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Symbolic Powers of Ideals

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 222)

Abstract

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.

Keywords

  • Symbolic powers
  • Differential operators
  • Uniform Symbolic Topologies
  • Packing problem

H. Dao—Partially supported by NSA Grant H98230-16-1-0012.

C. Huneke—Partially supported by the NSF Grant 1460638.

L. Núñez-Betancourt—Partially supported by the NSF Grant 1502282.

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Notes

  1. 1.

    Akesseh [1] and Walker [87, 88] has also made recent progress regarding Question 2.21.

  2. 2.

    The third and fourth authors [30] recently answered Question 2.21 affirmatively and proved Conjecture 2.22 for ideals defining F-pure rings.

  3. 3.

    See also [64].

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Acknowledgements

We thank Jeff Mermin for many helpful conversations concerning the packing problem, and in particular for discussions leading to Remark 4.19 and Corollary 4.20. We thank Jonathan Montaño, Andrew Conner, Jack Jeffries, and Robert Walker for helpful comments. Part of this work was done when the second and fifth authors were at the University of Virginia. They wish to thank this institution for its hospitality. Finally, the fifth author thanks the organizing committee for the ‘Brazil-Mexico 2nd meeting on Singularities’ in Salvador, Bahia, Brazil, where this project was initiated.

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Dao, H., De Stefani, A., Grifo, E., Huneke, C., Núñez-Betancourt, L. (2018). Symbolic Powers of Ideals. In: Araújo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J. (eds) Singularities and Foliations. Geometry, Topology and Applications. NBMS BMMS 2015 2015. Springer Proceedings in Mathematics & Statistics, vol 222. Springer, Cham. https://doi.org/10.1007/978-3-319-73639-6_13

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