Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: August 26, 2003
MathSciNet review: 2031043
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

\begin{displaymath}g= \sum_{\underline{n}\in\mathbf{N}^r} h(\underline{n})t^{\underline{n}}, \end{displaymath}


\begin{displaymath}h(\underline{n})=\ell(R/\Gamma(X,\mathcal{O}_X(-n_1E-1-\cdots-n_rE_r)). \end{displaymath}

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.

References [Enhancements On Off] (What's this?)

  • 1. S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic $p\ne 0$, Annals of Math. (2) 63 (1956), 491-526. MR 17:1134d
  • 2. M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485-496. MR 26:3704
  • 3. A. Borel, Linear Algebraic Groups, second ed., Springer-Verlag, New York, 1991. MR 92d:20001
  • 4. C. Banica and O. Stanaçila, Algebraic methods in the global theory of complex spaces, John Wiley and Sons, New York (1976). MR 57:3420
  • 5. A. Campillo, F. Delgado, and S. M. Gusein-Zade, The Alexander polynomial of a plane curve singularity, and the ring of functions on the curve (Russian), Uspekhi Mat. Nauk 54 (1999), no. 3, (327), 157-158; transl. in Russian Math. Surveys 54 (1999), 634-635. MR 2000h:32043
  • 6. A. Campillo and C. Galindo, The Poincaré series associated with finitely many monomial valuations, preprint.
  • 7. 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.
  • 8. S. D. Cutkosky, On unique and almost unique factorization of complete ideals II, Inventiones Math. 98 (1989), 59-74. MR 90j:14016a
  • 9. S. D. Cutkosky and V. Srinivas, On a problem of Zariski on dimensions of linear systems, Annals of Math. (2) 137 (1993), 531-559. MR 94g:14001
  • 10. H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368. MR 25:583
  • 11. A. Grothendieck, Téchnique de descente et théorèmes d'éxistence en géométrie algébrique, VI, Séminaire Bourbaki, 1961/62, Exposé 236, Secrétariat Math., Paris, 1962. MR 26:3561
  • 12. A. Grothendieck and J. Dieudonné, Eléments de Géometrie Algébrique, Inst. Hautes Etudes Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32. MR 36:177a, MR 36:177b, MR 36:177c, MR 29:1210, MR 30:3885, MR 33:7330, MR 36:178, MR 39:220
  • 13. A. Grothendieck and J. P. Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Math. 208, Springer-Verlag, New York (1971). MR 47:5000
  • 14. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York (1977). MR 57:3116
  • 15. M. Kato, Riemann-Roch theorem for strongly pseudoconvex manifolds of dimension $2$, Math. Ann. 222 (1976), 243-250. MR 54:594
  • 16. G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings I, Lecture Notes in Math. 339, Springer-Verlag, New York (1973). MR 49:299
  • 17. H. Laufer, On rational singularities, Amer. J. Math. 94 (1972), 597-608. MR 48:8837
  • 18. C. Lech, A note on recurring series, Arkiv Mat. 2 (1953), 417-421. MR 15:104e
  • 19. J. Lipman, Rational singularities with applications to algebraic surfaces and unique factorization, Inst. Hautes Etudes Sci. Publ. Math. 36 (1969), 195-279. MR 43:1986
  • 20. T. Matsusaka, The criteria for algebraic equivalence and the torsion group, Amer. J. Math. 79 (1957), 53-66. MR 18:602a
  • 21. M. McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995), 143-159. MR 96b:14020
  • 22. M. Morales, Calcul de quelques invariants des singularité de surface normale, in Knots, braids and singularities (Plans-sur-Bex, 1982), 191-203, Monograph Enseign. Math. 31, 1983. MR 85j:14003
  • 23. D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Etudes Sci. Publ. Math. 9 (1961), 5-22. MR 27:3643
  • 24. J. P. Murre, On contravariant functors from the category of preschemes over a field into the category of abelian groups, Inst. Hautes Etudes Sci. Publ. Math. 23 (1964), 5-43. MR 34:5836
  • 25. T. Oda, Convex bodies and algebraic geometry, Springer-Verlag, Berlin (1988). MR 88m:14038
  • 26. J. P. Serre, Algebraic Groups and Class Fields, Springer-Verlag, New York, 1988. MR 88i:14041
  • 27. J. P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier Grenoble 6 (1956), 1-42. MR 18:511a
  • 28. R. Stanley, Combinatorics and commutative algebra, Birkhäuser, Boston (1983). MR 85b:05002
  • 29. P. Vojta, Integral points on subvarieties of semi-abelian varieties I, Invent. Math. 126 (1996), 133-181. MR 98a:14034
  • 30. O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Annals of Math. 76 (1962), 560-616. MR 25:5065

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14B05, 14F05, 13A30

Retrieve articles in all journals with MSC (2000): 14B05, 14F05, 13A30

Additional Information

Steven Dale Cutkosky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Jürgen Herzog
Affiliation: FB 6 Mathematik und Informatik, Universität-GHS-Essen, Postfach 103764, D-45117 Essen, Germany

Ana Reguera
Affiliation: Univeristy of Valladolid, Departamento de Algebra, Geometría y Topología, 005 Valladolid, Spain

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

American Mathematical Society