Illinois Journal of Mathematics

Rational singularities and uniform symbolic topologies

Robert M. Walker

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Take $(R,\mathfrak{m})$ any normal Noetherian domain, either local or $\mathbb{N}$-graded over a field. We study the question of when $R$ satisfies the uniform symbolic topology property (USTP) of Huneke, Katz, and Validashti: namely, that there exists an integer $D>0$ such that for all prime ideals $P\subseteq R$, the symbolic power $P^{(Da)}\subseteq P^{a}$ for all $a>0$. Reinterpreting results of Lipman, we deduce that when $R$ is a two-dimensional rational singularity, then it satisfies the USTP. Emphasizing the non-regular setting, we produce explicit, effective multipliers $D$, working in two classes of surface singularities in equal characteristic over an algebraically closed field, using: (1) the volume of a parallelogram in $\mathbb{R}^{2}$ when $R$ is the coordinate ring of a simplicial toric surface; or (2) known invariants of du Val isolated singularities in characteristic zero due to Lipman.

Article information

Illinois J. Math., Volume 60, Number 2 (2016), 541-550.

Received: 23 May 2016
Revised: 9 February 2017
First available in Project Euclid: 11 July 2017

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Zentralblatt MATH identifier

Primary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]
Secondary: 14C20: Divisors, linear systems, invertible sheaves 14M25: Toric varieties, Newton polyhedra [See also 52B20]


Walker, Robert M. Rational singularities and uniform symbolic topologies. Illinois J. Math. 60 (2016), no. 2, 541--550. doi:10.1215/ijm/1499760021.

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  • W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact complex surfaces, 2nd ed., Springer, Berlin, 2004.
  • D. A. Cox, J. B. Little and H. K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, Am. Math. Soc., Providence, 2011.
  • L. Ein, R. Lazarsfeld and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241–252.
  • W. Fulton, Introduction to toric varieties, Annals of Math. Studies, vol. 131, Princeton University Press, Princeton, 1993.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Math., vol. 52, Springer, New York, 1977.
  • M. Hochster, Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337.
  • M. Hochster, Cohen–Macaulay rings and modules, Proceedings of the international congress of mathematicians, Helsinki, Finland, vol. I, Academia Scientiarum Fennica, Turku, 1980, pp. 291–298.
  • M. Hochster, Math 615 winter 2007 lecture 4/6/07. Available at
  • M. Hochster and C. Huneke, Comparison of ordinary and symbolic powers of ideals, Invent. Math. 147 (2002), 349–369.
  • M. Hochster and C. Huneke, Fine behavior of symbolic powers of ideals, Illinois J. Math. 51 (2007), no. 1, 171–183.
  • C. Huneke, Uniform bounds in Noetherian rings, Invent. Math. 107 (1992), 203–223.
  • C. Huneke, D. Katz and J. Validashti, Uniform equivalence of symbolic and adic topologies, Illinois J. Math. 53 (2009), no. 1, 325–338.
  • C. Huneke, D. Katz and J. Validashti, Uniform symbolic topologies and finite extensions, J. Pure Appl. Algebra 219 (2015), no. 3, 543–550.
  • M. R. Johnson, Containing symbolic powers in regular rings, Comm. Algebra 42 (2014), no. 8, 3552–3557.
  • J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 195–279.
  • M. P. Murthy, Vector bundles over affine surfaces birationally equivalent to a ruled surface, Ann. of Math. (2) 89 (1969), 242–253.
  • S. Takagi and K. Yoshida, Generalized test ideals and symbolic powers, Michigan Math. J. 57 (2008), 711–725.