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On the log canonical threshold and numerical data of a resolution in dimension 2

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Abstract

We show various properties of numerical data of an embedded resolution of singularities for plane curves, which are inspired by a conjecture of Nguyen and motivated by a conjecture of Igusa on exponential sums.

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References

  1. A’Campo, N.: La fonction zêta d’une monodromie. Comment. Math. Helv. 50, 233–248 (1975)

    Article  MathSciNet  Google Scholar 

  2. Cluckers, R., Mustaţǎ, M., Nguyen, K.: Igusa’s conjecture for exponential sums: optimal estimates for non-rational singularities, Forum of Mathematics. Pi 7, (2019). https://doi.org/10.1017/fmp.2019.3

  3. Cluckers, R., Veys, W.: Bounds for \(p\)-adic exponential sums and log-canonical thresholds. Am. J. Math. 138, 61–80 (2016)

    Article  MathSciNet  Google Scholar 

  4. Denef, J., Loeser, F.: Caractéristiques d’Euler-Poincaré, fonctions zêta locales, et modifications analytiques. J. Am. Math. Soc. 5, 705–720 (1992)

    MATH  Google Scholar 

  5. Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Alg. Geom. 7, 505–537 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Denef, J., Sperber, S.: Exponential sums mod \(p^n\) and Newton polyhedra, Bull. Belg. Math. Soc. Simon Stevin suppl. 55–63, (2001)

  7. Ein, L., Lazarsfeld, R., Smith, K., Varolin, D.: Jumping coefficients of multiplier ideals. Duke Math. J. 123, 469–506 (2004)

    Article  MathSciNet  Google Scholar 

  8. Eisenbud, D., Neumann, W.: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Mathmatics Studies., vol. 110. Princeton University Press, Princeton (1985)

    MATH  Google Scholar 

  9. Igusa, J.: Lectures on forms of higher degree (notes by S. Raghavan), Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, vol. 59. Springer, Berlin (1978)

    MATH  Google Scholar 

  10. Igusa, J.: An introduction to the theory of local zeta functions, Studies in Advanced Mathematics 14, AMS, (2000)

  11. Kashiwara, M.: \(B\)-functions and holonomic systems. Inv. Math. 38, 33–53 (1976)

    Article  Google Scholar 

  12. Mustaţǎ, M.: IMPANGA lecture notes on log canonical thresholds, in ‘Contributions to Algebraic Geometry’, EMS Series of Congress Reports, pp. 407–442. Zürich, Eur. Math. Soc. (2012)

  13. Némethi, A., Veys, W.: Generalized monodromy conjecture in dimension two. Geom. Topol. 16, 155–217 (2012)

    Article  MathSciNet  Google Scholar 

  14. Nguyen, K.H.: Uniform rationality of Poincaré series of \(p\)–adic equivalence relations and Igusa’s conjecture on exponential sums, PhD thesis, University of Lille (2018), arXiv:1903.06738

  15. Veys, W.: Zeta functions for curves and log canonical models. Proc. Lond. Math. Soc. 74, 360–378 (1997)

    Article  MathSciNet  Google Scholar 

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  1. Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Willem Veys

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  1. Willem Veys
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Correspondence to Willem Veys.

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The author is partially supported by KU Leuven Grant C14/17/083 and wants to thank Kien Nguyen for discussing the two–dimensional case.

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Veys, W. On the log canonical threshold and numerical data of a resolution in dimension 2. manuscripta math. 163, 1–11 (2020). https://doi.org/10.1007/s00229-019-01145-6

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  • DOI: https://doi.org/10.1007/s00229-019-01145-6

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