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 Römer.
| Original language | English |
|---|---|
| Pages (from-to) | 4663-4680 |
| Number of pages | 18 |
| Journal | Communications in Algebra |
| Volume | 48 |
| Issue number | 11 |
| DOIs | |
| State | Published - Nov 1 2020 |
Bibliographical note
Publisher Copyright:© 2020 Taylor & Francis Group, LLC.
Funding
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.
| Funders | Funder number |
|---|---|
| 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 |
Keywords
- Symbolic powers
- arithmetically Cohen-Macaulay
- codimension two
- locally complete intersection
- points in
ASJC Scopus subject areas
- Algebra and Number Theory