The Denef-Loeser series for toric surface singularities

Abstract

Let $H$ denote the set of formal arcs going through a singular point of an algebraic variety $V$ defined over an algebraically closed field $k$ of characteristic zero. In the late sixties, J. Nash has observed that for any nonnegative integer $s$, the set $j^s(H)$ of $s$-jets of arcs in $H$ is a constructible subset of some affine space. Recently (1999), J. Denef and F. Loeser have proved that the Poincar\'{e} series associated with the image of $j^s(H)$ in some suitable localization of the Grothendieck ring of algebraic varieties over $k$ is a rational function. We compute this function for normal toric surface singularities.