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Transactions of the American Mathematical Society

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Poincaré series of resolutions of surface singularities


Authors: Steven Dale Cutkosky, Jürgen Herzog and Ana Reguera
Journal: Trans. Amer. Math. Soc. 356 (2004), 1833-1874
MSC (2000): Primary 14B05, 14F05, 13A30
DOI: https://doi.org/10.1090/S0002-9947-03-03346-4
Published electronically: August 26, 2003
MathSciNet review: 2031043
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Abstract: Let $X\rightarrow \mathrm {spec}(R)$ be a resolution of singularities of a normal surface singularity $\mathrm {spec}(R)$, with integral exceptional divisors $E_1,\dotsc ,E_r$. We consider the Poincaré series \[ g= \sum _{\underline {n}\in \mathbf {N}^r} h(\underline {n})t^{\underline {n}}, \] where \[ h(\underline {n})=\ell (R/\Gamma (X,\mathcal {O}_X(-n_1E-1-\cdots -n_rE_r)). \] We show that if $R/m$ has characteristic zero and $\mathrm {Pic}^0(X)$ is a semi-abelian variety, then the Poincaré series $g$ is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.


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  • S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic $p\ne 0$, Annals of Math. (2) 63 (1956), 491-526.
  • Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. MR 146182, DOI https://doi.org/10.2307/2372985
  • Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
  • Constantin Bănică and Octavian Stănăşilă, Algebraic methods in the global theory of complex spaces, Editura Academiei, Bucharest; John Wiley & Sons, London-New York-Sydney, 1976. Translated from the Romanian. MR 0463470
  • S. M. Guseĭn-Zade, F. Del′gado, and A. Kampil′o, The Alexander polynomial of a plane curve singularity, and the ring of functions on the curve, Uspekhi Mat. Nauk 54 (1999), no. 3(327), 157–158 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 3, 634–635. MR 1728649, DOI https://doi.org/10.1070/rm1999v054n03ABEH000160
  • A. Campillo and C. Galindo, The Poincaré series associated with finitely many monomial valuations, preprint.
  • V. Cossart, O. Piltant, and A. Reguera, Divisorial valuations on rational surface singularities, Fields Inst. Comm. Vol. 32: “Valuation theory and its applications", Amer. Math. Soc., Providence, RI, 2002, 89-101.
  • Steven Dale Cutkosky, On unique and almost unique factorization of complete ideals, Amer. J. Math. 111 (1989), no. 3, 417–433. MR 1002007, DOI https://doi.org/10.2307/2374667
  • S. D. Cutkosky and V. Srinivas, On a problem of Zariski on dimensions of linear systems, Ann. of Math. (2) 137 (1993), no. 3, 531–559. MR 1217347, DOI https://doi.org/10.2307/2946531
  • Hans Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368 (German). MR 137127, DOI https://doi.org/10.1007/BF01441136
  • Séminaire Bourbaki, 14ième année: 1961/62. Fasc. 1, 2 et 3: Textes des Conférences, Exp. 223 à 240, Secrétariat mathématique, Paris, 1962 (French). 2ième édition, corrigée. MR 0146035
  • A. Grothendieck and J. Dieudonné, Eléments de Géometrie Algébrique, Inst. Hautes Etudes Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32. , , , , , ,
  • Alexander Grothendieck and Jacob P. Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, Vol. 208, Springer-Verlag, Berlin-New York, 1971. MR 0316453
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • Masahide Kato, Riemann-Roch theorem for strongly pseudoconvex manifolds of dimension $2$, Math. Ann. 222 (1976), no. 3, 243–250. MR 412468, DOI https://doi.org/10.1007/BF01362581
  • G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
  • Henry B. Laufer, On rational singularities, Amer. J. Math. 94 (1972), 597–608. MR 330500, DOI https://doi.org/10.2307/2374639
  • C. Lech, A note on recurring series, Arkiv Mat. 2 (1953), 417-421.
  • Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
  • T. Matsusaka, The criteria for algebraic equivalence and the torsion group, Amer. J. Math. 79 (1957), 53-66.
  • Michael McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995), no. 1, 143–159. MR 1323985, DOI https://doi.org/10.1007/BF01241125
  • M. Morales, Calcul de quelques invariants des singularités de surface normale, Knots, braids and singularities (Plans-sur-Bex, 1982) Monogr. Enseign. Math., vol. 31, Enseignement Math., Geneva, 1983, pp. 191–203 (French). MR 728586
  • David Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 5–22. MR 153682
  • J. P. Murre, On contravariant functors from the category of pre-schemes over a field into the category of abelian groups (with an application to the Picard functor), Inst. Hautes Études Sci. Publ. Math. 23 (1964), 5–43. MR 206011
  • Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
  • Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564
  • J. P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier Grenoble 6 (1956), 1-42.
  • Richard P. Stanley, Combinatorics and commutative algebra, Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1983. MR 725505
  • Paul Vojta, Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126 (1996), no. 1, 133–181. MR 1408559, DOI https://doi.org/10.1007/s002220050092
  • Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615. MR 141668, DOI https://doi.org/10.2307/1970376

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Additional Information

Steven Dale Cutkosky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
MR Author ID: 53545
ORCID: 0000-0002-9319-0717
Email: cutkoskys@missouri.edu

Jürgen Herzog
Affiliation: FB 6 Mathematik und Informatik, Universität-GHS-Essen, Postfach 103764, D-45117 Essen, Germany
MR Author ID: 189999
Email: mat300@uni-essen.de

Ana Reguera
Affiliation: Univeristy of Valladolid, Departamento de Algebra, Geometría y Topología, 005 Valladolid, Spain
Email: areguera@agt.uva.es

Received by editor(s): August 1, 2002
Published electronically: August 26, 2003
Additional Notes: The first author’s research was partially supported by NSF
Article copyright: © Copyright 2003 American Mathematical Society