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)

 
 

 

Jumping numbers on algebraic surfaces with rational singularities


Author: Kevin Tucker
Journal: Trans. Amer. Math. Soc. 362 (2010), 3223-3241
MSC (2000): Primary 14B05
DOI: https://doi.org/10.1090/S0002-9947-09-04956-3
Published electronically: December 17, 2009
MathSciNet review: 2592954
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.


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

  • [Art62] Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485-496. MR 0146182 (26:3704)
  • [Art66] -, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136. MR 0199191 (33:7340)
  • [BL04] Manuel Blickle and Robert Lazarsfeld, An informal introduction to multiplier ideals, Trends in commutative algebra, Math. Sci. Res. Inst. Publ., vol. 51, Cambridge Univ. Press, Cambridge, 2004, pp. 87-114. MR 2132649 (2007h:14003)
  • [ELSV04] Lawrence Ein, Robert Lazarsfeld, Karen E. Smith, and Dror Varolin, Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), no. 3, 469-506. MR 2068967 (2005k:14004)
  • [FJ04] Charles Favre and Mattias Jonsson, The valuative tree, Lecture Notes in Mathematics, vol. 1853, Springer-Verlag, Berlin, 2004. MR 2097722 (2006a:13008)
  • [FJ05] -, Valuations and multiplier ideals, J. Amer. Math. Soc. 18 (2005), no. 3, 655-684 (electronic). MR 2138140 (2007b:14004)
  • [Ful93] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry. MR 1234037 (94g:14028)
  • [How01] J. A. Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2665-2671 (electronic). MR 1828466 (2002b:14061)
  • [How03] -, Multiplier ideals of sufficiently general polynomials, 2003.
  • [HS03] Eero Hyry and Karen E. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math. 125 (2003), no. 6, 1349-1410. MR 2018664 (2006c:13036)
  • [Jär06] Tarmo Järvilehto, Jumping numbers of a simple complete ideal in a two-dimensional regular local ring, 2006.
  • [Kaw82] Yujiro Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann. 261 (1982), no. 1, 43-46. MR 675204 (84i:14022)
  • [KMM87] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283-360. MR 946243 (89e:14015)
  • [Laz04] Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004, Positivity for vector bundles, and multiplier ideals. MR 2095472 (2005k:14001b)
  • [Lib83] A. Libgober, Alexander invariants of plane algebraic curves, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 135-143. MR 713242 (85h:14017)
  • [Lip69] Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 195-279. MR 0276239 (43:1986)
  • [LV90] F. Loeser and M. Vaquié, Le polynôme d'Alexander d'une courbe plane projective, Topology 29 (1990), no. 2, 163-173. MR 1056267 (91d:32053)
  • [LW03] Joseph Lipman and Kei-ichi Watanabe, Integrally closed ideals in two-dimensional regular local rings are multiplier ideals, Math. Res. Lett. 10 (2003), no. 4, 423-434. MR 1995782 (2004m:13059)
  • [Mum61] David Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961), no. 9, 5-22. MR 0153682 (27:3643)
  • [ST06] Karen E. Smith and Howard M. Thompson, Irrelevant exceptional divisors for curves on a smooth surface, Algebra, Geometry and their Interactions Conference, Contemp. Math., vol. 448, pp. 245-254, Amer. Math. Soc., Providence, RI, 2007. MR 2389246
  • [ST07] Karl Schwede and Shunsuke Takagi, Rational singularities associated to pairs, 2007, To appear in the Michigan Mathematics Journal.
  • [Vaq92] Michel Vaquié, Irrégularité des revêtements cycliques des surfaces projectives non singulières, Amer. J. Math. 114 (1992), no. 6, 1187-1199. MR 1198299 (94d:14015)
  • [Vaq94] -, Irrégularité des revêtements cycliques, Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser., vol. 201, Cambridge Univ. Press, Cambridge, 1994, pp. 383-419. MR 1295085 (95f:14030)
  • [Sha94] Igor R. Shafarevich, Basic algebraic geometry. 1, second ed. Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid, Springer-Verlag, Berlin, 1994. MR 1328833 (95m:14001)
  • [VV07] Lise Van Proeyen and Willem Veys, Poles of the topological zeta function associated to an ideal in dimension two, Mathematische Zeitschrift, 260 (2008), 615-627. MR 2434472
  • [Vie82] Eckart Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1-8. MR 667459 (83m:14011)

Similar Articles

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

Retrieve articles in all journals with MSC (2000): 14B05


Additional Information

Kevin Tucker
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: kevtuck@umich.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04956-3
Received by editor(s): April 9, 2008
Received by editor(s) in revised form: August 27, 2008
Published electronically: December 17, 2009
Additional Notes: The author was partially supported by the NSF under grant DMS-0502170.
Dedicated: In memory of Juha Heinonen
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society