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

Producción científica: Article › revisión exhaustiva

7 Citas (Scopus)

Resumen

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.

Idioma original	English
Páginas (desde-hasta)	4663-4680
Número de páginas	18
Publicación	Communications in Algebra
Volumen	48
N.º	11
DOI	https://doi.org/10.1080/00927872.2020.1769120
Estado	Published - nov 1 2020

Nota bibliográfica

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

Financiación

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\u2013Linea intervento 2, Universit\u00E0 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\u2019s NSF grant DMS-1049268, \u201CNSF Junior Oberwolfach Fellows\u201D as well as from NSF grant DMS\u20131601024 and EPSCoR grant OIA\u20131557417. 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 \u201CIdeals of Linear Subspaces, Their Symbolic Powers and Waring Problems\u201D 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.

Financiadores	Número del financiador
Istituto Nazionale di Alta Matematica "Francesco Severi"
North Dakota State University
Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni
Università di Catania
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	1049268, 0811239, DMS–1601024, 1601024
Office of Experimental Program to Stimulate Competitive Research	OIA–1557417
Simons Foundation	636513, 317096, 309556
Narodowym Centrum Nauki	2014/15/B/ST1/02197, RGPIN-2019-05412

ASJC Scopus subject areas

Algebra and Number Theory

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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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AU - Cooper, Susan

AU - Fatabbi, Giuliana

AU - Guardo, Elena

AU - Lorenzini, Anna

AU - Migliore, Juan

AU - Nagel, Uwe

AU - Seceleanu, Alexandra

AU - Szpond, Justyna

AU - Tuyl, Adam Van

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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.

AB - 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.

KW - Symbolic powers

KW - arithmetically Cohen-Macaulay

KW - codimension two

KW - locally complete intersection

KW - points in

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Symbolic powers of codimension two Cohen-Macaulay ideals

Resumen

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Financiación

ASJC Scopus subject areas

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