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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 \[ g= \sum _{\underline {n}\in \mathbf {N}^r} h(\underline {n})t^{\underline {n}}, \] where \[ h(\underline {n})=\ell (R/\Gamma (X,\mathcal {O}_X(-n_1E-1-\cdots -n_rE_r)). \] 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
MR Author ID: 53545
ORCID: 0000-0002-9319-0717

Jürgen Herzog
Affiliation: FB 6 Mathematik und Informatik, Universität-GHS-Essen, Postfach 103764, D-45117 Essen, Germany
MR Author ID: 189999

Ana Reguera
Affiliation: Univeristy of Valladolid, Departamento de Algebra, Geometría y Topología, 005 Valladolid, Spain

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