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Poincaré series of resolutions of surface singularities

Authors:	Steven Dale Cutkosky Jü rgen Herzog Ana Reguera

Institution:	Department of Mathematics, University of Missouri, Columbia, Missouri 65211 ; FB 6 Mathematik und Informatik, Universität-GHS-Essen, Postfach 103764, D-45117 Essen, Germany ; Univeristy of Valladolid, Departamento de Algebra, Geometría y Topología, 005 Valladolid, Spain

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

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