Poincaré series of resolutions of surface singularities

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.