Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Log canonical thresholds of quasi-ordinary hypersurface singularities

Authors: Nero Budur, Pedro D. González-Pérez and Manuel González Villa
Journal: Proc. Amer. Math. Soc. 140 (2012), 4075-4083
MSC (2010): Primary 14B05, 32S45
Published electronically: April 6, 2012
MathSciNet review: 2957197
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The log canonical thresholds of irreducible quasi-ordinary hypersurface singularities are computed using an explicit list of pole candidates for the motivic zeta function found by the last two authors.

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

  • 1. S. S. Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575-592. MR 0071851 (17:193c)
  • 2. M. Aprodu and D. Naie, Enriques diagrams and log-canonical thresholds of curves on smooth surfaces. Geom. Dedicata 146 (2010), 43-66. MR 2644270 (2011f:14030)
  • 3. E. Artal Bartolo, Pi. Cassou-Noguès, I. Luengo, and A. Melle-Hernández, On the log-canonical threshold for germs of plane curves. Singularities I, Contemp. Math., 474, Amer. Math. Soc., Providence, RI, 2008, pp. 1-14. MR 2454343 (2009m:32050)
  • 4. -, Quasi-ordinary power series and their zeta functions, Mem. Amer. Math. Soc. 178 (2005), no. 841, vi+85 pp. MR 2172403 (2007d:14005)
  • 5. N. Budur, Singularity invariants related to Milnor fibers: survey. To appear in Zeta Functions in Algebra and Geometry, Contemp. Math., Amer. Math. Soc.
  • 6. T. de Fernex, L. Ein, and M. Mustaţă, Shokurov's ACC conjecture for log canonical thresholds on smooth varieties. Duke Math. J. 152 (2010), no. 1, 93-114. MR 2643057 (2011c:14036)
  • 7. J. Denef and F. Loeser, Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., vol. 201, Birkhäuser, Basel, 2001, pp. 327-348. MR 1905328 (2004c:14037)
  • 8. V. Egorin, Characteristic varieties of algebraic curves. Ph.D. Thesis, University of Illinois at Chicago, 2004, 80 pp. MR 2705805
  • 9. Y.-N. Gau, Embedded topological classification of quasi-ordinary singularities, Mem. Amer. Math. Soc. 74 (1988), no. 388, 109-129. With an appendix by Joseph Lipman. MR 954948 (89m:14002)
  • 10. P. D. González Pérez, Toric embedded resolutions of quasi-ordinary hypersurface singularities, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 6, 1819-1881. MR 2038781 (2005b:32064)
  • 11. P. D. González Pérez and M. González Villa, Motivic Milnor fibre of a quasi-ordinary hypersurface. arXiv:1105.2480v1.
  • 12. J.-i. Igusa, On the first terms of certain asymptotic expansions. Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 357-368. MR 0485881 (58:5680)
  • 13. L. H. Halle and J. Nicaise, Motivic zeta functions of abelian varieties, and the monodromy conjecture, Adv. Math. 227 (2011), no. 1, 610-653. MR 2782205
  • 14. -, Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties. arXiv:1012.4969.
  • 15. J. Kollár, Singularities of pairs. Algebraic geometry - Santa Cruz 1995, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997, pp. 221-287. MR 1492525 (99m:14033)
  • 16. T. Kuwata, On log canonical thresholds of reducible plane curves. Amer. J. Math. 121 (1999), no. 4, 701-721. MR 1704476 (2001g:14047)
  • 17. R. Lazarsfeld, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 49. Springer-Verlag, Berlin, 2004. MR 2095472 (2005k:14001b)
  • 18. J.  Lipman, Topological invariants of quasi-ordinary singularities, Mem. Amer. Math. Soc. 74 (1988), no. 388, 1-107. MR 954947 (89m:14001)
  • 19. M. Mustaţa, Singularities of pairs via jet schemes, J. Amer. Math. Soc. 15 (2002), 599-615. MR 1896234 (2003b:14005)
  • 20. W. Veys and W. Zuniga-Galindo, Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra. Trans. Amer. Math. Soc. 360 (2008), no. 4, 2205-2227. MR 2366980 (2008i:11140)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14B05, 32S45

Retrieve articles in all journals with MSC (2010): 14B05, 32S45

Additional Information

Nero Budur
Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley Hall, South Bend, Indiana 46556

Pedro D. González-Pérez
Affiliation: ICMAT, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040, Madrid, Spain

Manuel González Villa
Affiliation: ICMAT, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040, Madrid, Spain – and – Mathematics Center Heidelberg (Match), Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany

Keywords: Log canonical threshold, quasi-ordinary singularity
Received by editor(s): May 23, 2011
Published electronically: April 6, 2012
Additional Notes: The first author is supported by the NSA grant H98230-11-1-0169. The second and third authors are supported by MCI-Spain grant MTM2010-21740-C02.
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society