Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type

Abstract

In this paper, we give a formula for normal reduction number of an integrally closed \(\mathfrak m\)-primary ideal of a two-dimensional normal local ring \((A,\mathfrak m)\) in terms of the geometric genus pg(A) of A. Also, we compute the normal reduction number of the maximal ideal of Brieskorn hypersurfaces. As an application, we give a short proof of a classification of Brieskorn hypersurfaces having elliptic singularities.

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References

  1. 1.

    Cutkosky, S.D.: A new characterization of rational surface singularities. Invent. Math. 102(1), 157–177 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Goto, S., Nishida, K.: The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations. Mem. Am. Math. Soc. 110(526) (1994)

  3. 3.

    Hochster, M., Huneke, C.: Tight closure, invariant theory, and the Briançon-Skoda theorem. J. Am. Math. Soc. 3(1), 31–116 (1990)

    MATH  Google Scholar 

  4. 4.

    Hoa, L.T., Zarzuela, S.: Reduction number and a-invariant of good filtrations. Comm. Algebra 22(14), 5635–5656 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Huneke, C.: Hilbert functions and symbolic powers. Michigan Math. J. 34(2), 293–318 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  7. 7.

    Konno, K., Nagashima, D.: Maximal ideal cycles over normal surface singularities of Brieskorn type. Osaka J. Math. 49(1), 225–245 (2012)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Lipman, J.: Rational singularities with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. 36, 195–279 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Lipman, J., Teissier, B.: Pseudo-rational local rings and a theorem of Briançon-Skoda on the integral closures of ideals. Michigan Math. J. 28(1), 97–116 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Matsumura, H.: Commutative Ring Theory Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  11. 11.

    Meng, F.-N., Okuma, T.: The maximal ideal cycles over complete intersection surface singularities of Brieskorn type. Kyushu J. Math. 68(1), 121–137 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Okuma, T.: Cohomology of ideals in elliptic surface singularities. Illinois J. Math. 61(3–4), 259–273 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Okuma, T., Watanabe, K.-i., Yoshida, K.-i.: Good ideals and p g-ideals in two-dimensional normal singularities. Manuscripta Math. 150(3–4), 499–520 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Okuma, T., Watanabe, K.-i., Yoshida, K.-i.: Rees algebras and p g-ideals in a two-dimensional normal local domain. Proc. Am. Math. Soc. 145(1), 39–47 (2017)

    Article  MATH  Google Scholar 

  15. 15.

    Okuma, T., Watanabe, K.-i., Yoshida, K.-i.: A characterization of two-dimensional rational singularities via core of ideals. J. Algebra 499, 450–468 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Tomaru, T.: A formula of the fundamental genus for hypersurface singularities of Brieskorn type. Ann. Rep. Coll. Med. Care Technol. Gunma Univ. 17, 145–150 (1996)

    Google Scholar 

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Funding

This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) Grant Nos., 25400050, 26400053, 17K05216.

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Correspondence to Kei-ichi Watanabe.

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Okuma, T., Watanabe, Ki. & Yoshida, Ki. Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type. Acta Math Vietnam 44, 87–100 (2019). https://doi.org/10.1007/s40306-018-00311-4

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Keywords

  • Normal reduction number
  • Geometric genus
  • Hypersurface of Brieskorn type

Mathematics Subject Classification (2010)

  • 13B22
  • Secondary 14B05
  • 14J17