Symbolic powers of codimension two Cohen-Macaulay ideals

Susan Cooper; Giuliana Fatabbi; Elena Guardo; Anna Lorenzini; Juan Migliore; Uwe Nagel; Alexandra Seceleanu; Justyna Szpond; Adam Van Tuyl

doi:10.1080/00927872.2020.1769120

Symbolic powers of codimension two Cohen-Macaulay ideals

Susan Cooper, Giuliana Fatabbi, Elena Guardo, Anna Lorenzini, Juan Migliore, Uwe Nagel, Alexandra Seceleanu, Justyna Szpond, Adam Van Tuyl

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let I_X be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme (Formula presented.) and let (Formula presented.) denote its m-th symbolic power. We are interested in when (Formula presented.) We survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of when (Formula presented.) for all (Formula presented.) We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in (Formula presented.) (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.

Original language	English
Pages (from-to)	4663-4680
Number of pages	18
Journal	Communications in Algebra
Volume	48
Issue number	11
DOIs	https://doi.org/10.1080/00927872.2020.1769120
State	Published - Nov 1 2020

Bibliographical note

Funding Information:
Cooper acknowledges support from the NDSU Advance FORWARD program sponsored by the National Science Foundation, HRD-0811239, as well as financial support provided by NSERC. Fatabbi and Lorenzini were partially supported by GNSAGA (INdAM). Guardo acknowledges the financial support provided by Prin 2011 and GNSAGA (INdAM) and Piano della ricerca–Linea intervento 2, Università di Catania. Migliore and Nagel were partially supported by Simons Foundation grants (#309556 for Migliore, #317096 and #636513 for Nagel). Seceleanu received support from MFO’s NSF grant DMS-1049268, “NSF Junior Oberwolfach Fellows” as well as from NSF grant DMS–1601024 and EPSCoR grant OIA–1557417. Szpond was partially supported by the National Science Center, Poland, grant 2014/15/B/ST1/02197. Van Tuyl acknowledges the financial support provided by NSERC RGPIN-2019-05412. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop “Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems” organized by C. Bocci, E. Carlini, E. Guardo, and B. Harbourne. All the authors thank the MFO for providing a stimulating environment. We also thank Brian Harbourne and Tomasz Szemberg for their feedback on early drafts of the paper.

Publisher Copyright:
© 2020 Taylor & Francis Group, LLC.

Keywords

Symbolic powers
arithmetically Cohen-Macaulay
codimension two
locally complete intersection
points in

ASJC Scopus subject areas

Algebra and Number Theory

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10.1080/00927872.2020.1769120

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title = "Symbolic powers of codimension two Cohen-Macaulay ideals",

abstract = "Let IX be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme (Formula presented.) and let (Formula presented.) denote its m-th symbolic power. We are interested in when (Formula presented.) We survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of when (Formula presented.) for all (Formula presented.) We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in (Formula presented.) (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of R{\"o}mer.",

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author = "Susan Cooper and Giuliana Fatabbi and Elena Guardo and Anna Lorenzini and Juan Migliore and Uwe Nagel and Alexandra Seceleanu and Justyna Szpond and Tuyl, {Adam Van}",

note = "Funding Information: Cooper acknowledges support from the NDSU Advance FORWARD program sponsored by the National Science Foundation, HRD-0811239, as well as financial support provided by NSERC. Fatabbi and Lorenzini were partially supported by GNSAGA (INdAM). Guardo acknowledges the financial support provided by Prin 2011 and GNSAGA (INdAM) and Piano della ricerca–Linea intervento 2, Universit{\`a} di Catania. Migliore and Nagel were partially supported by Simons Foundation grants (#309556 for Migliore, #317096 and #636513 for Nagel). Seceleanu received support from MFO{\textquoteright}s NSF grant DMS-1049268, “NSF Junior Oberwolfach Fellows” as well as from NSF grant DMS–1601024 and EPSCoR grant OIA–1557417. Szpond was partially supported by the National Science Center, Poland, grant 2014/15/B/ST1/02197. Van Tuyl acknowledges the financial support provided by NSERC RGPIN-2019-05412. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop “Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems” organized by C. Bocci, E. Carlini, E. Guardo, and B. Harbourne. All the authors thank the MFO for providing a stimulating environment. We also thank Brian Harbourne and Tomasz Szemberg for their feedback on early drafts of the paper. Publisher Copyright: {\textcopyright} 2020 Taylor & Francis Group, LLC.",

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AU - Fatabbi, Giuliana

AU - Guardo, Elena

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AU - Migliore, Juan

AU - Nagel, Uwe

AU - Seceleanu, Alexandra

AU - Szpond, Justyna

AU - Tuyl, Adam Van

N1 - Funding Information: Cooper acknowledges support from the NDSU Advance FORWARD program sponsored by the National Science Foundation, HRD-0811239, as well as financial support provided by NSERC. Fatabbi and Lorenzini were partially supported by GNSAGA (INdAM). Guardo acknowledges the financial support provided by Prin 2011 and GNSAGA (INdAM) and Piano della ricerca–Linea intervento 2, Università di Catania. Migliore and Nagel were partially supported by Simons Foundation grants (#309556 for Migliore, #317096 and #636513 for Nagel). Seceleanu received support from MFO’s NSF grant DMS-1049268, “NSF Junior Oberwolfach Fellows” as well as from NSF grant DMS–1601024 and EPSCoR grant OIA–1557417. Szpond was partially supported by the National Science Center, Poland, grant 2014/15/B/ST1/02197. Van Tuyl acknowledges the financial support provided by NSERC RGPIN-2019-05412. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop “Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems” organized by C. Bocci, E. Carlini, E. Guardo, and B. Harbourne. All the authors thank the MFO for providing a stimulating environment. We also thank Brian Harbourne and Tomasz Szemberg for their feedback on early drafts of the paper. Publisher Copyright: © 2020 Taylor & Francis Group, LLC.

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N2 - Let IX be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme (Formula presented.) and let (Formula presented.) denote its m-th symbolic power. We are interested in when (Formula presented.) We survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of when (Formula presented.) for all (Formula presented.) We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in (Formula presented.) (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.

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Symbolic powers of codimension two Cohen-Macaulay ideals

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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