Revista Matemática Iberoamericana

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Volume 19, Issue 2, 2003, pp. 581–612
DOI: 10.4171/RMI/361

Published online: 2003-08-31

The Denef-Loeser series for toric surface singularities

Monique Lejeune-Jalabert[1] and Ana J. Reguera[2]

(1) Université de Versailles Saint-Quentin, Versailles, France
(2) Universidad de Valladolid, Spain

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.

Keywords: Arc spaces, Denef-Loeser series, toric surfaces

Lejeune-Jalabert Monique, Reguera Ana: The Denef-Loeser series for toric surface singularities. Rev. Mat. Iberoam. 19 (2003), 581-612. doi: 10.4171/RMI/361