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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
Posted:
April 6, 2012
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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.
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Tommaso
de Fernex, Lawrence
Ein, and Mircea
Mustaţă, Shokurov’s ACC conjecture for log
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Gau, Embedded topological classification of quasi-ordinary
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no. 388, 109–129. With an appendix by Joseph Lipman. MR 954948
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Pedro
D. González Pérez, Toric embedded resolutions of
quasi-ordinary hypersurface singularities, Ann. Inst. Fourier
(Grenoble) 53 (2003), no. 6, 1819–1881
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P. D. González Pérez and M. González Villa, Motivic Milnor fibre of a quasi-ordinary hypersurface. arXiv:1105.2480v1.
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J.
Igusa, On the first terms of certain asymptotic expansions,
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(58 #5680)
- 13.
Lars
Halvard Halle and Johannes
Nicaise, Motivic zeta functions of abelian varieties, and the
monodromy conjecture, Adv. Math. 227 (2011),
no. 1, 610–653. MR 2782205
(2012c:14050), http://dx.doi.org/10.1016/j.aim.2011.02.011
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-, Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties. arXiv:1012.4969.
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János
Kollár, Singularities of pairs, Algebraic
geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62,
Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR 1492525
(99m:14033)
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Takayasu
Kuwata, On log canonical thresholds of reducible plane curves,
Amer. J. Math. 121 (1999), no. 4, 701–721. MR 1704476
(2001g:14047)
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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)
- 18.
Joseph
Lipman, Topological invariants of quasi-ordinary
singularities, Mem. Amer. Math. Soc. 74 (1988),
no. 388, 1–107. MR 954947
(89m:14001)
- 19.
Mircea
Mustaţǎ, Singularities of pairs via jet
schemes, J. Amer. Math. Soc.
15 (2002), no. 3,
599–615 (electronic). MR 1896234
(2003b:14005), http://dx.doi.org/10.1090/S0894-0347-02-00391-0
- 20.
Willem
Veys and W.
A. Zúñiga-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), http://dx.doi.org/10.1090/S0002-9947-07-04422-4
- 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)
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Additional Information
Nero Budur
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, South Bend, Indiana 46556
Email:
nbudur@nd.edu
Pedro D. González-Pérez
Affiliation:
ICMAT, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040, Madrid, Spain
Email:
pgonzalez@mat.ucm.es
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
Email:
mgv@mat.ucm.es, villa@mathi.uni-heidelberg.de
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11416-9
PII:
S 0002-9939(2012)11416-9
Keywords:
Log canonical threshold,
quasi-ordinary singularity
Received by editor(s):
May 23, 2011
Posted:
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
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