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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cutkosky, S.D.: A new characterization of rational surface singularities. Invent. Math. 102(1), 157–177 (1990)
Goto, S., Nishida, K.: The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations. Mem. Am. Math. Soc. 110(526) (1994)
Hochster, M., Huneke, C.: Tight closure, invariant theory, and the Briançon-Skoda theorem. J. Am. Math. Soc. 3(1), 31–116 (1990)
Hoa, L.T., Zarzuela, S.: Reduction number and a-invariant of good filtrations. Comm. Algebra 22(14), 5635–5656 (1994)
Huneke, C.: Hilbert functions and symbolic powers. Michigan Math. J. 34(2), 293–318 (1987)
Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)
Konno, K., Nagashima, D.: Maximal ideal cycles over normal surface singularities of Brieskorn type. Osaka J. Math. 49(1), 225–245 (2012)
Lipman, J.: Rational singularities with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. 36, 195–279 (1969)
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)
Matsumura, H.: Commutative Ring Theory Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989)
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)
Okuma, T.: Cohomology of ideals in elliptic surface singularities. Illinois J. Math. 61(3–4), 259–273 (2017)
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)
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)
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)
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)
Funding
This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) Grant Nos., 25400050, 26400053, 17K05216.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40306-018-00311-4