: Univeristy of Valladolid, Departamento de Algebra, Geometría y Topología, 005 Valladolid, Spain
Email: areguera@agt.uva.es

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03346-4
PII: S 0002-9947(03)03346-4
Received by editor(s): August 1, 2002
Published electronically: August 26, 2003
Additional Notes: The first author’s research was partially supported by NSF
Article copyright: © Copyright 2003 American Mathematical Society