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Commutative Algebra pp 343–371Cite as

Symbolic Rees Algebras

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Abstract

We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.

Keywords

  • Symbolic powers
  • Symbolic Rees algebra
  • Containment problem

2010 Mathematics Subject Classification

  • Primary: 13A15
  • Secondary: 13H05

Dedicated to David Eisenbud on the occasion of his 75th birthday.

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Notes

  1. 1.

    The book [105] is a comprehensive reference on the subject of integral closure.

  2. 2.

    For details on Rees valuations and their applications the reader is invited to consult [105, §10.1].

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Acknowledgements

The first author is supported by NSF grant DMS-2001445, now DMS-2140355. The second author is supported by NSF grant DMS-2101225. We thank Craig Huneke for help with tracking down the history of the terminology “symbolic Rees algebra”, and José Gonzalez for discussions regarding Mori dream spaces. We also thank Thomas Polstra for pointing us to [66], Elena Guardo for finding a typo in a previous version of the paper, and Kazuhiko Kurano for his comments on a previous version and for pointing us to [109]. Finally, we thank Anurag Singh for very detailed comments on an earlier version of the survey.

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  1. Department of Mathematics, University of Nebraska – Lincoln, Lincoln, NE, USA

    Eloísa Grifo & Alexandra Seceleanu

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  1. Eloísa Grifo
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  1. Department of Mathematics, Cornell University, Ithaca, NY, USA

    Irena Peeva

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Grifo, E., Seceleanu, A. (2021). Symbolic Rees Algebras. In: Peeva, I. (eds) Commutative Algebra. Springer, Cham. https://doi.org/10.1007/978-3-030-89694-2_11

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