The Denef-Loeser series for toric surface singularities (Q1884051)

Consider an algebraic variety \(V\) defined over an algebraically closed field of characteristic 0, \(H\) the set of all formal arcs going through a singular point of \(V\) and \(j^s(H)\) the set of \(s-\)truncations of formal arcs (the first author calls it \(s-\)jets, other people call it truncations). Recently \textit{J. Denef} and \textit{F. Loeser} have attached a canonical rational functions to \(H\) [Invent. Math. 135, 201--232 (1999; Zbl 0928.14004)]. The existence of such functions is a finiteness property of the family \(\{ j^s(H)\}, s\in \mathbb N\). Denef and Loeser have computed these functions for analytically irreducible plane curve singularities. The main theorem in this paper consists of computing this series for affine normal toric sufaces. It is well known that such singularities are given by a cone \(\sigma=\{ \mathbb R_+ e_1+ \mathbb R_+(pe_1+qe_2)\}\) with \(p<q\) non zero natural numbers, without common divisors. The authors give explicitely the Denef-Loeser series in terms of the continued fraction expansion of \(q/p\). This paper contains a nice appendix on toric varietes and Nash blowing-up.