JournalsrmiVol. 19, No. 2pp. 581–612

The Denef-Loeser series for toric surface singularities

  • Monique Lejeune-Jalabert

    Université de Versailles Saint-Quentin, Versailles, France
  • Ana J. Reguera

    Universidad de Valladolid, Spain
The Denef-Loeser series for toric surface singularities cover
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Abstract

Let HH denote the set of formal arcs going through a singular point of an algebraic variety VV defined over an algebraically closed field kk of characteristic zero. In the late sixties, J. Nash has observed that for any nonnegative integer ss, the set js(H)j^s(H) of ss-jets of arcs in HH 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 js(H)j^s(H) in some suitable localization of the Grothendieck ring of algebraic varieties over kk is a rational function. We compute this function for normal toric surface singularities.

Cite this article

Monique Lejeune-Jalabert, Ana J. Reguera, The Denef-Loeser series for toric surface singularities. Rev. Mat. Iberoam. 19 (2003), no. 2, pp. 581–612

DOI 10.4171/RMI/361