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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Poincaré series of resolutions of surface singularities
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by Steven Dale Cutkosky, Jürgen Herzog and Ana Reguera PDF
Trans. Amer. Math. Soc. 356 (2004), 1833-1874 Request permission

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
  • Email: cutkoskys@missouri.edu
  • Jürgen Herzog
  • Affiliation: FB 6 Mathematik und Informatik, Universität-GHS-Essen, Postfach 103764, D-45117 Essen, Germany
  • MR Author ID: 189999
  • 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
  • Received by editor(s): August 1, 2002
  • Published electronically: August 26, 2003
  • Additional Notes: The first author’s research was partially supported by NSF
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 1833-1874
  • MSC (2000): Primary 14B05, 14F05, 13A30
  • DOI: https://doi.org/10.1090/S0002-9947-03-03346-4
  • MathSciNet review: 2031043