Geodesic
Universal Abelian Covers of Rational Surface Singularities and Multi-Index Filtrations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 3-10.

Voir la notice de l'article provenant de la source Math-Net.Ru

In previous papers, the authors computed the Poincaré series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincaré series were expressed as the integer parts of certain fractional power series, whose interpretation was not given. In this paper, we show that, up to a simple change of variables, these fractional power series are reductions of the equivariant Poincaré series for filtrations on the ring of germs of functions on the universal Abelian cover of the surface. We compute these equivariant Poincaré series.
Keywords: universal Abelian cover, rational surface singularity, Poincaré series. 
@article{FAA_2008_42_2_a1,
     author = {S. M. Gusein-Zade and F. Delgado and A. Campillo},
     title = {Universal {Abelian} {Covers} of {Rational} {Surface} {Singularities} and {Multi-Index} {Filtrations}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {3--10},
     publisher = {mathdoc},
     volume = {42},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a1/}
}
TY  - JOUR
AU  - S. M. Gusein-Zade
AU  - F. Delgado
AU  - A. Campillo
TI  - Universal Abelian Covers of Rational Surface Singularities and Multi-Index Filtrations
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2008
SP  - 3
EP  - 10
VL  - 42
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a1/
LA  - ru
ID  - FAA_2008_42_2_a1
ER  - 
%0 Journal Article
%A S. M. Gusein-Zade
%A F. Delgado
%A A. Campillo
%T Universal Abelian Covers of Rational Surface Singularities and Multi-Index Filtrations
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2008
%P 3-10
%V 42
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a1/
%G ru
%F FAA_2008_42_2_a1
S. M. Gusein-Zade; F. Delgado; A. Campillo. Universal Abelian Covers of Rational Surface Singularities and Multi-Index Filtrations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 3-10. http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a1/

[1] S. M. Gusein-Zade, F. Delgado, A. Kampilo, “Integrirovanie po eilerovoi kharakteristike po prostranstvu funktsii i polinom Aleksandera osobennosti ploskoi krivoi”, UMN, 55:6 (336) (2000), 127–128 | DOI | MR | Zbl

[2] A. Campillo, F. Delgado, S. M. Gusein-Zade, “Poincaré series of a rational surface singularity”, Invent. Math., 155 (2004), 41–53 | DOI | MR | Zbl

[3] A. Campillo, F. Delgado, S. M. Gusein-Zade, “Poincaré series of curves on rational surface singularities”, Comment. Math. Helv., 80:1 (2005), 95–102 | MR | Zbl

[4] A. Campillo, F. Delgado, S. M. Gusein-Zade, “On Poincaré series of filtrations on equivariant functions of two variables”, Moscow Math. J., 7:2 (2007), 243–255 | DOI | MR | Zbl

[5] H. B. Laufer, Normal two-dimensional singularities, Ann. of Math. Stud., 71, Princeton Univ. Press, Princeton, NJ; Univ. of Tokyo Press, Tokyo, 1971 | MR | Zbl

[6] W. D. Neumann, J. Wahl, “Universal Abelian covers of surface singularities”, Trends in singularities, Trends Math., Birkhauser, Basel, 2002, 181–190 | DOI | MR | Zbl

[7] T. Okuma, “Universal Abelian covers of rational surface singularities.”, J. London Math. Soc. (2), 70:2 (2004), 307–324 | DOI | MR | Zbl

[8] T. Okuma, The geometric genus of splice-quotient singularities, http://arxiv.org/abs/math/0610464 | MR

[9] H. Pinkham, “Singularités rationelles de surfaces”, Séminaire sur les singularités des surfaces, Lecture Notes in Math., 777, Springer-Verlag, Berlin–Heidelberg–New York, 1980, 147–178 | DOI | MR